in assignment model the allocation is made to the cell which has

Formation of machine-part cells using assignment allocation algorithm

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Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

For example:

Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

The cost of performing each task by each resource is also known (shown in cells of matrix)

• In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model.

→ Assignment Model is a special application of Linear Programming (LP).

→ The mathematical formulation for Assignment Model is given below:

→ Let, C i j \text {C}_{ij} C ij ​ denotes the cost of resources 'i' to the task 'j' ; such that

→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

→ Subjected to constraint;

(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:

(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;

(iii) x i j x_{ij} x ij ​ is '0' or '1'.

Types of Assignment Problem:

(i) balanced assignment problem.

• It consist of a suqare matrix (n x n).
• Number of rows = Number of columns

(ii) Unbalanced Assignment Problem

• It consist of a Non-square matrix.
• Number of rows ≠ \not=  = Number of columns

Methods to solve Assignment Model:

(i) integer programming method:.

In assignment problem, either allocation is done to the cell or not.

So this can be formulated using 0 or 1 integer.

While using this method, we will have n x n decision varables, and n+n equalities.

So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

But, here in assignment problems, the matrix is a square matrix (m=n).

So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.

The assignments which gives minimum cost is selected as optimal solution.

But, such trail and error becomes very difficult and lengthy.

If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.

(iv) Hungarian Method:

It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

It is also know as Reduced matrix method or Flood's technique.

There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n). (2) Problem should be of minimization type.

Suggested Notes:

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Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

• First, prefix name of what this variable represents.
• Second is the list of all the variables.
• Third is the lower bound on this variable.
• Fourth variable is the upper bound.
• Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

• Solving Blending Problem in Python using Gurobi
• Transshipment Problem in Python Using PuLP

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Formation of machine-part cells using assignment allocation algorithm

—In this paper, an iterative procedure viz. assignment allocation algorithm has been employed to cell formation problem by solving a nonlinear mathematical model. The algorithm has been applied to ten benchmark problems and through computational experimentation it has been observed that the algorithm provides better results for six of the problems that have been considered.

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6.20: Assignment- Cell Builder

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\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

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\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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Create a model of a eukaryotic cell using any material of your choice. In your model be sure to include all the organelles appropriate to your cell (either plant or animal). Once complete, take multiple photographs of your model from all angles. Include these images in a document that also contains the following in table format:

• A detailed key/legend that matches the model;
• Each organelle or part with its basic function;
• A disease or disorder that is associated with the malfunction of each cellular component
• How this organelle is visualized microscopically

Note for the disease information, you can list a disease in either animals or plants, regardless of what type of cell you are modeling. In other words, its okay to discuss a “human” disease even if you are making a plant model, provided the organelle is present in both types of cells.

Some suggestions for 3D models include Legos, a decorated cake with candy toppings, or standard Styrofoam base with appropriate pieces attached. You can also draw or illustrate a model. Here is an example of what you might make.

Basic Requirements (the assignment will not be accepted or assessed unless the follow criteria have been met):

• Assignment has been proofread and does not contain any major spelling or grammatical errors
• Assignment includes appropriate references
• Assignment includes photographs or images of created model from all angles.
• Assignment includes a key documenting how each organelle is represented in the model.
• Assignment includes a completed table such as the one illustrated in the example document.
• Assignment includes a disease caused by malfunction of each identified component in the model.
• Assignment includes at least 7 organelles in the model and table.

Contributors and Attributions

• Performance Assessments: Cell Builder. Authored by : Shelli Carter. Provided by : Columbia Basin College. Located at : https://www.columbiabasin.edu/ . License : CC BY: Attribution

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Transportation, Transshipment, and Assignment

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Model Assignments and the Function Allocation Diagram

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(2007). Model Assignments and the Function Allocation Diagram. In: ARIS Design Platform. Springer, London. https://doi.org/10.1007/978-1-84628-613-1_12

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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

ADVERTISEMENTS:

Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

• Open access
• Published: 28 September 2021

Understanding and mathematical modelling of cellular resource allocation in microorganisms: a comparative synthesis

• Hong Zeng 1 ,
• Reza Rohani 2 ,
• Wei E. Huang 2 &
• Aidong Yang   ORCID: orcid.org/0000-0001-5974-247X 2  

BMC Bioinformatics volume  22 , Article number:  467 ( 2021 ) Cite this article

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The rising consensus that the cell can dynamically allocate its resources provides an interesting angle for discovering the governing principles of cell growth and metabolism. Extensive efforts have been made in the past decade to elucidate the relationship between resource allocation and phenotypic patterns of microorganisms. Despite these exciting developments, there is still a lack of explicit comparison between potentially competing propositions and a lack of synthesis of inter-related proposals and findings.

In this work, we have reviewed resource allocation-derived principles, hypotheses and mathematical models to recapitulate important achievements in this area. In particular, the emergence of resource allocation phenomena is deciphered by the putative tug of war between the cellular objectives, demands and the supply capability. Competing hypotheses for explaining the most-studied phenomenon arising from resource allocation, i.e. the overflow metabolism, have been re-examined towards uncovering the potential physiological root cause. The possible link between proteome fractions and the partition of the ribosomal machinery has been analysed through mathematical derivations. Finally, open questions are highlighted and an outlook on the practical applications is provided. It is the authors’ intention that this review contributes to a clearer understanding of the role of resource allocation in resolving bacterial growth strategies, one of the central questions in microbiology.

Conclusions

We have shown the importance of resource allocation in understanding various aspects of cellular systems. Several important questions such as the physiological root cause of overflow metabolism and the correct interpretation of ‘protein costs’ are shown to remain open. As the understanding of the mechanisms and utility of resource application in cellular systems further develops, we anticipate that mathematical modelling tools incorporating resource allocation will facilitate the circuit-host design in synthetic biology.

Following the initial suggestion in 2009 by Molenaar et al . [ 1 ] that cellular growth strategies are dependent on not only metabolism but also the synthesis cost of proteins, extensive efforts have been made to investigate the biophysical importance of the allocation of macromolecular resources in supporting cell growth. Significant progresses were obtained in the past decade. Quantitative analysis of growth-dependent proteomic datasets elucidates that resource allocation plays a central role in dictating metabolism and gene expression for maximizing the rates of steady-state growth [ 2 , 3 , 4 , 5 ]. Resource allocation has also been shown to govern transitional growth kinetics upon the nutrient shift in a global manner [ 6 ].

Several review articles have been produced in the past few years to consolidate the knowledge base of resource allocation. These include an insightful revisit of the history of the idea of resource allocation in living organisms and the development of resource balance analysis (RBA) model [ 7 ]. Another work demonstrates the power of multiscale metabolic models and omics datasets in elucidating resource allocation principles [ 8 ]. Complementary to [ 8 ] which focuses on fine-grained models that integrate metabolic networks with gene expression, a separate review of phenomenological, i.e. coarse-grained resource allocation models presents the value in making quantitative predictions of microbial phenotypes with only a few adjustable parameters [ 9 ]. Furthermore, a recent review summarises the mathematical structures of models that can predict the overflow metabolism, unifies all models into one standard form and concludes that two growth-limiting constraints are essential for predicting the gradual switch from a high-yield to a low-yield pathway [ 10 ].

Complementary to the existing reviews, this paper adds to the synthesis of the development of this area by (1) offering a generalised conception of cellular resource allocation, (2) presenting the converging understanding of the role of resource application in achieving cellular objectives, (3) classifying and contrasting proposed root causes of proteomic resource allocation, (4) identifying the commonalities and contrasts of both predictive and descriptive mathematical models that incorporate resource allocation, (5) exploring the implications of resource allocation research for synthetic biology, and (6) finally highlighting open questions in both understanding and mathematical modelling. We posit that resource allocation can help answer central bioengineering questions such as how microbes determine growth strategies in a changing environment and what overarching governing principles can guide the design of microbial factories. As such, an application-oriented outlook is also given. Overall, we hope that this concise review will add clarities to the understanding of what has been achieved in this area and hence facilitate its future development.

Types of resources for living microbial cells and the general concept of resource allocation

External and internal resources.

Microorganisms are unicellular organisms that require a range of resources to maintain their viability and to grow and self-replicate. These resources can be sorted to external and internal resources. External resources are environmental provisions that can be utilised by the cell, e.g. chemical substrates (organic compounds, carbon dioxide), nutrients and light. Internal resources comprise those that a cell ‘owns’, such as genetic information, cellular machinery (e.g. ribosomes, RNA polymerases (RNAPs), enzymes and other RNA- or protein-based molecular catalysis) and spatial resources such as membranes and intracellular space. Generally, resource allocation may refer to the cell’s dynamic allocation of any types of internal resources for certain objectives. Possibly because ~ 60% of the dry cell weight are proteins [ 11 , 12 , 13 ] and ~ 85% of the extracellular resources are used for protein productions [ 14 ], the allocation of proteomic resources has become the most acknowledged and best-studied among all the resource types.

While in theory external resources can be infinitely supplied, internal resources generally have physical limitations. Fundamentally this limitations arise from the fact that cells are self-replicating systems, at least during steady-state growth, so only certain combinations of parameter values are permissible [ 15 , 16 ]. For example, the limited translational speed (~ 40 amino acids per second per ribosome [ 8 ]) intuitively calls for an adequate allocation strategy (of finite ribosomal machinery) for fast growth (or upon nutrient shift) where numerous additional copies of proteins are required. Besides, there are often close links between internal and external resources. Importing extracellular substrates, e.g. glucose, into the cytoplasm requires various transporters (internal resources) [ 17 ]. Free energy (internal resource) needed to fuel biological processes is generally extracted from organic compounds or light (external resources) via respiration (or photosynthesis) and is stored and transferred mostly via ATP molecules. Furthermore, cellular membranes isolate the cell from the outer environment, allowing a relatively stable and mild internal space to house fragile and delicate biological apparatus. Surface membranes also accommodate a variety of proteins that control the mass and information transfers between internal and external environments.

A putative decision-making process for resource allocation

We postulate that many of the observed dynamic allocations of cellular (internal) resources in microorganisms are attributable to the balance between the cellular demand and the supply capability (Fig.  1 ). At any point in time, a microbial cell normally pursues an intrinsic objective, e.g. maximisation of growth, due to evolution or culture history. However, the level at which the cell actually achieves this objective depends on the extent to which the corresponding demands for materials and energy are met. Within an engineered organism, additional demands can originate from burdens of synthetic gene-circuits. For instance, the expression of recombinant genes requires additional building molecules, energy and expression machinery [ 18 ]; sub-lethal antibiotic dosage provokes the need for more ribosomes to maintain growth [ 2 ]. With certain (possibly changing) external and internal environments, the cell can be viewed as constantly facing a question as to whether the external and internal resources available to the cell can fulfil the requirement to achieve its chosen objective at a minimal threshold level. If the demands of this threshold are sufficiently met by the available resources, the current objective is pursued. Otherwise, a ‘limited’ or ‘stressed’ state will materialise (note that the term ‘stressed state’ here includes stresses that are driven by the shortage of internal and/or external resources, which is more general than the environmental stress often used in biology), which can be detected by the cell and lead to certain responses through multiple regulatory mechanisms [ 14 , 19 ]. Under certain circumstances, the cell is able to respond to the stress by manipulating its physiology to bypass the limitation without interfering its intrinsic objective. However, if the discrepancy between demand and supply is irreconcilable, the cell may reconfigure its objective, e.g. gradually changing from growth to survival to compromise. The internal resources will be adjusted accordingly to fit to the changed objective.

Illustration of a putative decision-making process underlying the observed dynamic allocation of cellular internal resources. The evolution or culture history normally confers the cell an intrinsic objective, e.g. maximisation of growth, which forms objective-driven demands. For engineered organisms, additional demands can originate from synthetic burdens. If the demands can be met by available external and internal resources, the current objective will be achieved. Otherwise, the cell can detect limitations and devise response strategies, e.g. modulating its internal resources, or under certain circumstances, reconfigure its objective

It has previously been noted that growth rate-dependent regulation is not always dominant [ 8 ]. The above putative “active decision-making process” intends to explain the occurrence of resource allocation under both nutrient-scarce (where growth is often not the first priority) and nutrient-rich growth conditions (where the cell grows at fast rates). Reflecting on previous discussions in this broad area (e.g. in [ 14 ] and [ 8 ]), we posit that cellular objectives, burdens and limitations (or stresses) are closely entangled and together contribute to the tug of war between supply and demand; it seems neither logical nor feasible to discuss one concept independently without considering the others. Applying orthogonal perturbations [ 9 ] (and adaptive laboratory evolution (ALE) if stress-driven mutations are of interests [ 20 ]) would help capture the transition between distinct cellular objectives to uncover the comprehensive decision-making atlas underlying the changing phenotypes across different growth conditions.

Current understanding of resource allocation in microorganisms

Resource allocation decisions facilitate cellular objectives.

In the past decade, our knowledge in resource allocation thrives based on the study of bacteria’s ability to actively modulate their proteome compositions to maximise the steady-state growth rate [ 2 , 3 , 4 , 5 ]. A typical example of growth rate-driven reallocation of proteome is the preference of carbon-spilling fermentation pathway in fast-growing Escherichia coli . As growth rate increases, E. coli changes its metabolic strategy from carbon-efficient respiration to proteomic-efficient aerobic fermentation (for energy biogenesis) so that more proteomic resources can used for biomass synthesis to support rapid growth [ 4 ]. However, as pointed out in [ 8 ], growth rate-dependent regulations are not always dominant. The proteome of E. coli grown on pyruvate, glycerol and galactose is not optimized for fastest growth [ 21 ]. Besides, a proteomic study of ‘persisters’ (i.e. bacteria with transient antibiotic tolerance) [ 22 ] reveals that resource allocation plays a key role in coordinating metabolism towards maximizing energy yield, instead of biomass production. The specific proteome adjustment observed in persister cells, which is comparable with that in starved and stressed cells, was shown to be driven by increased ppGpp levels (a result of general stress or stringent response) rather than a mere consequence of reduced growth rate [ 22 ]. Furthermore, it has proven that strong environmental perturbations can alter the protein composition through directly or indirectly interfering protein synthesis processes [ 23 ].

Since mounting evidence suggests that variations in the proteome composition (a typical example of resource allocation as stated above) can result from the cell fighting against vicious threats in addition to coping with rapid growth, it is reasonable to depict the resource allocation phenomena along with the cellular objective given certain growth conditions. Growth conditions can change from an optimal state (where the external nutrients efficiently support maximal growth), through a nutrient-limited state (where a cell can still grow but at a reduced rate or needs to adjust its physiology, e.g. secretion of a by-product, to maintain the same growth rate) and finally to a growth-threatening state (e.g. under osmotic, pH and temperature stresses where the cell could only maintain slow growth or even completely stop growing [ 24 ]) (Fig.  2 ). The cellular objective can correspondingly shift from growth to survival. Accordingly, it can be postulated that resource allocation emerges to increase cellular fitness to the changing growth environments. As the growth condition becomes increasingly harsh, resource allocation will become less coupled with the growth rate and eventually function for ensuring survival.

