importance of assignment problem in operational research

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:

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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:

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

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Assignment Problem

5.1  introduction.

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.

5.2  GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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• Operations Research Problems

Statements and Solutions

• © 2014
• Raúl Poler 0 ,
• Josefa Mula 1 ,
• Manuel Díaz-Madroñero 2

Research Centre on Production Management and Engineering, Polytechnic University of Valencia, Alcoy, Spain

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Escuela Politécnica Superior de Alcoy, Universidad Politécnica de Valencia, Alcoy, Spain

Universitat politècnica de valència, alcoy, spain.

• Provides a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science
• Identifies different operations management problems in order to improve the decision making process concerning readers
• Addresses the following topics: Linear programming, integer programming, non-linear programming, network modeling, inventory theory, queue theory, tree decision, game theory, dynamic programming and markov processes

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Table of contents (10 chapters)

Front matter, linear programming.

• Raúl Poler, Josefa Mula, Manuel Díaz-Madroñero
• Integer Programming

Non-Linear Programming

Network modelling.

• Inventory Theory

Queueing Theory

Decision theory, games theory.

• Dynamic Programming
• Markov Processes

Back Matter

• Game Theory
• Linear and Non-Linear Programming
• Network Modeling
• Queue Theory

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Authors and Affiliations

Josefa Mula

Manuel Díaz-Madroñero

Bibliographic Information

Book Title : Operations Research Problems

Book Subtitle : Statements and Solutions

Authors : Raúl Poler, Josefa Mula, Manuel Díaz-Madroñero

DOI : https://doi.org/10.1007/978-1-4471-5577-5

Publisher : Springer London

eBook Packages : Engineering , Engineering (R0)

Copyright Information : Springer-Verlag London Ltd., part of Springer Nature 2014

Hardcover ISBN : 978-1-4471-5576-8 Published: 22 November 2013

Softcover ISBN : 978-1-4471-7190-4 Published: 23 August 2016

eBook ISBN : 978-1-4471-5577-5 Published: 08 November 2013

Edition Number : 1

Number of Pages : XV, 424

Number of Illustrations : 32 b/w illustrations, 55 illustrations in colour

Topics : Industrial and Production Engineering , Operations Research/Decision Theory , Game Theory, Economics, Social and Behav. Sciences

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Operational research as implementation science: definitions, challenges and research priorities

• Thomas Monks 1  

Implementation Science volume  11 , Article number:  81 ( 2015 ) Cite this article

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Operational research (OR) is the discipline of using models, either quantitative or qualitative, to aid decision-making in complex implementation problems. The methods of OR have been used in healthcare since the 1950s in diverse areas such as emergency medicine and the interface between acute and community care; hospital performance; scheduling and management of patient home visits; scheduling of patient appointments; and many other complex implementation problems of an operational or logistical nature.

To date, there has been limited debate about the role that operational research should take within implementation science. I detail three such roles for OR all grounded in upfront system thinking: structuring implementation problems, prospective evaluation of improvement interventions, and strategic reconfiguration. Case studies from mental health, emergency medicine, and stroke care are used to illustrate each role. I then describe the challenges for applied OR within implementation science at the organisational, interventional, and disciplinary levels. Two key challenges include the difficulty faced in achieving a position of mutual understanding between implementation scientists and research users and a stark lack of evaluation of OR interventions. To address these challenges, I propose a research agenda to evaluate applied OR through the lens of implementation science, the liberation of OR from the specialist research and consultancy environment, and co-design of models with service users.

Operational research is a mature discipline that has developed a significant volume of methodology to improve health services. OR offers implementation scientists the opportunity to do more upfront system thinking before committing resources or taking risks. OR has three roles within implementation science: structuring an implementation problem, prospective evaluation of implementation problems, and a tool for strategic reconfiguration of health services. Challenges facing OR as implementation science include limited evidence and evaluation of impact, limited service user involvement, a lack of managerial awareness, effective communication between research users and OR modellers, and availability of healthcare data. To progress the science, a focus is needed in three key areas: evaluation of OR interventions, embedding the knowledge of OR in health services, and educating OR modellers about the aims and benefits of service user involvement.

Peer Review reports

Operational research (OR) is the discipline of using models, either quantitative or qualitative, to aid decision-making in complex problems [ 1 ]. The practice of applied healthcare OR distinguishes itself from other model-based disciplines such as health economics as it is action research based where operational researchers participate collaboratively with those that work in or use the system to define, develop, and find ways to sustain solutions to live implementation problems [ 2 ]. The methods of OR have been used in healthcare since the 1950s [ 3 ] to analyse implementation problems in diverse areas such as emergency departments [ 4 – 6 ] and management policies for ambulance fleet [ 7 ]; acute stroke care [ 8 – 11 ], outpatient clinic waiting times [ 12 ], and locations [ 13 ]; cardiac surgery capacity planning [ 14 ]; the interface between acute and community care [ 15 ]; hospital performance [ 16 ]; scheduling and routing of nurse visits [ 17 ]; scheduling of patient appointments [ 18 ]; and many other complex implementation problems of an operational or logistical nature.

Implementation science is the study of methods to increase the uptake of research findings in healthcare [ 19 ]. Given the volume of OR research in healthcare implementation problems, it is remarkable that limited discussion of the discipline has occurred within the implementation science literature. A rare example of debate is given by Atkinson and colleagues [ 20 ] who introduce the notion of system science approaches for use in public health policy decisions. Their argument focused on two modelling methods, system dynamics and agent-based simulation, and the potential benefits they bring for disinvestment decisions in public health. To complement and extend this debate, I define the overlap between implementation science and OR. I have focused on the upfront role that OR takes when used as an implementation science tool. Although some detail of method is given, the full breath of OR is beyond the scope of this article; a detailed overview of all the methods can be found elsewhere [ 21 ]. I describe three roles for OR within implementation science: structuring an implementation problem, prospective evaluation of an intervention, and strategic reconfiguration of services. For each role, I provide a case study to illustrate the concepts described. I then describe the challenges for OR within implementation science at the organisational, interventional, and disciplinary levels. Given these challenges, I derive a research agenda for implementation science and OR.

OR to structure an implementation problem

The first role for OR in implementation science is to provide a mechanism for structuring an implementation problem. Within OR, problem structuring methods provide participatory modelling approaches to support stakeholders in addressing problems of high complexity and uncertainty [ 22 ]. These complex situations are often poorly defined and contain multiple actors with multiple perspectives and conflicting interests [ 23 ]. As such, they are unsuitable for quantitative approaches. Problem structuring methods aim to develop models that enable stakeholders to reach a shared understanding of their problem situation and commit to action(s) that resolve it [ 23 ]. Approaches might serve as a way to clearly define objectives for a quantitative modelling study [ 24 ], systematically identify the areas to intervene within a system [ 25 ], or may be an intervention to improve a system in its own right.

A case example—understanding patient flow in the mental health system

A mental health service provider in the UK provided treatment to patients via several specialist workforces. Here, I focus on two: psychology and psychiatric talking therapies (PPT) and recovering independent life (RIL) teams. Waiting times to begin treatment under these services were high (e.g. for RIL team median = 55 days, inter-quartile range = 40–95 days), and treatment could last many years once it had begun. The trust’s management team were eager to implement new procedures to help staff manage case load and hence reduce waiting times to prevent service users, here defined as patients, their families, and carers, from entering a crisis state due to diminishing health without treatment. Management believed that reasons for delays were more complex than lack of staff, but the exact details were unclear and there was much disagreement between the senior management. The implementation science intervention I detail was conducted as an OR problem structuring exercise.