The relationship between cellular objective and resource allocation with changing growth conditions. The environmental stress increases as growth conditions vary from optimal (sufficient nutrients to support the maximal growth) to nutrient-limiting, e.g. in the case of carbon, nitrogen or phosphorus limitation and finally to growth-threatening (strong perturbations on bacterial homeostasis). During this process, the growth rate reduces from the maximal value ( \(\lambda_{max}\) ) to almost zero (i.e. the cell stops growing). The signal molecule ppGpp accumulates as a result of the general stress or stringent response [ 22 ]. The cellular objective shifts from maximizing growth to ensuring bacterial survival. Subsequently, resource allocation becomes less coupled with the growth rate and eventually functions for ensuring survival

Pre-allocation and growth optimality

Apart from dynamically facilitating cell growth and survival, resource allocation can take the form of pre-allocation which reflects its evolutionary importance. For instance, E. coli is able to pre-allocate its proteome. It has been shown that a considerable amount of proteins in E. coli is expressed with no immediate benefit given specific growth conditions [ 25 ]. Besides, many catabolic genes for substrates that are not presented in the medium were found to be upregulated with decreasing carbon quality [ 3 , 5 ]. In addition to protein pre-allocation, carbon-limited cells have been shown to have a higher fraction of inactive ribosomes (i.e. free ribosomes not bond to mRNA) than phosphorus-limited cells, which allows rapid growth acceleration upon nutrient upshift [ 26 ]. Pre-allocation of ribosomal capacity was also observed in cells undergone famine-to-feast cycles and is considered beneficial to the overall gain of biomass [ 27 ]. Furthermore, a recent modelling work quantitatively shows that spare ribosomal capacity prevents metabolic overshoots and permits rapid response to nutritional upshifts [ 28 ].

While it is generally considered that constitutive pre-allocation of cellular resources (proteins and/or ribosomes) can provide preparatory advantages to hedge against sudden environmental changes [ 29 ], such strategic decision also imposes substantial burdens that prevent the cell to grow faster. This may reflect an evolutionary choice of maintaining higher robustness at the cost of slower growth rates. Despite the suggestion that the inability to grow at the fastest rate could be explained by limited regulatory capabilities [ 9 ], it is also possible that a seemingly ‘sub-optimal’ growth strategy that enables better response upon nutrient shifts is in fact the ‘optimal’ strategy selected by evolution. From the evolution point of view, it might be inappropriate to evaluate the optimality of a growth or resource allocation strategy by growth rate only. Instead, resource allocation may play a central role on multiple fronts including improving cell growth, ensuring survival and enabling high adaptability and ultimately confer the cell greater evolutionary advantages to compete against others in fluctuating environments.

RNA and space allocation

The notion of resource allocation introduced above can in principle manifest with other internal resources, such as RNA and subcellular space in addition to proteins. We did not find existing evidence for intrinsic RNA allocation independent from proteome allocation or ribosome allocation (possibly because ~ 85% of the RNA is ribosomal RNA [ 30 , 31 ]). Nevertheless, artificial controls of mRNA populations [ 32 ] and the activity of RNAP [ 33 ] to direct the cellular resources to synthetic circuits have been achieved in synthetic biology. On the utilisation of intracellular space, the experimental observations of maximal cell buoyant density [ 34 ] and limited membrane protein density [ 35 ] imply that the space allocation can happen either over the whole cell or within certain subcellular compartments. In light of natural strategies for the spatial organization of metabolism (e.g. organelles in eukaryotic cells or bacterial microcompartments (BMCs) in prokaryotic cells) [ 36 ], there could be more sophisticated strategies for dynamic spatial organisation yet to be discovered.

Growth rate-driven proteome allocation

Using proteome allocation to explain important biological phenomena.

Being the most acknowledged and best-studied type of resource allocation, growth rate-dependent proteome allocation provides insights for many well-known biological phenomena, e.g. the cyclic adenosine monophosphate (cAMP)-dependent carbon catabolite repression (CCR) [ 3 ], the overflow metabolism in fast-growing E. coli [ 4 ] and the change between diauxie and co-utilization of mixed carbon sources [ 37 ]. For CCR, it has been shown that [ 3 ] the physiological function of cAMP-mediated CCR is to ensure proteomic resources are invested as needed for bacterial growth under diverse nutrient conditions, e.g. more resources are directed to biosynthetic processes as growth rate increases (Fig.  3 a). For overflow metabolism, Basan et al . [ 4 ] proposed and validated that proteome allocation plays a critical role in regulating the proteomic resources invested between different energy pathways. They quantified that the protein cost per ATP produced by the fermentation pathway is about 67% of that by respiration, which constitutes the key driver of the activation of the fermentation pathway at rapid growth (Fig.  3 b). More recently, a coarse-grained model of optimal allocation of protein resources quantitatively explained why and how the cell chooses between diauxie- and co-utilization of substrates under mixed carbon sources [ 37 ]. Carbon sources were categorized into those introduced at the upper part of glycolysis (Group A sources) and those entering at other nodes of the metabolic network (Group B sources). Prioritised carbon utilization occurs among Group A sources, and usually the one that supports higher growth rate (associated with higher substrate quality and higher pathway efficiency) is preferred. The preference of carbon sources is usually regulated by catabolic repression which repressors inhibit the gene expressions for catabolism of unfavourite carbon sources [ 38 ]. Sometimes, when Group A and Group B sources are both present in the media, co-utilisation would arise if it is more economical for some precursor pools to take a shortcut of drawing carbon flux from Group B sources while for other precursor pools taking up Group A sources is more efficient (Fig.  3 c). Furthermore, the remarkable achievement of using resource reallocation to explain the overflow metabolism in E. coli sheds light on the mechanisms for other widely recognised overflow phenomena, e.g. the production of ethanol in Saccharomyces cerevisiae , i.e. the Crabtree effect [ 39 , 40 , 41 ] and the production of lactate in cancer cells, i.e. the Warburg effect [ 42 , 43 , 44 ].

Proteome allocation can explain important biological phenomena. a Illustration of the physiological function of cAMP-dependent carbon catabolite repression (CCR) in allocating proteomic resources to meet growth demand adapted from ref. [ 3 ]. Under carbon limitation, catabolic genes (namely mass fraction of catabolic proteins \(\phi_{C}\) ) are upregulated while anabolic genes (indicated by the mass fraction of anabolic proteins \(\phi_{A}\) ) are downregulated with decreased growth rates. b Illustration of the overflow metabolism in fast-growing E. coli adapted from ref. [ 4 ]. The fraction of total proteome allocated to fermentation ( \(\phi_{f}\) ) and respiration ( \(\phi_{r}\) ) is different between slow growth (low carbon uptake) and fast growth (high carbon uptake). The key driver of such modulation of proteome resources lies in the much lower protein investment per ATP flux (yellow arrow) of the fermentation pathway compared with respiration. c Illustration of the coarse-grained model of diauxie and co-utilization of carbon sources adapted from ref. [ 37 ]. For diauxie, two group A sources ( A1 and A2 ) can both supply precursor pools for biomass production but with different pathway efficiencies ( \(\varepsilon_{1}\) and \(\varepsilon_{2}\) ). The one with higher efficiency is preferred for maximal growth. If two precursor pools supply the biomass synthesis, each pool derives from an intermediate node M or N . Either intermediate node can draw flux from either of the two sources A and B . Co-utilization occurs under conditions where the efficiency for biomass production is highest when directly drawing carbon flux from source A to precursor Pool 1 and from source B to precursor Pool 2, i.e. the optimal overall efficiency would be \(\varepsilon_{a1} + \varepsilon_{a2} + \varepsilon_{b1} + \varepsilon_{b2}\)

Investigating the potential physiological root cause of the overflow metabolism

Although various studies [ 1 , 4 , 14 , 41 , 45 , 46 , 47 ] have proven that proteome allocation plays a crucial role in regulating the overflow metabolism in E. coli , they are not without controversies. In particular, the macromolecular crowding (also known as ‘molecular crowding’) hypothesis [ 45 , 46 ] and the constrained proteome allocation hypothesis [ 4 ] have been considered hard to reconcile with each other [ 9 ]. In this section, we discuss different hypotheses for explaining the overflow metabolism towards deciphering the potential physiological root cause.

Molecular crowding hypothesis is based on the notion that a cell has an upper limit or optimal macromolecular density [ 48 ]. It proposed that the hard bound on the intracellular macromolecule concentration (or equivalently the finite cell volume or the solvent capacity constraint) triggers the metabolic shift from full respiration to the overflow metabolism [ 45 , 46 ]. This proposition is supported by the observed change in the cell buoyant density of E. coli MG1655, which gradually increases when growth rate increases from 0.1 to 0.4 h −1 and stays roughly constant at higher growth rates [ 34 ]. Therefore, the physiological root cause of the overflow metabolism in molecular crowding is hypothesised to be the finite cell buoyant density, which in this work is further classified as space limitation (Table 1 ).

On the other hand, the researchers developing the constrained proteome allocation hypothesis observed significant changes in the proteome composition, in particular for energy biogenesis, upon the metabolic shift from normal growth to the overflow metabolism in E. coli NCM3722 [ 4 ]. As mentioned above they determined that the protein investment per ATP flux of fermentation is about twice as efficient as that of respiration. This leads to their critical argument that the overflow metabolism results from the cell’s preference of more proteomic efficient pathways at rapid growth. However, although the work showed an extrapolated upper bound (from the proteomic data) of the proteome fraction available for energy biogenesis, an indication is lacking as to what physiological constraint leads to this phenomenological limitation (more discussion is provided in the section below).

Reflecting on the molecular crowding hypothesis, Basan et al . [ 4 ] argued that the macromolecular density (or the cell volume) constraint is not a valid constraint [ 4 , 9 ] as (1) the cell volume varies widely between growth conditions with similar densities [ 49 ] and (2) they did not observe variations in cell density (within a wide range of growth rates) in their own measurements [ 50 ]. In a subsequent discussion, Vazquez and Oltvai from the molecular crowding ‘camp’ suggested that the prediction of the overflow metabolism by the constrained proteome allocation model in [ 4 ] is achieved through ‘implicit assumptions that expand beyond the hypothesis of proteome allocation alone’ [ 48 ]. More specifically, they showed mathematically that in addition to the differential proteome efficiencies between fermentation and respiration, the prediction of the acetate production flux also requires (1) “a non-zero density of non-metabolic macromolecules”, and (2) “an upper bound in the cell macromolecular density”. They further stated that molecular crowding “explains” the latter point and hence “is a key factor in explaining overflow metabolism”.

Part of the above debate is concerned with cell densities at different growth rates. It should be pointed out that the reported range of growth rates with constant cell density for E. coli NCM3722 (in the constrained proteome allocation hypothesis) is 0.3–2.0 h −1 [ 50 ], and the overflow metabolism of this strain occurs at growth rates above ~ 0.8 h −1 [ 4 ]. On the other hand, the reported growth rate range of E. coli MG1655 (in the molecular crowding hypothesis) is 0.1–0.7 h −1 with the acetate overflow occurring at growth rates above 0.4 h −1 [ 45 ]. For MG1655, the cell buoyant density increases with the growth rate within the range of 0.1 to 0.4 h −1 and plateaus at growth rates above 0.4 h −1 [ 34 ]. One can see several key differences between these two cases in terms of the strain, the range of growth rate, and the growth rate at the onset of overflow; these differences call for cautions when comparing alternative propositions.

Other mathematical models incorporating proteome allocation-derived constraints imply additional root cause of the overflow metabolism. FBA ME (membrane economics) [ 47 ] suggests that the simultaneous use of fermentation and respiration at high growth rates is an outcome of finite cytoplasmic surface area (available for respiratory membrane proteins), which is classified here as (local) space limitation (Table 1 ). The membrane economics hypothesis is reinforced by a recently proposed membrane real estate hypothesis [ 35 ], with the presentation of experimental evidence of the decrease in surface-to-volume ratio and limitation on membrane-protein packing capacity at increased growth rates. However, Basan questioned the membrane economics hypothesis in its inability to explain the overflow metabolism emerging in slow-growing cells expressing a large amount of useless proteins, where the membrane capacity should be sufficient [ 9 ]. Additionally, Resource Balance Analysis (RBA) [ 14 , 51 , 52 ] and Metabolic and macromolecular expression (ME) [ 53 , 54 ] models indicate that the change in the macromolecular composition (primarily proteins) and the occurrence of the overflow metabolism at increased growth rates are derived from limited synthesis capacity of macromolecules (e.g. limited translational rate) and limited efficiencies of molecular catalysis (e.g. limited enzymatic catalytic rates), which is classified here as machinery limitation (Table 1 ).

The proposals described above suggest that the root cause (i.e. fundamental physiological limitation) of the overflow metabolism and the accompanied proteome re-allocation is not a concluded matter. At least two competing explanations, i.e. space limitations (represented by molecular crowding [ 45 , 46 ] and membrane occupancy [ 35 , 47 ] hypotheses) and machinery limitations (represented by RBA [ 14 , 52 ], ME [ 53 , 54 ] and possibly implicitly reflected by constrained proteome allocation hypotheses [ 4 ]) have been supported by experiments and/or mathematical models at least to a certain degree (Table 1 ). Besides, it is worth noting that molecular crowding, constrained proteome allocation hypotheses, RBA and ME correspond to global regulations of the proteome whereas the membrane occupancy-based hypotheses can only explain local proteome adjustments.

The potential link between the proteome fractions and the partition of ribosomal machinery

Interested in the possible physiological root cause behind the constrained proteome allocation hypothesis, we conducted the following derivations to understand how the proteome composition can be linked to the ribosomal machinery. The time-dependent change of the concentration of protein \(i\) ( \(P_{i}\) ) can be modelled as protein generation (via ribosome translation) minus dilution and degradation rate.

\(r_{i}\) is the generation rate of protein \(i\) , which is proportional to the ribosome abundance ( \(R\) ), i.e. \(r_{i} \propto R\) [ 55 ]. \(\lambda P_{i}\) is the protein dilution term with \(\lambda\) being the specific growth rate. \(D_{i}\) is the degradation rate of protein \(i\) [ 56 , 57 ]. We define the partition (allocation) of ribosomal machinery \(\theta_{i}\) as the production rate of protein \(i\) divided by the overall production rate of proteins.

At steady-state exponential growth, \(dP_{i} /dt = 0\) and \(\lambda\) is constant, Eq.  1 implies

Substituting Eq.  3 into Eq.  2 gives

The average \(D_{i}\) for non-ribosomal proteins (NRPs) has been reported to be ~ 0.1 per total NRPs per hour [ 55 ]. In growing E. coli , about 40% mass of total proteins is stable while the half-life of most of the remaining proteins is between 15 and 30 h, corresponding to degradation rate of 0.05–0.02 h −1 [ 58 ]. During steady-state exponential growth, especially for acetate-producing E. coli where the specific growth rate can readily reach beyond 0.8 h −1 [ 4 ], therefore it is reasonable to assume \(\lambda \gg D_{i}\) . The protein degradation term thus can be considered negligible compared with the growth dilution effect. Equation  4 can be simplified as

where \(\phi_{i}\) denotes the proteomic fraction of sector \(i\) [ 4 ]. It is worth noting that Eq.  5 is generally valid for fast-growing cells. Cells under strong burdens or environmental perturbations may not strive for maximal growth. Equation  5 should not be applied to these scenarios.

A direct interpretation of Eq.  5 is that the division of ribosomal machinery is approximately equal to the proteome composition if the cell is growing in the exponential phase (i.e. steady-state rapid growth). More importantly, if the proteome fractions can be seen as a direct proxy of the partition of ribosomal machinery, the hard proteomic constraint proposed by Basan et al.[ 4 ] can result from translational limitations, not necessarily belonging to a special case of the molecular crowding hypothesis. it should be noted that \(D_{i}\) has been reported to vary considerably between different proteins in a cell, e.g. in E. coli [ 58 ] and in Lactococcus lactis [ 59 ]. At slow growth where dilution becomes comparable with protein degradation, the discrepancy in \(D_{i}\) of individual proteins could become impactful which would in turn affect the validity of the approximate equivalence between proteome composition and ribosome occupancy. Nevertheless, this analysis shows that the proteome composition and observed proteome allocation constraints [ 4 ] may closely link to the allocation of the ribosome machinery, and this dependency is particularly strong in steady-state rapid growth. It is worth noting that a formulation similar to Eq.  5 has been shown in a recent study [ 60 ], which proposes that at steady state the relative strength of resources recruitment of a given protein equals its relative mass in the cell; this relevant work also links the maximum growth rate at steady state with the fraction of ribosomes being used to build new ribosomes relative to the total number of ribosomes, which resembles the fraction \(\theta_{i}\) shown in Eq.  5 . On the other hand, the above mathematical derivation shows the approximate equality between \(\theta_{i}\) and \(\phi_{i}\) , but it does not provide any evidence for the root source of the ‘limitation’ observed over \(\phi_{i}\) , which can potentially derive from either (1) the shortage of ribosomal machinery \(\theta_{i}\) , which triggers the cell’s regulation of protein synthesis, or (2) the shortage of space, which triggers the cell’s regulation of protein synthesis; as part of the response, \(\theta_{i}\) will be adjusted, leading to the change in \(\phi_{i}\) . Therefore, further elucidation remains necessary to increase our understanding on the potential causal links.

Network-based mathematical models of cell growth and metabolism incorporating resource allocation

Predictive and descriptive models.

We classify network-based metabolic models incorporating resource allocation (RA) into two major categories, i.e. predictive and descriptive RA models (Fig.  4 ). Predictive RA models are primarily Flux Balance Analysis (FBA [ 61 ])-based models for recapitulating growth phenotypes, which can be further classified into coarse-grained and fine-grained RA models, as distinguished in previous reviews [ 8 , 9 ].

Network-based metabolic models incorporating resource allocation (RA). RA models were classified into predictive and descriptive models. Predictive RA models include coarse-grained models that incorporating phenomenological constraints of macromolecular expressions and fine-grained models that integrate metabolic models with detailed matrices of macromolecular expression processes. Descriptive models are EFMA-based RA models that investigate resource allocation phenomena through converting the metabolic network into EFMs and combining EFMs with macromolecular expression information

Coarse-grained RA models incorporate resource allocation as phenomenological constraints to constrain the solution space of metabolic fluxes. The constraint can be in the form of imposing an upper bound ( \(A\) ) to the sum of the product of metabolic flux ( \(v_{i}\) ) and its ‘cost’ ( \(c_{i}\) ), i.e. \(\mathop \sum \limits_{i} c_{i} v_{i} \le A\) [ 41 , 45 , 46 , 47 , 62 , 63 , 64 ], where \(i\) denotes a flux to be included in the constraint. Alternatively, the constraint can be expressed as the maximum reaction rate ( \(v_{max}\) ) being a function of the enzyme abundance ( \(E\) , limited by an upper bound based on absolute proteomics) and the turnover rate ( \(k_{cat}\) , queried from BRENDA [ 65 ]), i.e. \(v_{max} = f\left( {E,k_{cat} } \right)\) in GECKO [ 66 ]. The simple form of phenomenological RA constraints allows coarse-grained RA models easy to construct. In contrast, fine-grained RA models integrate metabolic models with a macromolecular expression module, where the transcriptional and translational processes are described in the mechanistically detailed manner. Examples of fine-grained RA models are RBA [ 14 , 51 , 52 ] and ME [ 53 , 54 ], as well as the more sophisticated whole-cell model of Mycoplasma genitalium [ 67 ]. The enhanced mechanistic nature makes fine-grained models useful in fundamental studies and biological discovery, e.g. identifying the ‘core proteome’ that must be expressed to sustain cell growth [ 68 ] and to compute the metabolic cost of the production of virulence factors for plant pathogens [ 69 ]. It is worth noting that, although fine-grained models are more informative, their construction tends to be more demanding which could limit their applications to complex systems. In contrast, coarse-grained models offer a compromise between the level of details and the range of applicability. Therefore, the choice between these two types of models may depend on the application context. In addition to static coarse-grained and fine-grained RA models, efforts have been made to fuse enzyme production costs into dynamic frameworks. For example, deFBA [ 70 ] offers a dynamic optimization approach that explicitly includes detailed description of biomass composition and accounts for related enzyme capacity constraints. It was developed to solve dynamic flux optimization problems for metabolic networks coupled with gene expression. Extended from classical dynamic FBA (dFBA [ 71 ]) and static RA models, the deFBA method can predict the dynamics of both metabolic fluxes and biomass composition during metabolic adaptations.

Distinguished from the above predictive models, resource allocation have been incorporated with another stoichiometric modelling approach of metabolic networks (additional to FBA), i.e. Elementary Flux Mode Analysis (EFMA [ 72 ]). EFMA extracts the most ‘essential’ components of a metabolic network, termed the Elementary Flux Mode (EFM), which comprises a minimal set of enzymes that could operate a non-decomposable set of fluxes at steady state [ 72 ]. By overlaying biomass-producing EFMs with enzyme information, e.g. molecular weight, amino acid sequence, EFMA-based RA models were able to associate the occurrence of the overflow metabolism with the cell’s preference of low elemental requirements to construct a functional metabolic pathway [ 73 ] and growth rate-yield trade-offs [ 74 ]. Furthermore, by deciphering the observed optimal metabolic flux distributions into biomass-producing EFMs combined with resource allocation, it has been mathematically proved that the number of active metabolic pathways (i.e. EFMs) is at most equal to the number of biophysical constraints [ 75 ]. For models intended to describe normal and overflow growth states of a cell (corresponding to two active EFMs), at least two biophysical constraints are required. In general, EFMA-based RA models [ 73 , 74 , 75 ] do not focus on predicting growth phenotypes, therefore is considered here as descriptive. They are particularly useful for providing a posteriori explanations of resource allocation associated phenotypic patterns, showing how (mathematically) the observed phenomena are consistent with the topology of the metabolic network. Besides, similar to fine-grained models, it could be challenging to construct EFMA-based models for complex systems.

Key to the predictive power of fine-grained models: adopting variable catalytic rates

In addition to predicting the maximum growth rate and metabolic fluxes as achieved in coarse-grained RA models, fine-grained RA models like RBA [ 14 , 51 ] and ME [ 54 , 76 ] are able to predict the abundance of metabolic enzymes, ribosomes and other RNA- or protein-based macromolecules. The improved predictive power is achieved not only through the inclusion of macromolecular expression module, but also results from more delicate treatments of the variable efficiency of molecular machinery.

In RBA, the prediction of cellular configurations in a specific growth condition (e.g. condition X) is achieved by a two-step procedure [ 52 ]. The first step is the calibration of the apparent catalytic rate \(k_{Ei}\) using flux data (i.e. growth rate, uptake and exertion fluxes) and proteomic data measured from cells grown in condition X. The second step is the growth simulation with the goal of growth rate maximization. To run RBA growth simulations, \(k_{Ei}\) are set to the calibrated values (obtained in step one) and the extracellular nutrient concentrations are set corresponding to condition X. If the large datasets (flux and proteomic data) required for \(k_{Ei}\) identification (in step one) of another growth condition of interest, e.g. condition Y, is not available, the RBA growth simulation for condition Y is enabled through linear regression and projection of \(k_{Ei}\) . To do this, several rounds of \(k_{Ei}\) calibration (step one mentioned above) needed to be run using available fluxes and proteomic datasets obtained from other growth conditions (with different measured growth rates). The resulting multiple sets of calibrated \(k_{Ei}\) are then related to the measured growth rates via a linear function, i.e. \(k_{Ei} = a_{i} + \lambda b_{i}\) , where \(a_{i}\) and \(b_{i}\) are linear coefficients. With this estimated linear correlation, RBA growth simulations for cells grown in condition Y can be performed by (a) setting \(k_{Ei}\) to the value predicted by the linear equation (noted that \(\lambda\) of condition Y should have been measured) and (b) setting the extracellular nutrient concentrations according to condition Y.

In ME [ 54 ], two different types of growth simulation are considered: strictly nutrient-limited (SNL) simulation and proteome-limited simulation. In SNL growth simulations, metabolic enzymes are assumed to be operating below their maximal capacity, i.e. \(k_{eff} < k_{cat}\) , where \(k_{eff}\) is the effective catalytic rate (equivalent to the apparent catalytic rate \(k_{Ei}\) used in RBA); \(k_{cat}\) is the maximal catalytic rate, which is a genuine constant and is set to be proportional to the enzyme’s solvent accessible surface area (SASA). In SNL growth simulations, \(k_{eff}\) is a free variable and is predicted together with the maximum growth rate, metabolic fluxes and macromolecular abundances. In proteome-limited growth simulations, metabolic enzymes are assumed to be operating at their maximal rate, i.e. \(k_{eff} = k_{cat}\) . For both SNL and proteome-limited simulations, the growth conditions are set by specifying the maximum glucose (or another growth-controlling nutrient) uptake rate.

The above description shows that (1) both RBA and ME treat the apparent catalytic rate of metabolic enzymes (i.e. \(k_{Ei}\) or \(k_{eff}\) ) as growth rate- or condition-dependent variable, although the concrete modelling treatments are different; (2) the variable catalytic rate plays an important role in predicting the macromolecular states. The validity of this treatment is supported by the qualitative consistency between the experimental data-derived linear correlation \(k_{Ei} = a_{i} + \lambda b_{i}\) [ 52 ] and the predicted and measured linear \(\frac{{k_{eff} }}{{k_{cat} }}vs.\lambda\) profiles of phosphotransferase system (PTS) activity [ 54 ]. Besides, this idea might also be useful in improving the model prediction in coarse-grained RA models like CAFBA [ 62 ], where the enzyme turnover rate ( \(k_{cat}\) ) was assumed constant.

Other considerations

Cost, benefit and trade-offs.

Resource allocation models discussed above generally associate the physiological burden of proteins (often termed as the metabolic cost, protein cost or enzyme cost) with the protein synthesis process. However, the study of the lac operon model system [ 77 ] reveals that the activity of lac permease LacY, not the production or misfolding of the primary protein LacZ, accounts for the major physiological burden (quantified as relative reduction in growth rate due to operon expression) to the cell. The counter-intuitive result found in lac operon offers a reminder that future effort on resource allocation, in particular the modelling of ‘protein costs’, needs to take more caution to avoid ambiguous or even misleading interpretation of its biological basis.

In parallel to the cost, the benefit for protein expressions has been defined as increased growth rate [ 77 ], increased ATP production rate per protein [ 4 ], increased energy efficiency (growth rate divided by ATP production rate) [ 55 ], increased product yield [ 64 ] and increased adaptability to the changing environments [ 25 , 27 , 28 , 29 ]. Examining different sets of cost and benefit can lead to the study of different trade-offs, such as those between pathway protein cost and yield [ 73 , 78 ], growth rate and yield [ 74 , 79 ] and unused enzymatic and ribosomal capacity and additional storage [ 80 ]. These trade-offs may affect growth rates through distinct mechanisms and thus deserve separate attentions.

Bacterial growth in rich media

Most of the resource allocation studies focus on changes in the phenotypic patterns of cells grown on minimal media. Recent progress in the batch growth of E. coli in undefined rich medium reveals the critical role of amino acid catabolism in regulating the central carbon metabolism (e.g. inhibiting glucose uptake and increasing acetate overflow) for faster growth [ 81 ]. The reported intricate trade-off between decomposing expensive resources (e.g. methionine) and the potential benefit of gaining various carbon and nitrogen sources needed for growth offers insights that support extending the resource allocation principle (which was originally established for cells grown on minimal media) to systems with complex nutritional environments.

Limitation of protein-cost constraints

Compared with classic metabolic models, the key advance of resource allocation models is the inclusion of the costs of macromolecular expressions on top of the metabolism. Taking ME as an example, pathways with higher metabolic efficiency (high-yield) are usually ‘longer’ than less efficient (low-yield) pathways. Therefore, high-yield pathways are always coupled with higher protein costs. When the growth rate is maximised, this feature (i.e. high-yield-high-cost) governs the prediction of the metabolic switch from high-yield to low-yield pathways at increased growth rates. Although protein-cost constraints have been shown to efficiently improve the prediction of metabolic fluxes, this effectiveness can be easily spoiled if two alternative pathways hold similar protein costs. For example, pyruvate dehydrogenase (Pdh) and pyruvate formate lyase (Pfl) both convert pyruvate into acetyl-CoA with a cost of one enzyme. ME predicts the use of Pfl instead of Pdh, whilst the use of the latter was reported by fluxomics [ 54 ]. This outlier can be rectified by adding in regulatory rules, i.e. Pfl is activated only under anoxic conditions [ 82 , 83 ]. However, the regulatory network is currently not in the scope of ME [ 84 ] and to our knowledge nor has it been systematically modelled in other resource allocation models. It would be valuable to construct a model that comprises both metabolic, expression machinery and regulatory networks, paving the way towards more reliable predictive modelling. Apart from adding regulations, such flux prediction issues could be amended by introducing a global view of the system (through e.g. adding appropriate constraints that describe the dynamics of the entire system) beyond the local behaviour of individual reactions.

Implications for synthetic biology

Synthetic circuits compete with their host (and other circuits) for cellular resources required for their respective functions. This resource competition often leads to poor performance and unexpected behaviour of designed systems [ 85 , 86 , 87 ]. However, our growing understanding of microbial resource allocation can help us engineer systems with improved performance and predictability.

A first problem is that the introduction of synthetic circuits often results in growth defects because cellular resources are diverted away from biomass production. In the short term this can lead to loss of productivity; in the long term it can lead to complete loss of function due to evolution [ 88 ]. One approach to resolving this problem is to design synthetic circuits that use less cellular resources or avoid using the most limiting resources. The simplest way of doing this is by decreasing synthetic protein expression to relieve pressure on gene expression in general. This can be achieved either by swapping regulatory components or integrating constructs into the chromosome [ 89 ]. Alternatively, protein parts implementing memory or regulatory devices can be replaced with DNA or RNA parts, reducing load on translation which is usually the most limiting process in prokaryotic gene expression [ 90 ]. Many other functions, however, cannot be implemented without high enzyme concentrations. In such cases it is still possible to minimise the demand of synthetic circuits by using strong promoters and high copy number plasmids but with weak ribosome binding sites, again to limit the burden on translational machinery [ 91 ].

On the other hand, microbial hosts can be designed to increase their supply of cellular resources. In particular, it has been suggested that ‘lean-proteome’ strains with additional proteomic budgets can be generated by removing unnecessary but highly expressed genes [ 81 ]. According to the proteome allocation theory developed by Scott et al . [ 2 ], such strains should be capable of both improved growth and heterologous protein expression, and indeed significant increases in growth rate and biomass yield are observed upon deletion of non-essential genes in Bacillus subtilis [ 92 , 93 ] and E. coli [ 94 ]. In a similar vein, a recent study takes advantage of the transcriptional regulatory network of E. coli to reduce unnecessary enzyme expression by deleting upstream transcriptional factors (rather than deleting the enzymes themselves). With only three genetic interventions, an optimized strain is generated with a demonstrably higher proteome budget and an increased production from a heterologous metabolic pathway [ 95 ]. One limitation of these approaches is that they only look at freeing up proteomic resources of completely unutilized genes, whereas significant proteomic resources may also be wasted on under-utilized genes [ 25 ].