A system dynamics (SD) model was constructed to aid management target their interventions. SD is a subset of system thinking—the process of understanding how things within a system influence one another within the whole. SD models can be either qualitative or quantitative. In this case, a purely qualitative model was created. Figure  1 illustrates stock and flow notation that is commonly used in SD. The example is the concept of a simple waiting list for a (generic) treatment. It can be explained as follows. General practitioners (GPs) refer service users to a waiting list at an average daily rate, while specialist clinicians treat according to how much daily treatment capacity they have. The variable waiting list is represented as a rectangular stock: an accumulation of patients. The waiting list stock is either depleted or fed by rate variables, referring and treating, represented as flows (pipes with valves) entering and leaving the stock. Figure  1 also contains two feedback loops that are illustrated by the curved lines. The first loop is related to the GP reluctance to refer to a service with a long waiting time. As the waiting list for a service increases in number, so does the average waiting time of service users and so does the pressure for GPs to consider an alternative service (lowering the daily referral rate). The second loop is related to specialist clinicians reacting to long waiting lists by creating a small amount of additional treatment capacity and increasing admission rates.

Example system thinking for a waiting list—stock and flow notation. Notation guide. Rectangles represent stocks which are acculations of quantity of interest; Pipes with valves represent flows which feed or deplete stocks; arrows represent how one aspect of a system positively or negatively influences another

A preliminary version of the SD model was created using a series of interviews with clinicians and managers from the three services. This was followed by a group model building workshop that involved all senior management. Group model building is a structured process that aims to create a shared mental model of a problem [ 26 ]. The workshop began with a nominal group exercise. The group were asked to individually write down what they believed were the key factors that affected patient waiting times. The group were specifically asked to focus on strategic issues as opposed to detailed process-based problems. After all individual results had been shared, the group were asked to (i) hypothesise how these factors influenced each other and (ii) propose any missing variables that may mediate influence. For example, available treatment capacity is reduced by non-clinical workload. Non-clinical workload is increased by several other factors (discussed below in results) and so on.

Figure  2 illustrates one of the qualitative SD models developed in collaboration with the mental health trust. It uses the same stock and flow notation illustrated in Fig.  1 . The model shown is focussed on the RIL teams. Several insights were gained in its construction. First, it was clear to all parties that that this was not a simple demand and treatment capacity problem. For example, a great deal of non-core work takes place due to monitoring of ‘discharged’ service users within social care. The fraction of service users who undergo monitoring is determined by the degree of trust between clinicians and social care teams. When trust is low, the fraction of service users monitored increases and vice versa. A similar soft issue can be found in the discharge of complex patients, i.e. those that require a combination of medication, management by GPs in the community, and social care input. In this case, there is a delay while GPs build confidence that it is appropriate for a patient to be discharged into their care. While this negotiation takes place, a patient still requires regular monitoring by a mental health clinician. Other systemic issues are also visible. For example, the long delays in beginning treatment lead to clinicians spending time contacting patients by phone before they were admitted. This all takes time and reinforces the delay cycle.

A simplified version of the RIL team patient flow model

The results of the modelling were used to inform where interventions could be targeted. For example, a more detailed qualitative SD study to identify the trust issues between clinicians, social services, and general practitioners.

OR as a tool for prospective evaluation

The second role of OR within implementation science is as a prospective evaluation tool. That is, to provide a formal assessment and appraisal of competing implementation options or choices before any actual implementation effort, commitment of resources or disinvestment takes place. Informally, this approach is often called what-if analysis [ 21 ]. A mathematical or computational model of a healthcare system is developed that predicts one or more measures of performance, for example, service waiting times, patients successfully treated, avoided mortality, or operating costs. The model can be set up to test and compare complex interventions to the status-quo. For example, decision makers may wish to compare the number of delayed transfers of care in a rehabilitation pathway before and after investment in services to prevent hospital admissions and disinvestment in rehabilitation in-patient beds. The approach has been applied widely in the areas outlined in the introduction to this article.

A case example—emergency medicine capacity planning

As a simple case example of prospective evaluation, consider the emergency department (ED) overcrowding problems faced by the United Kingdom’s (UK) National Health Service (NHS). The performance of NHS EDs is (very publically) monitored by recording the proportion of patients who can be seen and discharged from an ED within 4 hours of their arrival. The UK government has set a target that 95 % of service users must be processed in this time. In recent years, many NHS EDs have not achieved this benchmark. The reasons for this are complex and are not confined to the department [ 27 ] or even the hospital [ 15 ]. However, given the high public interest, many EDs are attempting to manage the demands placed on them by implementing initiatives to reduce waiting times and optimise their own processes.