A second problem is that resource competition couples the behaviour of circuits and hosts together in ways that can be difficult to predict. Synthetic circuits depend on host cellular resources such as ribosomes, amino acids and ATP to operate. The performance of synthetic circuits also depends on host physiological states, e.g. growth rate can affect the dilution of circuit components. However, formal models based on our growing biological understanding and tailored mathematical and computational methods are increasingly being used to accurately predict the outcome of circuit-host coupling [ 96 ]. For example, Weiße et al . develop a simple mechanistic model linking the expression of coarse-grained genes to microbial growth [ 97 ]. Their model not only recapitulates well established growth laws such as the linear relationship between ribosome levels and growth rate, but is also used to predict the amplitude and frequency of oscillations generated by a repressilator with varying transcriptional rates. Similarly, Liao et al . develop a multi-scale model of a fine-grained synthetic circuit operating inside a coarse-grained cell, and use their model to predict the bulk and single-cell dynamics of a bistable switch across nutritional and inductional parameters [ 98 ].

Alternatively, circuit-host coupling can be minimised by engineering microbial systems with orthogonal resource pools dedicated to synthetic functions [ 99 ]. For example, orthogonal transcriptional and degradation machinery have been implemented in E. coli using heterologous RNA polymerase and proteases taken from T7 bacteriophage and Mesoplasma florum bacterium respectively [ 100 , 101 ]. However, the same strategy cannot be applied to make orthogonal translational machinery because the structure and function of ribosomes are highly conserved across species. Instead, orthogonal ribosomes have been implemented using synthetic 16S rRNA with high affinity for a non-canonical ribosome binding site. Despite the significant progresses in this area, it should be noted that these strategies achieve partial decoupling at best, as many important resource types such as amino acids and ATP are still shared with the cellular host and have no orthogonal counterpart [ 102 ].

The study of cell growth and resource allocation contributes to the rising consensus on the key driver(s) of the phenotypic patterns of microorganisms. Cell growth and metabolic strategies are no longer considered to be dependent on the metabolic network only, but also subject to the cell’s active allocation of internal resources. Various resource allocation-based hypotheses have been developed to explain important biological phenomena, including (but not limited to) the pre-allocation of proteins and ribosomes, carbon catabolite repression, diauxie and co-utilisation of mixed carbon sources and the overflow metabolism. One important indication from these studies is that the biological importance of resource allocation can be diverse and may encompass maximising the growth rate, enabling rapid response to nutritional shifts and ensuring survival under harsh conditions. This thus supports the argument that the central role of resource allocation is to improve the overall cell’s fitness to the ever-changing environmental conditions. Resource allocation can facilitate different cellular objectives to provide the cell with different competitive advantages: pre-allocation offers the cell high adaptability to sudden nutritional shifts [ 25 , 26 , 27 , 28 ]; stress-driven proteome adjustments enable the cell strong vitality (survival capability) under harsh conditions [ 22 ]; and the utilisation of pathways with lower protein cost (e.g. the overflow metabolism) allows the cell to grow at fast rates [ 78 ]. Meanwhile, various mathematical models have been proposed to validate resource allocation associated hypotheses and experimental observations. We classified network-based resource allocation models into predictive models (including coarse-grained models and fine-grained models that are both generally FBA-based) and descriptive models that focus on elucidating the network topology through EFMA. Detailed overviews of each class of models can be found in [ 8 , 9 , 10 ]. Here, we complement previous work by emphasising the key feature of different resource allocation models and deciphering important modelling treatments (i.e. variable enzyme catalytic efficiencies in RBA and ME).

Despite these prominent achievements in this area, several important questions remain open. Firstly, among resource allocation-derived hypotheses for explaining the overflow metabolism, two competing explanations, i.e. the space limitation and machinery limitation, seem to be independently correct, making the physiological root cause of the overflow metabolism remain unclear. Secondly, the study in lac operon model system [ 77 ] raises the concern in the interpretation of the ‘protein cost’, which might not be limited to protein synthesis but also protein activation/function. Thirdly, previous research in resource allocation (both theoretical and experimental) mainly focuses on cell growth in minimal medium; the connection between cell growth and resource allocation in rich media remains an un-reclaimed field.

Finally, we would like to emphasise the importance of understanding resource allocation for synthetic biology, especially as we move towards implementing increasingly complex synthetic circuits. Widespread adoption of engineered microbes depends on our ability to design performant and predictable systems, which necessarily entails accounting for resource allocation. The realisation of this goal will likely involve a mixture of circuit, host and integrative design.

As the understanding of the mechanisms and utility of resource application in cellular systems further develops, we anticipate that mathematical modelling tools incorporating resource allocation will also further evolve to improve their mechanistic soundness. Together, these will facilitate the widening of the application of resource application mechanisms and principles in synthetic biology.

Availability of data and materials

Not applicable.

Abbreviations

bacterial microcompartments

cyclic adenosine monophosphate

carbon catabolite repression

dynamic flux balance analysis

elementary flux mode

elementary flux mode analysis

flux balance analysis

metabolic and macromolecular expression

non-ribosomal proteins

pyruvate dehydrogenase

pyruvate formate lyase

resource allocation

resource balance analysis

RNA polymerases

solvent accessible surface area

strictly nutrient-limited

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H.Z. is sponsored by the China Scholarship Council (CSC) through a PhD scholarship. The funding body played no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.

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Zeng, H., Rohani, R., Huang, W.E. et al. Understanding and mathematical modelling of cellular resource allocation in microorganisms: a comparative synthesis. BMC Bioinformatics 22 , 467 (2021). https://doi.org/10.1186/s12859-021-04382-3

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• Published: 09 September 2019

Probabilistic cell-type assignment of single-cell RNA-seq for tumor microenvironment profiling

• Allen W. Zhang   ORCID: orcid.org/0000-0002-7606-089X 1 , 2 , 3 ,
• Ciara O’Flanagan 1 ,
• Elizabeth A. Chavez 4 ,
• Jamie L. P. Lim 1 , 2 ,
• Nicholas Ceglia 2 ,
• Andrew McPherson 1 ,
• Matt Wiens 1 ,
• Pascale Walters 1 ,
• Tim Chan 1 ,
• Brittany Hewitson 1 ,
• Daniel Lai   ORCID: orcid.org/0000-0001-9203-6323 1 ,
• Anja Mottok 4 , 5 ,
• Clementine Sarkozy 4 ,
• Lauren Chong 4 ,
• Tomohiro Aoki   ORCID: orcid.org/0000-0001-6782-8361 4 , 6 ,
• Xuehai Wang 7 ,
• Andrew P Weng 7 ,
• Jessica N. McAlpine 8 ,
• Samuel Aparicio   ORCID: orcid.org/0000-0002-0487-9599 1 , 6 ,
• Christian Steidl 4 ,
• Kieran R. Campbell   ORCID: orcid.org/0000-0003-1981-5763 1 , 9 , 10 &
• Sohrab P. Shah   ORCID: orcid.org/0000-0001-6402-523X 1 , 2 , 6  

Nature Methods volume  16 ,  pages 1007–1015 ( 2019 ) Cite this article

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• Cancer genomics
• Cancer microenvironment
• Computational models
• Statistical methods

Single-cell RNA sequencing has enabled the decomposition of complex tissues into functionally distinct cell types. Often, investigators wish to assign cells to cell types through unsupervised clustering followed by manual annotation or via ‘mapping’ to existing data. However, manual interpretation scales poorly to large datasets, mapping approaches require purified or pre-annotated data and both are prone to batch effects. To overcome these issues, we present CellAssign, a probabilistic model that leverages prior knowledge of cell-type marker genes to annotate single-cell RNA sequencing data into predefined or de novo cell types. CellAssign automates the process of assigning cells in a highly scalable manner across large datasets while controlling for batch and sample effects. We demonstrate the advantages of CellAssign through extensive simulations and analysis of tumor microenvironment composition in high-grade serous ovarian cancer and follicular lymphoma.

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Cell type and gene expression deconvolution with BayesPrism enables Bayesian integrative analysis across bulk and single-cell RNA sequencing in oncology

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Acknowledgements

We thank V. Svensson for his feedback on this manuscript. We also thank W. W. Wasserman, B. H. Nelson, P. T. Hamilton and A. Miranda for helpful discussions. A.W.Z. is funded by scholarships from the Canadian Institutes of Health Research (CIHR) (Vanier Canada Graduate Scholarship, Michael Smith Foreign Study Supplement) and a BC Children’s Hospital (UBC) MD/PhD studentship. K.R.C. is funded by postdoctoral fellowships from the CIHR (Banting) no. 01353-000, the Canadian Statistical Sciences Institute and the UBC Data Science Institute no. 201803. S.P.S. is a Susan G. Komen scholar. We acknowledge the generous funding support provided by the BC Cancer Foundation. In addition, S.P.S. receives operating funds from the CIHR (grant no. FDN-143246), Terry Fox Research Institute (grant nos. 1021 and 1061) and the Canadian Cancer Society (grant no. 705636). This work was supported by Cancer Research UK (grant no. C31893/A25050 to S.A. and S.P.S.). S.P.S. is supported by the Nicholls-Biondi endowed chair and the Cycle for Survival benefitting Memorial Sloan Kettering Cancer Center. C.S. is an Allen Distinguished Investigator supported by the Allen Frontiers Group no. 12829.

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Department of Molecular Oncology, British Columbia Cancer Research Centre, Vancouver, British Columbia, Canada

Allen W. Zhang, Ciara O’Flanagan, Jamie L. P. Lim, Andrew McPherson, Matt Wiens, Pascale Walters, Tim Chan, Brittany Hewitson, Daniel Lai, Samuel Aparicio, Kieran R. Campbell & Sohrab P. Shah

Computational Oncology, Department of Epidemiology and Biostatistics, Memorial Sloan Kettering Cancer Center, New York, NY, USA

Allen W. Zhang, Jamie L. P. Lim, Nicholas Ceglia & Sohrab P. Shah

BC Children’s Hospital Research, Vancouver, British Columbia, Canada

Allen W. Zhang

Centre for Lymphoid Cancer, British Columbia Cancer Research Centre, Vancouver, British Columbia, Canada

Elizabeth A. Chavez, Anja Mottok, Clementine Sarkozy, Lauren Chong, Tomohiro Aoki & Christian Steidl

Institute of Human Genetics, Ulm University and Ulm University Medical Center, Ulm, Germany

Anja Mottok

Department of Pathology and Laboratory Medicine, University of British Columbia, Vancouver, British Columbia, Canada

Tomohiro Aoki, Samuel Aparicio & Sohrab P. Shah

Terry Fox Laboratory, British Columbia Cancer Research Centre, Vancouver, British Columbia, Canada

Xuehai Wang & Andrew P Weng

Department of Obstetrics & Gynaecology, University of British Columbia, Vancouver, British Columbia, Canada

Jessica N. McAlpine

Department of Statistics, University of British Columbia, Vancouver, British Columbia, Canada

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UBC Data Science Institute, University of British Columbia, Vancouver, British Columbia, Canada

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Contributions

A.W.Z., K.R.C. and S.P.S. designed the study. A.W.Z., K.R.C. and S.P.S. wrote the manuscript. A.W.Z., C.O.F., E.A.C., J.L.P.L., A. McPherson, A. Mottok, N.C., L.C., M.W., T.A., A.P.W., J.N.M., S.A., C. Steidl, K.R.C. and S.P.S. reviewed the manuscript. A.W.Z., S.A., C. Steidl, K.R.C. and S.P.S. interpreted the data. B.H., D.L., L.C. and C. Sarkozy curated the data. A.W.Z., K.R.C., N.C., M.W., P.W., T.C. and X.W. analyzed the data. A.W.Z., K.R.C. and S.P.S. developed the model. C.O.F., E.A.C. and J.L.P.L. performed the single-cell processing. A. Mottok, J.N.M., C. Steidl and C. Sarkozy performed the case identification. K.R.C., S.P.S., C. Steidl and S.A. supervised the study.

Corresponding authors

Correspondence to Kieran R. Campbell or Sohrab P. Shah .

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S.P.S. and S.A. are founders, shareholders and consultants of Contextual Genomics.

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Integrated supplementary information

Supplementary figure 1 simulation performance across a range of proportions of differentially expressed genes, using differential expression parameters derived from comparing naïve cd8+ and naïve cd4+ t cells..

( a ) Accuracy and cell-level F1 score ( Methods ) for varying proportions of differentially expressed genes per cell type. All methods were provided with expression data for the same set of marker genes. *, **, *** denote FDR-adjusted p-vaues (Wilcoxon signed-rank test) for pairwise comparisons between CellAssign and other methods < 0.05, 0.01, 0.001 respectively. Nine simulated datasets were generated for each parameter setting. Dotted lines separate marker-based, unsupervised, and supervised methods. ( b ) Correspondence between true simulated log fold change values and log fold change (δ) values inferred by CellAssign. R refers to the Pearson correlation between true and inferred logFC values for CellAssign. ( c and d ) Performance of CellAssign where a certain proportion of entries in the marker gene matrix are flipped at random, using ( c ) 5 and ( d) 20 marker genes per cell type. Nine simulated datasets were generated for each parameter setting. In all boxplots, lower and upper hinges denote the 1st and 3rd quartiles, with whiskers extending to the largest value less than 1.5 × the inter-quartile range.

Supplementary Figure 2 Simulation performance across a range of proportions of differentially expressed genes, using differential expression parameters derived from comparing B and CD8+ T cells.

( a ) Accuracy and cell-level F1 score ( Methods ) for varying proportions of differentially expressed genes per cell type. CellAssign was provided with a set of marker genes ( Methods ); all other methods were provided with all genes. Asterisks indicate FDR-adjusted statistical significance (Wilcoxon signed-rank test) for pairwise comparisons between CellAssign and other methods. Nine simulated datasets were generated for each parameter setting. ( b ) Accuracy and cell-level F1 score for varying proportions of differentially expressed genes per cell type. All methods were provided with the same set of marker genes. Nine simulated datasets were generated for each parameter setting. *, **, *** denote FDR-adjusted p-vaues (Wilcoxon signed-rank test) for pairwise comparisons between CellAssign and other methods < 0.05, 0.01, 0.001 respectively. ( c ) Correspondence between true simulated log fold change values and log fold change (δ) values inferred by CellAssign. R refers to the Pearson correlation between true and inferred logFC values for CellAssign. n = 1000 single-cells were simulated. ( d ) Performance of CellAssign where a certain proportion of entries in the marker gene matrix are flipped at random. Nine simulated datasets were generated for each parameter setting. Dotted lines separate marker-based, unsupervised, and supervised methods. In all boxplots, lower and upper hinges denote the 1st and 3rd quartiles, with whiskers extending to the largest value less than 1.5 × the inter-quartile range.

Supplementary Figure 3 Simulation performance (accuracy and cell-level F1 score) across a range of proportions of differentially expressed genes, where clusters from “unsupervised” algorithms are assigned to respective cell types through correlation to the transcriptomes of purified cell types.

Dotted lines separate marker-based, unsupervised, and supervised methods. ( a ) & ( b ) Simulating data based on parameters for B and CD8+ T cells, and running algorithms for whole-transcriptome and marker gene data only, respectively. ( c ) & ( d ) Simulating data based on parameters for naïve CD4+ and CD8+ T cells, and running algorithms for whole-transcriptome and marker gene data only, respectively. In all boxplots, lower and upper hinges denote the 1st and 3rd quartiles, with whiskers extending to the largest value less than 1.5 × the inter-quartile range. *, **, *** denote FDR-adjusted p-values (two-sided Wilcoxon signed-rank test) for pairwise comparisons between CellAssign and other methods < 0.05, 0.01, 0.001 respectively. In all cases, n = 1000 single-cells were simulated.

Supplementary Figure 4 Performance (accuracy and cell type-level F1 score, Methods) of CellAssign and the best-performing clustering methods evaluated by 6 on FACS-purified H7 human embryonic stem cells in various stages of differentiation.

t-SNE plots of ( a ) ground-truth FACS annotations; ( b ) CellAssign-derived annotations; ( c ) SCINA-derived annotations; ( d ) SC3 clusters (using all genes); ( e ) Seurat clusters (resolution = 0.8, using all genes); ( f ) Seurat clusters (resolution = 0.8, using the same marker gene set used by CellAssign); ( g ) Seurat clusters (resolution = 1.2, using all genes); ( h ) Seurat clusters (resolution = 1.2, using the same marker gene set used by CellAssign).

Supplementary Figure 5 Expression of select marker genes in HGSC single cell RNA-seq data.

( a ) Expression (log normalized counts) of PECAM1 (for endothelial cells), CD3D (for T cells), CD79A (for B cells), KLHDC8A (for ovary-derived cells), ACTA2 (for myofibroblasts and smooth muscle), MYH11 (for smooth muscle), and MCAM (for vascular cell types including endothelial cells, vascular smooth muscle, and pericytes). ( b ) Expression (log normalized counts) of marker genes expressed in epithelial ovarian cancers but not in normal ovarian tissue. Expression values were winsorized between 0 and 4.

Supplementary Figure 6 Comparison of clusters from CellAssign and state-of-the-art unsupervised clustering approaches 6 on HGSC single cell RNA-seq data.

( a ) Expression (log normalized counts) of key marker genes of hematopoietic subpopulations CD3D (for T cells), CD79A (for B cells), and CD14 (for monocytes/macrophages). Expression values were winsorized between 0 and 4. UMAP plots of ( b ) CellAssign-derived annotations; ( c ) SC3 clusters (using all genes); ( d ) Seurat clusters (resolution = 0.8, using all genes); ( e ) Seurat clusters (resolution = 0.8, using the same marker gene set used by CellAssign); ( f ) Seurat clusters (resolution = 1.2, using all genes); ( g ) Seurat clusters (resolution = 1.2, using the same marker gene set used by CellAssign).

Supplementary Figure 7 Cluster-specific HLA expression in HGSC epithelial cells.

( a ) Expression (log normalized counts) of HLA class I genes in all HGSC cells. Expression values clipped from 0 to 8. ( b ) Expression of HLA class I genes across cell types in all HGSC cells. Epithelial (1): epithelial cells from cluster 1. Epithelial (other): epithelial cells from all other clusters. Lower and upper hinges denote the 1st and 3rd quartiles, with whiskers extending to the largest value less than 1.5 × the inter-quartile range. n = 4847 single-cells total (15 B cells, 24 T cells, 99 Monocyte/Macrophage, 361 endothelial, 300 vascular smooth muscle cells, 438 ovarian myofibroblast, 809 ovarian stromal 750 epithelial (1), 2051 epithelial (other)).

Supplementary Figure 8 Cluster-specific phenotypes in HGSC epithelial cells.

( a ) Hallmark pathway enrichment results for epithelial clusters 3 (n = 161 single cells) vs. 1 (n = 750 single cells) from the left ovary sample. ( b ) Gene-level differential expression for epithelial clusters 3 vs. 1, with statistical testing performed using the findMarkers function in the scran R package. ( c and d ) Hallmark pathway enrichment results for epithelial clusters ( c ) 2 vs. 0; and ( d ) 2 vs. 4. ( e ) Expression (log normalized counts) of select hypoxia-associated markers in HGSC epithelial cells. Expression values were winsorized between 0 and 4.

Supplementary Figure 9 Cell-type specific expression in follicular lymphoma.

( a ) Expression (log normalized counts) of select marker genes CD2 (for T cells), MS4A1 (for B cells), CD8A and GZMA (for CD8+ T cells), CD4 (for T follicular helper cells and other CD4+ T cells) and CXCR5 and ICA1 (for T follicular helper cells). Expression values were winsorized between 0 and 3. ( b ) Heatmap of marker gene expression, labeled by maximum probability CellAssign-inferred cell types.

Supplementary Figure 10 Comparison of clusters from CellAssign and state-of-the-art unsupervised clustering approaches 6 on follicular lymphoma single cell RNA-seq data (showing only T cell subtypes).

Comparison of clusters from CellAssign and state-of-the-art unsupervised clustering approaches 6 on follicular lymphoma single cell RNA-seq data (showing only T cell subtypes).

Supplementary Figure 11 Expression (log normalized counts) of κ and λ light chain constant region genes in nonmalignant B cells.

Class assignments were determined by CellAssign ( Methods ).

Supplementary Figure 12 Expression (log normalized counts) of selected marker genes ( CD2 , CD3D , and CD3E for T cells; CD79A , MS4A1 , and CD19 for B cells) in scvis embedding of reactive lymph node data.

Expression values were winsorized between 0 and 3.

Supplementary Figure 13 Differential gene regulation for FL1018 and FL2001.

Differential expression results using scran’s findMarkers for malignant vs. nonmalignant B cells in ( a ) FL1018 and ( b ) FL2001. Comparisons was performed accounting for timepoint and potential interactions between malignant status and timepoint using a multivariate linear model described in Methods . Genes upregulated among malignant cells have logFC values > 0. P-values were adjusted with the Benjamini-Hochberg method. Significantly enriched Reactome pathways (BH-adjusted P -value ≤ 0.05) among the top 50 most highly upregulated genes (ranked by log fold change) in ( c ) FL1018 and ( d ) FL2001. Up to 30 pathways are shown in either plot ( Methods ). Differentially expressed genes (found using scran’s findMarkers) for ( e ) T follicular helper and ( f ) other CD4 T cells between T2 vs. T1. Genes upregulated in T2 have log fold change values > 0. The activation marker CD69 is highlighted. P -values were adjusted with the Benjamini-Hochberg method.

Supplementary Figure 14 Fitting single cell RNA-seq simulation models to the Zheng PBMC 68k dataset, using cell type annotations provided in 51 (n = 66205 single cells), and of FACS prufied data from Koh et al. (n = 369 single cells).

( a ) Log fold change values computed from differential expression analysis between naïve CD8+ and naïve CD4+ T cells. ( b ) ‘Null’ log fold change values computed by randomly splitting naïve CD8+ T cells into equally sized halves 10 times. ( c ) Quantile-quantile (QQ) plot comparing observed log fold change values between naïve CD8+ and naïve CD4+ T cells and posterior predictive samples from the splatter model ( Methods ). ( d ) Quantile-quantile (QQ) plot comparing observed log fold change values between naïve CD8+ and naïve CD4+ T cells and posterior predictive samples from the modified model ( Methods ). ( e ) Log fold change values computed from differential expression analysis between human embryonic stem cells (hESCs) and day 3 somite cells (ESMT). ( f ) ‘Null’ log fold change values computed by randomly splitting anterior primitive streak cells into equally sized halves 10 times. ( g ) Quantile-quantile (QQ) plot comparing observed log fold change values between hESC and ESMT cells and posterior predictive samples from the splatter model ( Methods ). ( h ) Quantile-quantile (QQ) plot comparing observed log fold change values between hESC and ESMT cells and posterior predictive samples from the modified model ( Methods ).

Supplementary Figure 15 Benchmarking results for CellAssign across a range of simulated data set sizes (number of cells), number of cell types being inferred, and number of marker genes per cell type.

( a ) Runtime (to convergence, defined as a relative change in log-likelihood < 10 −3 between successive iterations, as a function of data set size and the number of marker genes used per cell type, on simulated data ( Methods ). Two cell types were used. ( b ) Runtime (to convergence, defined as a relative change in log-likelihood < 10 −3 between successive iterations, as a function of the number of cell types and the number of marker genes used per cell type, on simulated data. One thousand cells were used. n = 5 simulated datasets were generated for each parameter setting. In all boxplots, lower and upper hinges denote the 1st and 3rd quartiles, with whiskers extending to the largest value less than 1.5 × the inter-quartile range.

Supplementary information

Supplementary information.

Supplementary Figs. 1–15 and Supplementary Note.

Reporting Summary

Supplementary table 1.

Performance measures on simulated data

Supplementary Table 2

Marker gene matrices used in analysis

Supplementary Table 3

Pathway enrichment results for follicular lymphoma and HGSC data

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Zhang, A.W., O’Flanagan, C., Chavez, E.A. et al. Probabilistic cell-type assignment of single-cell RNA-seq for tumor microenvironment profiling. Nat Methods 16 , 1007–1015 (2019). https://doi.org/10.1038/s41592-019-0529-1

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Evaluation of a channel assignment scheme in mobile network systems

• Nahla Nurelmadina 1 ,
• Ibtehal Nafea 1 &
• Muhammad Younas 2  

Human-centric Computing and Information Sciences volume  6 , Article number:  21 ( 2016 ) Cite this article

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The channel assignment problem is a complex problem which requires that under certain constraints a minimum number of channels have to be assigned to mobile calls in the wireless mobile system. In this paper, we propose a new scheme, which is based on double band frequency and channel borrowing strategy. The proposed scheme takes into account factors such as limited bandwidth of wireless networks and the capacity of underlying servers involved in processing mobile calls. It aims to ensure end-to-end performance by considering the characteristics of mobile devices. This is achieved by determining the position of users (or mobile stations) in wireless mobile systems. The proposed scheme is simulated in order to investigate its efficiency within a specific area of a large city in Saudi Arabia. Experimental results demonstrate that the proposed scheme significantly improves the performance of mobile calls as well as reduces the blocking when the number of mobile call increases.

Mobile devices and particularly mobile phones have been used for a variety of purposes ranging from voice calls through to sending SMS/emails to online banking and shopping. Mobile phones generally use cellular network system as one of the main communication network. The rate of increase in the popularity of mobile phone usage has far outpaced the availability of usable frequencies which are necessary for the communication between mobile users and the base stations of cellular networks. This constitutes an important bottleneck in the provided capacity of mobile cellular systems. Careful design of a network is necessary to ensure efficient use of limited frequency resources. One of the most important issues in the design of a cellular radio network is to determine a spectrum-efficient and conflict-free allocation of channels among the cells while satisfying both the traffic demand and the electromagnetic compatibility (EMC) constraints [ 1 ]. This is usually referred to as channel assignment or frequency assignment. The problem of channel assignment becomes increasingly important, i.e., how do we assign calls to available channels so as to improve performance and to minimize interference. This paper proposes a new scheme for channel assignment, which is called Double Band Frequency Channel Borrowing (DBFCB) scheme. The objective of the proposed scheme is to optimize channel utilization, improve performance and to reduce the blocking probability of calls in a wireless mobile network system.

The proposed scheme is systematically developed and validated through various simulation experiments. It has been applied to the central area of a large city, Madina Monwara, in the Kingdom of Saudi Arabia, using two bands (900, 1800 MHz). Channel borrowing techniques were simulated to investigate the efficiency of this scheme and to make sure that the scheme is viable. The theoretical analysis of the tele traffic was validated through MATLAB simulation analysis [ 2 ]. The simulation model is based on the number of users within a specific area which has BTSs. This is based on data collected from the famous local mobile operator, Zain Telecom Company‎.

The main contributions of the proposed scheme are to reduce the call blocking and call dropping probabilities. Such probabilities generally increase with the increase in the number of mobile users. Thus reduction in call blocking/dropping will enable improved service provisioning in mobile wireless network. In addition, DBFCB algorithm improves response time by using both benchmark and heavy traffic demands with the same known constraints.

The remainder of the paper is structured as follows. ‘‘ Mobile network architecture and communication ’’ section describes an architecture of a mobile network and a mobile communications process. ‘‘ Related work ’’ section reviews and analyses related work. ‘‘ The proposed scheme ’’ section presents the proposed scheme. ‘‘ Modelling of the proposed scheme ’’ section describes modelling of the proposed scheme. ‘‘ Experimental results ’’ section describes the experimental results and analysis. ‘‘ Conclusion ’’ section presents the conclusion.

Mobile network architecture and communication

This section describes the fundamental principles and concepts of wireless mobile network systems. It first presents a generalized architecture of mobile networks and describes its main components. It then describes mobile communications process.

Mobile network architecture

Figure  1 shows an mobile network architecture. The process of call handling in mobile network is carried out in different steps. First, a mobile device (making call) establishes a connection with the access point which is the base station. If the connection is successful the base station responds to the call of mobile device. Radio frequency connection establishment is triggered by sending a channel request message. This message requests the Base station system (BSS) for allocation of radio resources for radio connection setup. The mobile device then waits for an assignment of the access channel. At this point the mobile device is listening to the access channel for a reply. The BSS allocates a channel to the mobile device. This channel allocation assigns a frequency and a timeslot on that frequency. After the mobile device receives this message, it will only use the specified resources for communication within the mobile network.

The main components, of Fig.  1 , involved in mobile call handling, are explained as follows.

Mobile switching center (MSC) It provides call control and telephony switching services between telephone and data systems, and it also provides access to the fixed Public Switched Telephone Network. The MSC manages handoff and switching processes between cells. It communicates with each relevant BS (Base Station) in order to drop the call from the old BS and to set up a new one in the new BS (as a part of the handoff process). MSCs also orchestrate the process of creating new voice calls. An MS initiates a call by using a reverse control channel to make a request. The MSC has then to grant the request, after which a pair of voice channels is assigned to the call. The MSC includes one database for storing location information and call details of a mobile terminal. The MSC is also connected to a second database in which information about a subscriber registered in its mobile communication service is stored. The base stations route the communications to the MSC via a serving BSC. The MSC routes the communications to another subscribing wireless unit via a BSC/base station path or via the PSTN/Internet/other network to terminating destination. Between MSCs, circuit connections provide the handover mechanism that services calls as users roam from one service zone to another.

Home location register (HLR) It is a central master database within the GSM network, which maintains a permanent store of subscribers’ information, and location information for the mobile network. The HLR provides information on the services (subscribed) to the network users. It is also an important source of data to support the roaming process which enables incoming calls that are to be routed to the location of the subscriber.

AC or AUC This is the Authentication Center which contains a secured database handling authentication and encryption keys. It is also a key component of the HLR. It validates the mobile SIM (Security Information Management) card which attempts to connect to a mobile network. It verifies a mobile device by sending a randomly generated number to the mobile device. The mobile device then performs a calculation against it with a number it has stored and sends the result back. If the switch gets the number it expects then the call proceeds. The AC stores all data needed to authenticate a call and to encrypt both voice traffic and signaling messages [ 3 ].

Base station system (BSS) All radio-related functions are performed in the BSS, which consists of base station controllers (BSCs) and the base transceiver stations (BTSs) [ 3 ].

BSC It provides all the control functions and physical links between the MSC and BTS. It is a high-capacity switch that provides functions such as handover, cell configuration data, and control of radio frequency (RF) power levels in base transceiver stations. A number of BSCs are served by an MSC.