Our case study took place at a large ‘underperforming’ hospital in the UK. The management team were divided in their view about how to reduce waiting times. One option was to implement a clinical decision-making unit (CDU). A CDU is a ward linked to the ED that provides more time for ED clinicians to make decisions about service users with complex needs. However, at times of high pressure, a CDU can also serve as buffer capacity between the ED and the main hospital. That is, a CDU provides space for service users at risk of breaching the 4-h target once admitted; service users are no longer at risk of breach. The question at hand was if a CDU were implemented, how many beds are required in order for the ED to achieve the 95 % benchmark?

Figure  3 illustrates the logic of a computer simulation model that was developed to evaluate the implementation of a CDU on ED waiting times. A computer simulation model is a simplified dynamic representation of the real system that in most cases is accompanied by an animation to help understanding. In this case, the simulation mimicked the flow of patients into an ED, their assessment, and treatment by clinicians and then flow out to different parts of the hospital or to leave the hospital entirely. The scope of the modelling included the hospital’s Acute Medical Unit (AMU) that admits medical patients from the ED. In Fig.  3 , the rectangular boxes represent processes, for example, assessment and treatment in the ED. The partitioned rectangles represent queues, for example, patient waiting for admission to the AMU. The model was set up to only admit patients to the CDU who had been in ED longer than 3.5 h and only then if there was a free bed. Once a patient’s CDU stay was complete, they would continue on their hospital journey as normal, i.e. discharged home, admitted to the AMU or admitted to another in-patient ward.

Emergency department and clinical decision-making unit model. Notation guide. Rectangles represent processes; partitioned rectangles represent queues; ellipses represent start and end points; arrows represent the direction of patient flow

In the model, the various departments and wards are conceptualised as stochastic queuing systems subject to constraints. This means that the variability we see in service user arrival and treatment rates (e.g. sudden bursts in arrivals combined with more complex and hence slower treatments) combined with limited cubicle and bed numbers results in queues. There are three reasons why prospective evaluation is appropriate for these systems. First, capacity planning for such complex systems based on average occupancy fails to take queuing into account and will substantially underestimate capacity requirements [ 28 ]. Second, the processing time, i.e. the time taken to transfer a patient to a ward and then to make a clinical decision, within a CDU is uncertain, although it is likely to be slower than the high pressure environment of the ED. Third, as the same ED and AMU clinicians must staff the CDU, the (negative or positive) impact on their respective processing times is uncertain.

The model developed was a discrete-event simulation [ 29 ] that mimics the variation in service user arrival and treatment rates in order to predict waiting times. The uncertainty in CDU processing time was treated as an unknown and varied in a sensitivity analysis . The limits of this analysis were chosen as 2 and 7 h on average, as these were observed in similar wards elsewhere.

The model predicted that the number of CDU beds would need to be between 30 and 70 in order to achieve the ED target (for reference, the ED had 10 cubicles for minor cases and 18 cubicles for major cases). This result illustrated that even if a decision was made in 2 h on average with no negative effect on ED or AMU processing time, the CDU would need to be at least the same size as the ED overall. It also highlighted that the CDU impact on ED performance was highly sensitive to processing time.

The benefit of evaluating the CDU implementation upfront was that it ruled the CDU out as a feasible intervention before any substantial resource had been mobilised to implement it. The hospital could not safely staff a 30-bedded CDU or indeed provide space for that size of ward. As such, the modelling helped the management team abandon their CDU plan and consider alternative solutions with minimal cost and no disruption to the service.