BTS It handles the radio interface to the mobile station. The BTS is the radio equipment (transceivers and antennas) needed to service each cell in the network. A group of BTSs are controlled by a BSC.

Mobile communication

Each mobile device uses a separate, temporary radio channel in order to communicate with the cell site. The cell site talks to many mobile devices at once, using one channel per mobile device. Channels use a pair of frequencies for communication (see Fig.  2 )—one frequency (the forward link) for transmission from the cell site and one frequency (the reverse link) for the cell site to receive calls from the mobile device. Mobile devices must stay near the base station to maintain communications. The basic structure of mobile networks includes telephone systems and radio services. Mobile radio service operates in a closed network and has no access to the telephone system. But mobile telephone service allows interconnection with the telephone network.

Mobile communication system

Related work

Various techniques and models have been developed in order to improve the performance of mobile calls and related services in mobile wireless networks. Different factors contribute to the performance aspects such as network traffic, bandwidth, computing devices, and the wireless signals between the mobile devices and nearby base stations of cellular radio networks.

Various channel assignment schemes have been widely investigated with a goal to maximize the frequency reuse. The channel assignment schemes in general can be classified into three strategies: fixed channel assignment (FCA), dynamic channel assignment (DCA) and the hybrid channel assignment (HCA) [ 4 , 5 ]. In FCA, a set of channels are permanently allocated to each cell based on pre-estimated traffic intensity. In this case, the co-channel interference (Transmission on same frequency), adjacent channel interference (Transmission on close frequencies), and the co-site channel interference lead to the main problem, i.e., it does not adapt to changing traffic conditions and user distribution. Moreover, the frequency planning becomes more difficult in a microcellular environment as it is based on the accurate knowledge of traffic and interference conditions. The main problem of FCA is the poor channel utilization wherein some users are unable to find any channel to use.

In DCA, there is no permanent allocation of channels to cells. Rather, the entire set of available channels is accessible to all the cells, and the channels are assigned on a call-by-call basis in a dynamic manner. This means that base station chooses frequencies depending on the frequencies already used in neighboring cells. But the issue with the DCA is to handle more traffic in a particular cell [ 6 , 7 ].

Kyasanur et al. [ 8 ] propose to improve the capacity of multi-channel wireless networks. This work exploits multiple interfaces but with the constraint that the number of available channels is greater than the number of available interfaces. It also proposes a strategy that maintains the autonomy of IEEE 802.11 such that it is not required to be modified.

Rajagopalan et al. [ 9 ] take into account quality of service parameters such as residual bandwidth, number of subscribers, duration of calls, frequency of calls and their priority. This work is based on the optimization of dynamic channel allocation using genetic algorithm (GA). It attempts to allocate channels to users such that overall congestion in the network is minimized by reusing already allocated frequencies. This work utilizes GA in order to ensure optimization. The optimized channels are then compared with non-optimized channels in order to check the efficiency of the proposed algorithm.

Shindeet al. [ 10 ] propose a multi-channel allocation model. It uses an evolutionary strategy with an allocation distance in order to enable efficient use of frequency spectrum. The problem of determining an optimal allocation of channels to mobile users that minimizes call blocking and call dropping probabilities is also emphasized in this work.

In order to ensure efficient and smooth service provisioning in the presence of network congestion, link failures, and mobile service station failures, Boukerche et al. [ 11 ] propose that the cellular network be divided into hexagonal cells as shown in Fig.  3 . This approach divides the cells into five groups of varying sizes. The request for a channel can be granted if the requesting cell receives the reply from all members of a group. However, this algorithm may not work properly if the replies received by the requesting cell do not satisfy the above mentioned criteria. The algorithm is successful in the scenarios when the area of coverage is divided into hexagonal cells and the reuse distance is fixed for all cells.

Frequency reuse (channel allocation)

The proposed scheme

The assignment of channels to cells or mobile devices is one of the fundamental resource management issues in a mobile communication system as it involves different cellular components, handover scenarios, and the complex roles of the base station (BS) and the mobile switching center (MSC). In order to appropriately plan a mobile cellular radio network it is necessary to allocate channels to base stations (BS) so as to ensure that the network can carry sufficient traffic while avoiding interference problem [ 12 ].

In a mobile communication system the total number of channels made available (free) to a system depends on the allocated spectrum and the bandwidth of each channel. However, in the current mobile communication system, the available frequency spectrum is limited and the number of mobile users is increasing. Hence the channels must be reused as much as possible in order to increase the system capacity. Thus it is important to allocate channels to cells or mobile devices in such a way so as to minimize the dropping probability of incoming and outgoing calls and the probability that the carrier-to interference ratio of any call falls below a pre-specified value; i.e. the blocking probability which is one of the most important quality of service (QoS) parameters in the channel assignment schemes.

The overall objective is to serve the maximal number of network users over limited transmission resources. The transmission resource is an available radio spectrum which consists of a limited number of frequencies or (channels). Channel assignment problem involves assigning frequencies to each radio cell in such a way that a set of constraints is satisfied [ 13 ]. These include the limited number of available frequencies in the radio spectrum as well as the traffic constraints corresponding to the minimum number of frequencies indispensable for covering communication between mobile devices moving in a particular cell. In addition, the electromagnetic compatibility constraints (EMC) may happen between channels in the same cell (co-site channel constraint), interference between neighboring cells (adjacent channel constraint) and interference between other cells utilizing the same channel (co-channel constraint) [ 14 ].

This paper proposes a new scheme (or algorithm) in order to optimize the frequency assignment and to enable the reuse of same frequency by sufficiently distant cell. This is to maximize the number of communication (calls) but with a limited number of frequencies. The proposed scheme is called dual band frequency channel borrowing (DBFCB). In Simple Borrowing, channel assignments are borrowed from the adjacent cells and are returned to that cell after it has become free. When a new call initiates and reaches to a cell, and if currently, all the permanent channels allocated to the cell are busy, then channels are borrowed from adjacent cell provided the channels are available (in adjacent cell) and minimum reusable distance constraint is met. In Channel Borrowing algorithms, a database is maintained for the record of channels as per their status either currently in use, borrowed or free. Mobile switch center (MSC), taking care of the channel borrowing activities, runs the channel borrowing procedure, so that channels available are borrowed from the cell having relatively more free channels. Channel borrowing is done under minimum reusable distance constraint. The performance may be reduced for ongoing connections, due to increase of overheads in the base stations of the cellular Mobile system [ 15 ].

The main steps of the working mechanism of the DBFCB scheme are illustrated as follows. These steps are diagrammatically shown in Fig.  4 .

Flow chart of the dual band frequency channel borrowing (DBFCB) scheme

When a mobile user wants to communicate with another user or a base station, it must first obtain a channel from one of the base stations that hears it. That is, when a user (mobile device) wants to starts a call, the base station (BS) is identified [ 16 ]. BS is then made aware about user’s location.

Based on the location, users close to the BS get higher priority compare to users who are away from the BS.

When a call request occurs within a cell, the channel allocation (with frequency 900 MHz) of this cell are tested.

The channels are tested in an order starting from the first channel of the list. This is to look for the availability of a free channel.

If a free channel is found, it is assigned to the call associated with the user (mobile device).

If no free channel can be found and all the channels are busy then a channel allocation (with frequency 1800 MHz) is borrowed from the adjacent cell. The adjacent cell is required to have the largest number of channels available for borrowing.

If all channels in the adjacent cell are busy then it borrows channels from the next cell (with frequency 1800 MHz), if available.

Modelling of the proposed scheme

This section explains the main elements which are involved in order to model the proposed scheme. Based on these the proposed scheme is then tested and evaluated through simulation experiments [ 17 ].

Modelling of the geographical area

In order to test the proposed scheme we model the (simulated) geographical area with respect to a real geographical area of one of the major cities in Saudi Arabia, called Madina Monwara. This city attracts a large number visitors and thus providing a good venue for testing the proposed scheme. It represents the user mobility and traffic behavior within a certain area such as the Haram Area in the city, as shown in Fig.  5 . For the proposed scheme, this area represents one cluster (as in related studies of modelling city areas [ 18 ]). In line with the related studies, the area under consideration (as in Fig.  5 ) exhibits specific characteristics such as population distribution, and distribution of MAPs (Movement Attraction Point).

Geographical area in the city of Madina Monwara, Saudi Arabia

Population distribution Population of people in a geographical area can be grouped into different classes including: visitors, cars, and local working people. The classification of groups is based on the mobility behavior of a population. However, in the proposed scheme, we consider a representative sample of people which are mobile users (making mobile calls). This is because mobile communication systems focus merely on the mobility behavior of mobile users.

Movement attraction points (MAP) MAPs represent locations that attract the population movements and at which people spend considerable time. Examples are work places, residences, shopping centers, etc. Each MAP characterizes the people group type it attracts. The proposed scheme considers the MAP (shown in Fig.  5 ) which is the main attraction for visitors in the city of Madina Monwara. Other types of MAPs include residences, work places, shopping centers, etc.

Traffic modelling

We consider the arrival of both incoming and outgoing calls. The call arrival rate refers to the total number of incoming and outgoing calls during busy hour conditions. The call arrival process follows Poisson distribution. For high mobility users, the rate of incoming calls is assumed to be higher than the corresponding outgoing calls.

Consider the scenario in wireless mobile network consisting of two cells in a series. New calls arrive in the first cell with Poisson rate and are served for a time interval that is negative exponentially distributed with mean calls carried in the first cell (block call). After completion of service, calls are offered to the second cell with a fixed handoff probability. These calls are serviced in the second cell for time intervals that are negative exponentially distributed with mean. For simplicity, we assume that cell receives no new calls and also generates no block calls to be given to the first cell. The blocking experienced by the new calls of mobile network in the first cell is given by the Erlang. The traffic load, in Erlang, is the product of the call arrival time and the call duration [ 19 ]. The call arrival time represents the cumulative sum of calls inter-arrival time, which follows a Poisson distribution with an average time (λ). Note that we characterize the joint probability distribution of the number of calls in the cells in such a way that we take into account that the users perform random motions. The inter-arrival time define the time period between two consecutive calls.

During the first part of simulation, λ was kept constant in order to investigate the performance at a certain time period with a fixed traffic load. In the second part, the traffic load varied with the simulation time, thus the performance was according to the traffic load. The call duration is chosen as a negative exponentially distributed because for all calls the arrival time and call duration are treated as independent random variables.

Experimental results

The proposed scheme is simulated using the MATLAB software [ 20 ]. The simulation model is divided into three parts. The first part deals with simulation parameters, such as the size of simulation area. The second part deals with the traffic generation parameters, such as inter-arrival time, call arrival time, call duration time and random variable generation (e.g., mobile location in the simulation window). The third part deals with the channel assignment mechanism.

Simulation model

The simulation model consists of a fixed window with four-overlapped cells. Each cell consists of two bands frequency, 900 MHz and 1800 MHz. The simulation area is equal to 4 Km 2 . Every cell covers 1 Km 2 ; assume that the cell type used can cover up to 1 Km 2 , macrocell. As shown in Fig.  6 , the simulation area is divided into four cells, each associated with one BTS (Base Transceiver Station). The coordinates for each BTS are as follows.

Distribution of mobile users in the simulation area

BTS(1) in X-pos starts from 0 to 1000 m and in y-pos from 0 to 1000 m.

BTS(2) in X-pos starts from 0 to 1000 m and in y-pos starts from 1000 m to 2000 m.

BTS(3) in X-pos starts from 1000 m to 2000 m and in y-pos starts from 1000 m to 2000 m.

BTS(4) in X-pos starts from 1000 to 2000 m and in y-pos starts from 0 to 1000 m.

The main parameters considered in the simulation are number of cells, number of channels, population size and the maximum number of iterations.

Simulation results

The proposed algorithm was investigated using four different cases. Each simulation was run ten times in order to obtain an average value in each case.

Case 1 The number of mobile users in the simulation area is 5000 and the channels are 72 in each cell in the entire BTS. The algorithm was investigated under extremely high traffic intensity. The average holding time call was adjusted to 60 s and the average arrival time was adjusted to 1 s. The simulation results are shown in Fig.  7 . The blocking call values show that all channels were consumed; i.e., value of channel availability is zero because all channels are busy and there is no free channel at BST. The negative value of channel availability means that new calls have no free channels. The positive value of channel availability means that free channels are available.

Channel consumption results at the end of simulation time in case 1

The results show, that in the case of frequency 900 band (1 to 4 BTS) all channels were consumed in 4 Base station that means no free channel (all channels locations were busy in cell) use the other frequency 1800 band.

In the case of frequency 1800 band (5 to 8 BTS), the following observations were made:

BTS 1 consumed all channels and 25 new calls were blocked no channel free available;

BTS 2 consumed all channels and 23 new calls were blocked no channel free available;

BTS 4 consumed all channels and 7 new calls were blocked no channel free available;

BTS 3 consumed 66 channels and 6 new channels were available channel free available.

Case 2 In this case, the algorithm was investigated under high traffic intensity. The average holding time was adjusted to 180 s and the average arrival time was adjusted to 1 s. Figure  8 shows the simulation results:

Channel consumption result at the end of simulation time in case 2

The results show that in frequency 900 band, all channels were consumed and no channels were free in the band 900 of all 4BTS. In using the other frequency 1800 band, the following observations were made:

BTS 1 consumed all channels and 4 calls were blocked. All channel in this base station are busy and thus 4 new calls were blocked;

BTS 2 consumed all channels and no free channel was available. All channels were busy in the cell and thus 6 new calls were blocked;

BTS 4 consumed all channels. 23 calls were blocked as there was no free channel available;

BTS 3 consumed 59 channels. 7 channels were available channel so calls are not blocked.

Case 3 In this case, the average holding time was adjusted to 180 s and the average arrival time was adjusted to 30 s. The simulation results are shown in Fig.  9 .

Channel consumption result at the end of simulation time in case 3

The results indicate that in frequency 900 band:

BTS 1 consumed 4 channels and 68 channels are available. Thus no call was blocked;

BTS 2 consumed 3 channels and 69 channels are available. Thus no call was blocked;

BTS 3 consumed 4 channels and 68 channels are available. Thus no call was blocked;

BTS 4 consumed 6 channels and 65 channels are available. Thus no call was blocked.

On the other hand, in frequency 1800 band, there were no channels consumed. All channels were available.

The algorithm was investigated under medium traffic intensity. Total channels in each cell were 72.

Case 4 The average holding time was adjusted to 180 s and the average arrival time was adjusted to 120 s. Figure  10 shows the simulation results.

Channel consumption result at the end of simulation time in case 4

According to the results gathered with frequency 900 band, the following observations were made:

BTS-1 consumed 1 busy channel and 71 channels were available, and no call attempted was blocked;

BTS-2 consumed 1 busy channel and 71 channels were available, and no call attempted was blocked;

BTS-3 consumed 2 busy channels and 70 channels were available, and no call attempted was blocked;

BTS-4 consumed no channel and all channel were available, and no call attempted was blocked;

But in the case of 1800 band there were no channels consumed. All channels were available and no call attempted was blocked. The algorithm in case 4 had lower call blocking as compared to the other cases. This shows the improvement of the proposed scheme in reducing the call blocking.