OR as a tool for strategic reconfiguration

The previous section described an implementation science approach to evaluate a small number of competing options at an operational level. In some instances, particularly in healthcare logistics and estate planning, a more strategic view of a system is needed to shortlist or choose options for reconfiguration. In such implementation problems, there may be a large number of options reaching into the hundreds, if not hundreds of thousands of competing alternatives. To analyse these problems, mathematical and computational optimization techniques are required. For example, if a provider of sexual health services wanted to consolidate community clinics from 50 to 20 and there are 100 candidate locations, then there are in the order of 10 20 configurations to consider. OR’s implementation science role is to provide tools that identify options that help meet a strategic objective. For example, this might be maintaining equitable patient access to services across different demographics groups or modes of transportation while increasing service quality and reducing cost.

A case example—where should TIA outpatient clinics be located?

As a simple exemplar, consider a rural region in the UK that provided a 7-day transient ischemic attack (TIA) service through outpatient clinics in the community. Clinics ran at five locations but with only one location open per day. Magnetic resonance imaging (MRI) was available at three locations. Service users attending clinics without imaging but who require access to an MRI make an additional journey to the closest location with imaging capacity.

Service users are booked into clinic appointments across the week as they are referred to the TIA service by their diagnosing clinician, typically the patients local GP or an attending emergency department physician. The diagnosing clinician risk stratifies service users as high or low risk of a major stroke. High-risk service users require to be seen within 24 h of symptom onset and low-risk patients within 7 days [ 30 ].

The healthcare providers had concerns that splitting the clinics across five sites increased the variation in care received by service users and wished to consolidate to one to three clinic locations. Hence, there were two complicating factors when assessing equitable access: how many locations and which ones. There were also concerns that one location—clinic X—on the coast of the region was extremely difficult for high-risk TIAs to reach on the same day as diagnosis. There would also be political implications for any closure at clinic X. In total, there were 25 combinations of clinics for the providers to consider for both the low- and high-risk TIA groups, i.e. 50 options to review.

A discrete-choice facility location model was developed to evaluate the consequences of different TIA clinic configurations and inform the decision-making process for the reconfiguration of the service. Location analysis is a specialised branch of combinatorial optimisation and involves solving for the optimal placement of a set of facilities in a region in order to minimise or maximise a measure of performance such transportation costs, travel time, or population coverage [ 31 ]. In this, case an analysis was conducted separately for high-risk and low-risk TIAs. The analysis of high-risk TIAs aimed to minimise the maximum travel time of a service user from their home location to the closest clinic (as these service users must be seen the same day). The low-risk analysis minimised the weighted average travel time to their closest clinic. The weighted average measure allows for locations with the highest level of demand to have the greatest impact on results, diminishing the impact of outlying points. In general, if there are n demand locations and on a given day the travel time x from locations i to the nearest clinic, then the weighted average travel \( \overline{x} \) time is given by the simple formula depicted in Eq. ( 1 ). Table  1 illustrates the use of the equation with two fictional locations. For each location, the number of patients who travel and the travel time for patients to a hospital is given. In the table, the weighted average is compared to the more familiar mean average.

The model demonstrated that clinics most central to the region were all good choices to provide equitable patient access. A three-clinic solution provided the most equitable solution for service users. The problematic clinic X on the coast of the region was not included in an optimal configuration; however, it could be included in a three-clinic solution without substantial effect on travel times if scheduled infrequently. This latter result allowed the decision makers to move on from the strategic debate about location and focus on the more detailed implementation issues of scheduling and capacity planning for clinics. This was again addressed upfront using a computer simulation study to evaluate a small number of competing options for scheduling the clinics.

Lessons for implementation science

Each of the three roles emphasises the use of OR to conduct implementation science upfront before any action to alter a care pathway or service has been taken. Many OR scholars argue that the benefit of constructing a model upfront is that it forces decision makers to move from a world of imprecise language to a world of a precise language (sometimes referred to as a common language [ 32 ]) and ultimately develop a shared understanding of the problem; although as I will argue later, there is very limited empirical evidence supporting this proposition. Such a shared understanding increases the likelihood if implementation will actually go ahead and importantly if it will be sustained or normalised.