In mobile network systems, assigning a channel to a call in a cell in order to achieve high spectral efficiency is crucial to maintaining call quality and reducing call blocking. This paper proposed a new scheme in order to improve channel assignment problem in the mobile network systems. The proposed scheme takes into account double band frequency channel borrowing. It shows greater response to both benchmark and heavy traffic demands and it enhances network performance with optimum load on the network. The algorithm was evaluated using MATLAB that simulated the network and user distribution behavior in a specific (and busy) area of the city of Madina Monwara in Saudi Arabia. Various experiments were conducted. The results showed that the proposed scheme has the capability of reducing the probability of call blocking and call dropping. It also optimizes channel utilization in mobile network systems. The results also show the effectiveness of the algorithm in borrowing and assigning channels in a high traffic intensity and crowded area. Overall the proposed algorithm reduces the blocking rates of calls and improves the response time even under heavy traffic conditions.

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Authors’ contributions

NN and IN carried out related studies and analysis of the literature. NN, IN and MY participated in the design and development of the proposed scheme which is based on double band frequency and channel borrowing strategy. NN and IN collected simulation data and carried out experiments. MY participated in its design and coordination and helped to draft the manuscript. All authors have read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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Nurelmadina, N., Nafea, I. & Younas, M. Evaluation of a channel assignment scheme in mobile network systems. Hum. Cent. Comput. Inf. Sci. 6 , 21 (2016). https://doi.org/10.1186/s13673-016-0075-0

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Home > Books > Knowledge Management Strategies and Applications

Knowledge‐Based Assignment Model for Allocation of Employees in Engineering‐to‐Order Production

Submitted: 04 May 2016 Reviewed: 08 June 2017 Published: 21 November 2017

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In today’s rapidly changing business environment, it is necessary to react promptly in response to the product changes that happen constantly in an Engineering‐to‐Order production environment. Very often, there is not sufficient time to educate employees regarding new and necessary knowledge. If we insist on the standardization of a process execution, the process always requires appropriate knowledge from among available employees. In this chapter, an option for adjusting processes to available knowledge is studied. Following calculations, it was concluded that a partial corruption of a perfect process leads to a better knowledge alignment of employees. At first, with the corruption of a perfect process, its efficiency is decreased, but with better knowledge alignment, process efficiency is consequently increased to a level better than the original one. The optimization model presented in this chapter is based on a modified classic assignment problem and it includes a numerical example based on the data of ETO company. We proved our findings from the aspects of balance, employee capacity load and process efficiency.

• knowledge allocation
• optimization model
• activity‐cutting principle

Author Information

Matjaz roblek *.

• Faculty of Organizational Sciences, University of Maribor, Kranj, Slovenia

Benjamin Urh

*Address all correspondence to: [email protected]

1. Introduction

Global competitiveness requires constant innovations of products and processes, which inherently require changes on the part of production companies. Management of these changes is especially important for those companies for which the production of new products is a regular business, that is, for which every customer requirement is so unique that it requires for the integration of research and development (R&D) department employees to a certain level. Linking of sales, R&D and production in such way is called an ‘Engineering‐to‐Order production strategy’ (ETO). Products in ETO production have a complex structure and a customer‐specified production that is treated as a project. These projects are generally unique and were never previously executed. Therefore, it is impossible that they be handled with existing standard project activities. Problems with the allocation of employees appear in the first activities of the ETO production project, in which activities require a high level of innovation, and the project requires a proper knowledge allocation prior to capacity allocation. Of course, the management needs both allocation views, but the knowledge aspect is more important when dealing with new product or technology changes. The typical question before executing each ETO project is: Do we have appropriate knowledge to do that?

Knowledge is an element of the employees and also an element of the activities of business processes [ 1 ]. In Make‐to‐Stock (MTS), production activities are highly specialized and require a small set of required knowledge. In ETO production, employees execute many activities with a large set of required knowledge. Due to salary requirements, the human‐resource‐required knowledge is linked to the work position definitions [ 2 ]. The management goal is to optimize the required knowledge of work positions and the current knowledge of employees. With every product or process change, the knowledge structure of the work position is changed. If changes are permanent, there will be a continuous searching for new appropriate employees. However, what if the process of change was adjusted so that it took into consideration currently available knowledge? These employees are the only source that is available at the time a new product requires new knowledge in the process. What if the capacity load of each employee’s knowledge and not just the employee’s capacity in general were taken into consideration?

2. Literature review

In literature, this kind of optimization problem is classified as the worker assignment problem [ 3 ]. Applications of this problem are matching employees on work positions, where the required knowledge of work positions is compared to the actual knowledge of known employees [ 4 ]. The optimal solution (objective function) depends on the global minimum of the current knowledge deficit or the global maximum of the current knowledge surplus.

In a real environment, production processes are complicated and diverse. Almost every product and its production technology require modification of its objective function or modification of the entire optimization problem. Even if there is production of the same product in different locations, there will be modification needs, despite work standardization efforts. During process execution (over several years), the optimization problem also changes because of expected and unexpected events, such as production errors, economic opportunities and new arrangements. These events are sometimes very important for optimization. In the case of the presence of a more important and/or urgent business event, their importance for optimization disappears, and their priorities for optimization are changed. Therefore, there are many specific solutions for the worker assignment problem in the literature. Some solutions are case specific while other are made in an attempt to be universally applicable. Depending on the complexity of the worker assignment problem, researchers implement different optimization methods: mathematic programming models (linear, non‐linear, integer), genetic algorithms and heuristics.

The following research has been used as a background for the worker assignment problem in this chapter. From the perspective of tasks, Azizi and Liang [ 5 ] developed an integrated approach to the worker assignment problem. Their dominant assignment problem includes workforce flexibility acquisition and task rotation. They used a constructive‐search heuristic method and set the objective to minimizing the total cost including the incremental cost of new training cost, flexibility cost and productivity loss cost. The learning effect in the worker assignment model was also the subject of research in a project task scheduling problem [ 6 ]. They used a mixed non‐linear integer program, solved by a proposed genetic algorithm. The objective function was to minimize outsourcing costs. From the task perspective, there is optimization model of task allocation and knowledge worker scheduling [ 7 ]. The purpose of this model is to assign knowledge workers to every task and arrange them (the tasks) in order to minimize the total time required to finish all projects. Their optimization is based on the Ant Colony algorithm as an optimization technique [ 8 ]. Nembhard [ 9 ] uses a heuristic approach for assigning workers to tasks that is based on individual learning rates.

There are also worker assignment models originating in production layout and shifts. McDonald et al. [ 10 ] developed a worker assignment model to evaluate a lean manufacturing cell, using a binary integer programming model that is solved using a branch‐and‐bound approach. The objective of this model is to minimize net present costs (initial training costs, incremental training costs, inventory costs and cost of poor quality). Previously, a model of worker assignment considering technical and human skills in cellular manufacturing was developed [ 11 ]. It is classified as mixed‐integer programming problem. The objective of the model is to maximize profit, where profit has three components: productivity, quality costs and training costs. Ingolfsson et al. [ 12 ] combined integer programming and the randomization method to schedule employees by using an integer programming heuristic to generate schedules; they used the randomization method to compute service levels. They described a method to find low cost shift schedules with a time‐varying service level that is always above a specified minimum.

There are worker‐assigning models that deal with the satisfaction of workers. Brusco and Johns [ 13 ] defined a model of staffing a multi‐skilled workforce with varying levels of productivity. They applied integer linear programming model with the objective of minimizing workforce staffing costs subject to the satisfaction of minimum labour requirements across the planning horizon of a single work shift. Mohan [ 14 ] created a model of scheduling part‐time personnel with availability restrictions and preferences to maximize employee satisfaction. He proposed an integer programming model to maximize employee satisfaction (while considering their seniority and availability) and to meet the demand requirements for each shift. A branch‐and‐bound algorithm was used for this.

From the perspective of competencies [ 15 ], there is a competence‐driven staff assignment approach that is based on a stochastic working status model. This model seeks to minimize employee wages and maximize strategic gains of the company from the increment of desirable competencies. The authors used a genetic algorithm as the optimization method. Competencies are also used in a model that seeks to maximize a weighted average of economic gains from projects and strategic gains from the increment of desirable competencies. As a sub‐problem, the scheduling and staff assignment for a candidate set of selected projects is also optimized [ 16 ]. The authors used non‐linear mixed‐integer program formulation for the overall problem and then proposed heuristic solution techniques composed of a greedy heuristic for the scheduling and staff assignment, and alternative ‘meta’ heuristics for the project selection.

Recent studies are showing that the worker assignment problem is still important subject of research. Grosse et al. [ 17 ] designed a framework for integrating human factors into planning models. Crawford et al. [ 18 ] showed application of worker assignment problem in project scheduling and they innovated optimization approach using hyper‐cube framework. A similar problem that discuses assignment of health care staff to tasks using fuzzy evaluation method was presented by Mutingi et al. [ 19 ]. Olivella et al. [ 20 ] gave emphasis on the cross‐training goals, while Senjuti et al. [ 21 ] optimized the assignment of tasks to workers by proposing efficient adaptive algorithms. Current efforts are dealing with additional variables in creating the perfect optimization framework (knowledge, cross‐training, etc.), or in finding the best optimization algorithms for solving worker assignment problem. They still assume that tasks are allocated to workers as ‘they are'. Our effort was to study the effect of task redefinition in the meaning of splitting tasks on smaller parts with the goal of better knowledge alignment. From the organizational view, especially when the creative job must be done (like in ETO companies), the list of required tasks is created according to the available knowledge of workers, and the new definition of tasks is a subject of optimization output. This was our main theoretical issue that is described as real business example as follows:

At first, there is an optimal worker assignment on the work position requirements of ETO company.

Then, one or many workers leave the company at their own initiative. Because of the high level of customer demand, there is no time to re‐educate the existing employees, and management will not approve recruiting new employees.

The quality of process output (product) must remain at the same quality level. It is assumed that the quality can be reached only with proper knowledge.

The quantity of process output may be reduced.

This is a typical example of a company that needs to increase the use of its internal sources. Many cases have been found in practice in ETO companies in which the management solved the problem of outgoing knowledge with reorganization of internal employees rather than with the simple extension of employees’ existing capacities, for example, overtime work [ 22 ]. We also set two assumptions that were not subjects of this research: first, we accepted that in ETO production, business processes are constantly changing and, therefore, knowledge requirements are also changing. Second, because these are simulations, the relation between knowledge and the process efficiency was accepted: if employees have proper knowledge for the execution of activities, then these activities are performed faster. This has an impact on better efficiency of the whole process if that activity is simultaneously a process bottleneck [ 23 ].

The key solution of adjusting processes to the current knowledge lies in the theory of business process management [ 24 ], in which the main problem of achieving a short process throughput time lies in the waiting times among different work positions that are the consequence of unbalanced work. This problem is insignificant if the entire process is executed by only one employee who occupies one work position, because there are no work position breaks [ 25 ]. This works only in small companies. Large business systems are complicated: they have many business processes with diverse knowledge requirements (e.g. ETO production) and require many employees with different types and levels of knowledge. Work is divided into activities between different work positions. Each work position has its own knowledge requirements. In this case, management needs control over the specific knowledge and over the number of the work position changes, and must keep them at the ‘desired’ minimum level so that the optimal process efficiency and the work balance are reached. The problem is also in the required and actual capacity of the specific knowledge. The process output quantity reflects the frequency of activity executions [ 26 ]. From a previous description of the principle of minimization work position breaks, when the capacity of one employee is exceeded, an additional employee who can perform all activities in the process is required. Such a broadly educated employee is too expensive, and this solution is thus irrational. Therefore, the process is divided into activities (tasks) among many work positions with the least expensive employees. Management creates work positions with a simple and complex knowledge structure. However, dividing work in too many work positions slows down the process: the throughput time is extended because of the additional waiting time each time the work position is switched.

Regarding the theory of work position breaks, work position knowledge structure and employee knowledge capacity, we modified our previously published model [ 22 ]. Figure 1 shows the steps of upgraded conceptual model. In the new model, we are measuring the effect of the partial corruption of a perfect process regarding better current knowledge alignment from the perspective of employee capacity load and from that of process efficiency; with corruption of the process, we are decreasing its efficiency due to new additional work position breaks, but with better knowledge alignment we are again increasing the process efficiency.

Figure 1.

Knowledge‐based assignment conceptual model.

3.1. Measuring optimal knowledge alignment

We can observe in practice that if the current knowledge deficit is below the required knowledge, the result is less efficient work. Surprisingly, even an excess of actual knowledge over the required level of knowledge has the same result of over‐educated and intelligent employees becoming bored when they are executing routine activities [ 22 ]. Therefore, we modified a classic assignment linear integer problem of Kolman and Beck [ 3 ]. In the original optimization model ( Eq. (1) ), the value c ij represents the added value if employee i is allocated to work position j and the optimization function maximizes a profit.

We replaced the added value with the minimal knowledge deficit/surplus (absolute) gap of n key required knowledge K k . That means if we allocate an employee with his/her actual knowledge that is nearest to required knowledge on the work position (neither below nor above) then we have attained optimal knowledge alignment. The idea is to minimize the overall absolute key knowledge gap in the processes of the specific company ( Eq. (2) ).

where i… n = number of compared employees; j… n = number of different work positions; k… n = number of compared key knowledge; and |K k | = absolute difference between required and actual knowledge K .

In case of a new required ETO production change, this model can be used in the following situations:

If there is an ‘open’ set of available employees, all potential candidates in the optimization function can be matched. If the candidate knowledge gap is excessive (the appropriate level was not a subject of this research) the candidate is inappropriate for the work position because the performed work will be less efficient. This action has certain inherent costs (hiring, firing).

If there is time to provide additional education to employees, then the knowledge deficit can be decreased with additional knowledge. This action has additional education and training costs.

Existing employees can also be re‐assigned on existing work positions so that the company knowledge alignment is optimal.

Are these all the possible management actions?

3.2. Measuring the corruption of a perfect process

As an innovation, the effect of a partial corruption of a perfect process was tested, including its impact on a better knowledge alignment with the limitation that the set of employees must remain untouched. The hypothesis was that with a corruption of the process, a better knowledge alignment can be achieved and, consequently, the process efficiency can be increased, despite a simultaneous decrease of its efficiency due to new additional work position breaks. Moreover, there must be a point in the process corruption procedure after which the inefficiency of the process exceeds the benefits of better knowledge alignment.

The effect of work position breaks in the process is measured by structural index K wpb ( Eq. (3) ) [ 27 ]. This is a common key performance indicator in the theory of analysing business processes.

C wp counts all work position breaks in a specific process. P a counts all activities in that process. In this theory, the process slightly stops each time the next process activity is performed by different employee (on a different work position). This is one of practical causes for additional waiting time in the structure of throughput time of the process. There can be up to n − 1 work position breaks in a process of n sequential activities. According to the total number of all process activities, a small number of work position breaks means that the process is more efficient.

In practice, poor work quality can be found in the process due to inappropriate knowledge alignment. This generates additional feedback loops, activities are repeated and the result is additional work position breaks. Determining the causes of additional activity breaks is not a subject of this research.

3.3. Linking knowledge optimization and work position breaks

From the perspective of real business in ETO production, especially in this time of global economic crisis, accessibility to newly required knowledge is greatly limited due to extra educational costs. Downsizing also means that processes must be executed with fewer employees but at the same time the level of product quality must remain equal to previous process executions. Management typically reacts with reorganization of employees on activities. Furthermore, because we cannot split ‘the human body', his or her structure of knowledge and the time capacity of that knowledge cannot be optimal for current (ideal) process. In the theory, the problem can be easily solved if we have all current employees with all required knowledge of the process.