It is important to emphasise that the three case studies illustrate the simpler end of what can be achieved in using OR for upfront implementation science. This is partly a stylistic choice in order to aid reader understanding, for example, many optimisation problems are hugely complex, but also because in my experience simpler models tend to be accepted and used more in healthcare. Simpler models also need less input data and hence can be built and run quickly.

Along with the three case studies, OR is in general grounded in the use of models to improve upfront decision-making in complex implementation problems. Although there is a significant overlap between OR and implementation research, there are differences. For example, OR would not provide the rich contextual information collected in a process evaluation.

Implementation science challenges for OR

Implementation science poses a number of challenges for OR. I propose that these lie at three levels: disciplinary, organisational, and interventional. Table  2 summarises these key challenges.

Challenges at a disciplinary level

This article describes three roles for OR within implementation science. An irony is that OR interventions themselves are poorly understood with barely any published evaluation of practice or impact [ 33 – 36 ]. Limited examples can be found in Monks et al. [ 37 ], Pagel et al. [ 38 ], and Brailsford et al. [ 39 ]. The explanation for this can be found at a disciplinary level. That is, academic OR is predominately driven and rewarded by the development of theory for modelling methodology as opposed to understanding interventions and the issues they raise for practice. As such, a discipline that promotes the use of evidence for decision-making in healthcare cannot confidently answer the question does OR in health work ? I am regularly challenged on this point by healthcare professionals.

A second disciplinary challenge is to systematically involve service users in the co-design of OR interventions. To date, evidence of service user involvement is limited (see Walsh and Holstick [ 40 ] for an example). There is also confusion between service users framed as research participants (typically treated as a data source to parameterise models with behavioural assumptions) and co-designers of research objectives and methods, although there has been an effort to clarify the important difference [ 41 ].

Challenges at the organisational level

The three roles of OR outlined above are widely applicable across healthcare implementation problems. However, before OR can be used within practice, users of the research, in this case, healthcare managers, clinicians and service users, must be aware of the approaches. This is currently a substantial barrier to a wide scale adoption in health services [ 42 – 44 ] and stands in stark contrast to domains such as manufacturing and defence where it is used frequently to generate evidence before action [ 45 ]. The implication of low awareness of OR in health is that it is often difficult to engage senior decision makers in the complex operational and logistical problems that matter the most for service users.

Challenges at an interventional level

Fifty years ago, Churchman and Schainblatt [ 46 ] wrote about a ‘dialectic of implementation’ in the journal management science. In this paper, the two authors advocated that a position of mutual understanding between a researcher and manager was necessary in order to implement results of a study. That is, the researcher must understand the manager’s position, values, and implementation problem in order to tackle the correct problem in the right way. The manager must understand the method that the researcher has applied, at least at the conceptual level, in order to scrutinise, challenge, and implement results. The concept of mutual understanding is an elegant one, but in practice, achieving it is a challenge for both sides. As a simple example from a researcher perspective, it is difficult to assess if the users of a model understand why a model is producing certain results [ 42 ]. That is, do users understand how the model works or are they simply accepting the results based on some heuristic, such as ‘these are the results I want’ or ‘I trust the person telling me the results’? Given the disciplinary challenge outlined above, to date, there is limited validated guidance about how to manage such complex interventions within OR.

The computer software used in the three case studies have been available for considerable time, but appropriate data to parameterise the quantitative models used to illustrate the second and third roles are potentially not collected routinely. All models require data from the system studied. The TIA clinic study had relatively low requirements: individual service user-level data detailing date of clinic attendance, clinic attended, the risk classification of patient, and a home location of the patient—much of which is collected routinely by a health system for financial reporting purposes. Simulation modelling studies such as that described in the emergency department case study have high data requirements, including fine-grained timings of processes such as triaging and doctor assessment. It is unlikely such data are collected routinely as they have no use in financial reporting.

An agenda for OR in implementation science

Given the organisational, interventional, and disciplinary issues outlined in the ‘ Implementation science challenges for OR ’ section, I propose the following agenda for OR within implementation science.