In ETO production, there are many specialists (e.g. electrical engineers, mechanical engineers, software engineers) with one or two dominant fields of knowledge of very high quality or strength, and few employees with wide spectra of high quality knowledge (senior engineers, mechatronics), because the latter are too expensive. However, they are also key employees for the ETO production; they have the big picture over each new product, and they can control the efficiency and quality of the overall production process. They are never ‘bottlenecks’ in the process with regard to knowledge, but they can be problematic with regard to the available time capacity of his/her specific required knowledge, because they are involved in many processes (ETO projects).

This phenomenon is also a result of the accumulation of many small organizational changes in processes over time. When the company was established (or after process re‐engineering project), processes and work positions were optimally designed for execution, employees were carefully selected and their knowledge was appropriate for knowledge requirements of work positions ( Figure 2 ).

Figure 2.

Explanation of cutting activities when employee leaves the process.

Over time, new activities were slowly added to work positions, thus generating newly required knowledge. These changes were so small at the beginning that the management did not recognize them as knowledge problems or capacity problems. They had no effect on the employees except that the work position received one or two new key pieces of knowledge that employees had to obtain. After a few years of small changes, the work position and their key knowledge structure had expanded in such a way that the management and the employee did not know which pieces of knowledge of the work position were key for business success (e.g. a designer in ETO production is working 30% of his capacity on designing, 40% of the time he is occupied with routine paper work and another 30% he is attending meetings; if we require 100% design work, then this person’s design knowledge is a capacity bottleneck).

For such cases, we created a process and knowledge algorithm that is connected with a Key performance indicators (KPI) that measures process corruption as follows:

We must have input data of current processes (As‐Is), their activities and times, current work positions, required knowledge, current employees and their actual knowledge.

Then, we test the impact of employee reduction on the knowledge structure of process. We can start with required knowledge that is recognized as a process bottleneck or with knowledge that is missing at the new activity executor.

In first case, we reduce the process activity until only work with knowledge that was bottlenecked remains (i.e. knowledge that is available by only one employee). The removed parts of activity with removed knowledge are distributed among other employees in the process until the optimal knowledge alignment is reached ( Eq. (2) ). If some knowledge is insufficient with one employee, the part of activity requiring this knowledge is given to an employee who can cover it successfully. Then, we repeat this procedure until optimal process knowledge alignment is reached.

At the same time, we measure the impact of the activity‐cutting principle on the process ( Eq. (3) ). Because the better knowledge alignment improves the process efficiency, and the activity‐cutting principle reduces the process efficiency, the algorithm serves as a ‘trading’ point when we are balancing and allocating employee knowledge on activities within his/her available time capacity ( Figure 3 )

The final result (output) is a new process (To‐Be) that is feasible.

Figure 3.

Possible outputs of algorithm for optimal knowledge alignment in ETO production.

Such a reorganized process is reengineered on the basis of knowledge.

4. Input data

4.1. processes, process activities, work positions and required knowledge.

In ETO production, at first sight, almost every product has its own and unique production process (routing). The fact is that activities (operations) among different processes are almost the same with regard to required knowledge. They differ mostly in the time required for execution. Because each product has its unique structure (bill of material), the process is named in practice as a project and its operations are named as activities. However, from the top‐down approach, each project in ETO production has almost the same set and the same sequence of project phases (with many sub‐activities), for example, (1) preparation, (2) design, (3) construction, and (4) testing. Therefore, it can be assumed that we have a standard form of the process (with activities) for almost all new products.

The same process activity could appear in a structure of many different processes and it is usually performed by the same work position (e.g. the same quality control activity with the same control parameters and tools for the whole product group). Moreover, one work position executes many activities. Until the system is well organized, a work position aggregates activities with approximately the same required set of knowledge. We defined that the required knowledge of a specific work position is represented as a set of knowledge from all executed activities. The sets of required knowledge of specific activity and their strength (Likert scale from 1 to 5; 5 meaning very important) are defined by the company’s internal and external experts. If a specific piece of knowledge is required for the execution of many activities, the model uses its maximal value as a required strength.

Complex work positions have a wide range of required knowledge, many of unimportant strength. Reducing the amount of various required knowledge can simplify the calculations. Simplification was achieved with the definition of key knowledge K k for each work position. If the strength of specific knowledge is above a specific level, it is treated as key knowledge of that work position.

In practice, the above‐described idea of capturing process activities and their required knowledge can be used for documenting As‐Is processes and, more importantly, for predicting future products, To‐Be processes and their expected required knowledge. This is of great importance for planning required knowledge of future ETO production. We can analyse the following:

Which activity among all activities of specific process is the most important from the key knowledge aspect, for example, to find the activity that is the ‘knowledge bottleneck’ in a process. Then we can combine this information with activity throughput rate and find an activity that is the real‐time capacity bottleneck in the process.

Which process (from among all of them) is the most important from the aspect of key knowledge, for example, for ranking all processes on the basis of the knowledge required (i.e. which process is currently the most important/crucial for the company from the knowledge view; this is important information for any ETO company in addition to the information regarding which process is crucial from capacity aspect).

In ETO production, each work position typically executes many different activities in many different processes. Therefore, we are interested which work position has the highest required strength of all key knowledge, for example, we can use this information as a basis for creating salary grades.

Which work positions in the company are exceptional from the knowledge aspect; a work position that has only one key type of knowledge but with a high required strength (e.g. CNC programmer) and which work positions are universal, that is, have many key types of required knowledge (e.g. ETO project manager).

Which type of knowledge is dominant (repeats at every executed activity) for the specific process (short‐term view) and for the whole company (long‐term view).

If we have proper data on all the above mentioned entities (processes, activities, work positions, knowledge requirements with required strength) for the present time, and if we have good knowledge requirements (definitions) of new products (especially required technology and activities), we can then simulate all future knowledge requirements in advance. Therefore, we can determine differences, for example, which work position must be knowledge‐reconstructed in the future; consequently, we can define projected mandatory changes in a structure of actual knowledge (employees).

4.2. Employees, actual knowledge and knowledge gap

Employees represent the basis for gathering current knowledge. There are many approaches to prove that an employee possesses specific knowledge and what the quality of it is (strength, level). In our approach, the 360° feedback method [ 28 ] was used. We used a list of all key required knowledge and assessed all employees (Likert scale from 0 to 5; 0 means knowledge not available). We gave employees the opportunity to extend this explicit knowledge with their tacit knowledge. In the context of our model, the term ‘tacit’ means the knowledge of an employee that is currently unknown to the company. Knowing about tacit knowledge is essential information when new processes have requirements for new types of knowledge. In practice, for optimization, it is also recommended that we have the knowledge data about potential candidates for employees.

The last step of input data preparation is a calculation of the key knowledge gap: each employee is compared to all work positions. We used the criterion c ij , explained in Eq. (2) . Any deviation of actual knowledge over and below the required knowledge is considered to be inappropriate and will lower process efficiency ( Table 1 ).

Table 1.

Matching required and actual knowledge.

Table 1 shows a numerical example of matching the actual knowledge from k 1 to k 10 of employee E 1 on activities from a 1 to a 7 of work position W 1 (e.g. Product Manager of ETO project). The example is based on the real data of ETO company, Iskratel. Negative values (grey cells) represent deficits of employee knowledge strength compared to the required knowledge of a work position. The top rows represent activities of the work position with a sum of negative values. We can identify activities that the employee is not suitable to execute (e.g. a 1 , a 2 , a 3). The left column represents the required knowledge with the sum of negative values. We can identify the lack of employee knowledge (e.g. k 4 , k 8).

In practice, we could integrate in our model the effect of learning and forgetting knowledge over time (decreasing knowledge strength if employee is not using that type of knowledge in processes for a long time). Because of model simplicity, this was not a subject of this research.

We demonstrated the capabilities of our model on a small section of the real process that was described in Figure 3 . This numerical example is based on the data of company Iskratel. We performed simulations of this example with the same tools as the calculations of real cases ( Tables 2 and 3 ). Definitions of processes were recorded in the repository of Aris Toolset software [ 29 ]. Definitions of actual and required knowledge were recorded with MS Share Point and MS SQL. All data were then exported to the MS Excel analytical tool and solved with the WhatsBest [ 30 ] add‐on. MS Excel was also used as reporting tool.

Table 2.

Input data of simulation scenarios.

Table 3.

Simulation results.

5.1. Input data of simulation scenarios

We prepared four simulation scenarios as follows:

Scenario 0: As‐Is situation. In the current state, there are three employees assigned to their own work positions, and the processing of four activities with four different types of knowledge.

Scenario 1: employee on work position w 2 left the company. His/her activity a 2 is assigned to w 1 and a 3 to w 3 . This is typical management decision that does not generate an additional work position switch in the sequence of activities.

Scenario 2: use of our algorithm: achieving better knowledge alignment. Employee on w 3 has no knowledge K 3 that is required for execution of activity a 3 ; therefore, we split activity on a ′ 3 and a ″ 3 .

Scenario 3: is same as scenario 2, with one additional activity cut: we are searching for better balance of capacities between w 1 and w 3 . We split activity a 2 and we add knowledge K 2 to work position w 3 .

We can observe the things as follows:

In scenario 1, the result of management action on knowledge distribution among work positions: Knowledge K 1 and K 2 are moved from w 2 to w 1 . Knowledge K 3 and K 4 are moved from w 2 to w 3 . In case this is the same knowledge, we used the maximal strength as the required strength.

In scenario 2, the result of optimization algorithm: according to As‐Is situation, we moved from w 2 to w 1 knowledge K 1 and K 3 . This caused the rise of the strength of both types of knowledge for w 1 . We moved from w 2 to w 3 only knowledge K 4 , because the newly required strength is below the current required strength so it remains as it was for w 3 .

In scenario 3, the new activity cut did not cause any change in knowledge requirements (and strength) of w 1 and w 3 according to scenario 2.

5.2. Simulation results

We can see in Scenario 2 (implementing activity‐cutting principle) that we decreased the knowledge gap in Scenario 1. Now, we must ‘merge’ the results of optimal knowledge alignment to determine the impact of using the activity‐cutting principle on classic production optimization parameters (Scenario 3). Otherwise, we will break some lean manufacturing principles, for example, work balancing or eliminating waiting times. We added additional input data of As‐Is process in Table 4 .

The first assumption (i) in our evaluation is the amount of time that is added to process throughput time each time we change the work position (sending work from me to you etc.). In a real case, this could be measured exactly but in our demonstration we assumed a fixed value of 3 min.

The second assumption (ii) in our evaluation is the amount of time that is added to process throughput time because of non‐optimal knowledge alignment. In the As‐Is process, we know that we have 0.8 by the Likert non‐optimal knowledge alignment. If the times in this table were measured without being aware of this knowledge gap then the real throughput time is longer. In a real case, we could measure this by comparing the knowledge gap and the difference between planned and real production times (we have to exclude other causes for time extension first). In our demonstration, we assumed that every 0.1 of knowledge gap adds 1% to planned process throughput time.

6. Discussion

The main specialty of our model is that we permit changes of the process because the actual knowledge is not appropriate for it. However, we do not allow changes in the sequence of activities; we allow only changes in the sequence of using employees. The results are new partial activities in the process; consequently, the process workflow is jumping forwards and backwards between employees.

In our model, we removed all unnecessary knowledge from the work positions that were process ‘bottlenecks’ and replaced it with the new process structure; this was done by taking into consideration the availability of the actual knowledge of employees. The entire individual employee time capacity is now focused only on the utilization of knowledge that is bottlenecked. Other required knowledge in the process that is also present in other employees is removed from that work position. Employee capacity is now free of all non‐bottleneck knowledge, and this raises its capacity availability.

In our simulations, we used process time indicators to verify our assumption, even if we know, on the basis of real projects [ 31 , 32 ], that the best improvements in the ETO production are achieved on the process quality indicators. Time indicators are improved indirectly as a result of better product quality: fewer aftermarket repairs means less additional invested time in the total production time of the specific product. The starting point of all scenarios is the departure of one employee from the original process (Scenario 0). In Scenario 1, we reacted by implementing the lean manufacturing principle of capacity balancing: the work of the lost employee is divided among remaining employees on the basis of capacity levelling without additional work position breaks. This is a common management decision, and it is expressed as a load capacity per shift (%) indicator in Table 4 . This decision produced the knowledge gap of 1.7 ( Table 3 ).

In Scenario 2, we used our model with the activity‐cutting principle, and we reduced the knowledge gap by 0.4 or 23.5% ( Table 3 ). Most time indicators were also improved ( Table 5 ), except for the unbalanced load capacity per shift (%) indicator, and a lower process throughput rate (from 9 to 8 products per shift). Both indicators would have negative impact in mass or serial production, but according to the requirements of the ETO production it is more important that we achieved the desired quality of knowledge for production process because there are no repetitions (rather only unique, one‐time process executions). Management can balance these indicators and make the decision that is adopted for a specific process ‘case'.

Table 4.

Production parameters of As‐Is process.

Table 5.

The impact of activity‐cutting principle on production parameters in scenarios from 1 to 3.

In Scenario 3, we tested the total ignorance of the Lean Manufacturing principles, and we performed additional activity cuts for searching for even better knowledge alignment. We did not achieve a lower knowledge gap ( Table 3 ); we also worsened all time indicators according to Scenario 2 ( Table 5 ). This indicated that there is a point in the repetition of activity‐cutting procedure after which the process becomes so inefficient that is better to hire a new employee if the knowledge gap is still too high for achieving the appropriate quality of ETO products. Where that point is, what the gap should be and whether its value is of universal use or case sensitive are all subjects of future research.

7. Conclusions

In Make‐to‐Stock, Assemble‐to‐Order and Make‐to‐Order production, assignment models for the allocation of employees assume that tasks of production processes (or routings) are of a fixed structure. Managers believe they found the most ‘efficient’ process of producing products and, therefore, all current optimization models are searching for appropriate employees for that process. Small deviations between the required and the actual knowledge are resolved with alternative routing; its structure is also known and fixed in advance. All of this is possible because extra time is invested for testing and preparing optimal processes for many repetitions. Extra time is also invested for finding employees with proper knowledge for that processes. This is the case of known theoretical and practical solutions of worker assignment problem.

However, in ETO production, and consequently in all knowledge‐intensive processes or case‐like processes, we determined that processes are structured around the available knowledge of employees. Otherwise, the cost of searching for missing knowledge in the form of a new employee could exceed all the added value to the business. Process ‘cases’ are never the same and each process ‘repetition’ requires a process structure that is adapted to the actual knowledge and its capacity in the company; the bottleneck is not the capacity of the employee but the capacity of his/her specific actual knowledge. With the activity‐cutting principle in our assignment model, we proved that we can release the ‘hidden’ time capacity of employee who is the bottleneck so that we could remove all activities and consequently the knowledge that is also available with other employees from the work position. We recommend that this principle can be an option of all assignment models for the allocation of employees for ETO production and all other knowledge‐intense companies. This is our main contribution to the theory of modelling worker assignment problem.

Of course, this research raises additional questions for our future work, especially in the field of practical application: is knowledge the right category in our assignment model or is it better to use all measureable work habits and personal skills [ 33 ]? There are also assumptions in Table 4 that will need additional research and explanation. Nevertheless, our concept of redefining tasks with the goal of reaching optimal worker knowledge alignment could be used as a ‘smart’ reorganization principle for dynamic and real‐time redefinition of processes in companies, where the standardization of tasks is not the main factor of reaching efficiency.

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