Priority 1: creating the evidence base

At the forefront of the research agenda is the need to evaluate the impact of OR on complex interventions. The focus here should be on the consumers of research as opposed to the modellers and the process they follow [ 47 , 48 ]. There is a need to understand how stakeholders make sense of an OR intervention and how the results of studies are used to assist decision-making. Recent research offers some promise in progressing this aim. PartiSim [ 49 ] is a participative modelling framework that aims to involve stakeholders in structured workshops throughout a simulation study. Structured frameworks like PartiSim provide an opportunity to study the user side of OR more efficiently, as the modelling steps are known upfront. Another area showing promise is the recent emergence of Behavioural OR [ 50 ]. One of the core aims of Behavioural OR is to analyse and understand the practice and impact of OR on context (e.g. [ 51 – 53 ]).

Priority 2: raising demand and the liberation of OR

Much of the challenge in the use of OR as an implementation science technique that I outline is rooted in the lack of organisational awareness and experience of the approach. But what if this challenge were to be resolved? To examine this further, consider a counterfactual world where all health service users, managers, and clinicians are well versed in the three implementation science roles of OR and all have free access to a substantial evidence base detailing the efficacy of the approach. In this world, where OR is an accepted implementation science approach, the constraint has now moved from demand to supply of modelling services. Current supply is predominately provided by the (relatively) small specialist consultancy and research communities. There is a great need to liberate OR from its roots as the tool of the ‘specialist’ and transfer knowledge to research users. Two initial efforts to achieve this priority include the Teaching Operational Research for Commissioning in Health (TORCH) in the UK [ 54 ] and the Research into Global Healthcare Tools (RIGHT) Project [ 55 ]. TORCH successfully developed a curriculum for teaching OR to commissioners, although it has yet to be implemented on a wide scale or evaluated. The RIGHT project developed a pilot web tool to enable healthcare providers select an appropriate OR approach to assist with an implementation problem. Both of these projects demonstrate preliminary efforts at liberating OR from the traditional paradigm of specialist delivery.

The liberation of OR has already taken place in some areas in the form of Community OR . The three case studies illustrated interventions where the collaboration puts the emphasis on a modeller to construct the model and provide results for the wider stakeholder group. Alternatively, service users could develop or make use of OR methods to analyse a problem themselves. Community OR changes the role of an operational researcher from a modeller to a facilitator in order to aid those from outside of OR to create appropriate systematic methodology to tackle important social and community-based issues. In a rare example of community OR in healthcare [ 40 ], two examples illustrate where service users take the lead. In the first example, users of mental health services used system methods to produce a problem structuring tool to evaluate the impact of service users on NHS decision-making. In the second example, service users developed and applied an idealised planning approach for the future structure of mental health services. These approaches are qualitative in nature but are systematic and in-line with an OR implementation science approach.

Priority 3: PPI education for OR modellers

The first two priorities listed might be considered long-term goals for the OR implementation science community. An immediate priority that is arguably achievable over the short term is Patient and Public Involvement (PPI) education for OR modellers. The co-design of healthcare models with decision makers is often held up as a critical success factor for modelling interventions [ 42 ]. For ethical and practical reasons, co-design of OR modelling interventions should also include service users [ 41 ]. Education need not be complicated and could at first be done through widely read OR magazines and a grass roots movement delivered through master degree courses.

Conclusions

Operational research offers improvement scientists and individuals who work in complex health systems the opportunity to do more upfront system thinking about interventions and change. OR's upfront role within implementation science aims to answer questions such as where best to target interventions, will such an intervention work even under optimistic assumptions, which options out of many should we implement, and should we consider de-implementing part of a service in favour of investing elsewhere. As OR becomes more widely adopted as an implementation science technique, evaluation of the method through the lens of implementation science itself becomes more necessary in order to generate an evidence base about how to effectively conduct OR interventions. It is also necessary to liberate OR from its traditional roots as a specialist tool.

Operational research (OR) is a mature discipline that has developed a significant volume of methodology to improve health services. OR offers implementation scientists the opportunity to do more upfront system thinking before committing resources and taking risks. OR has three roles within implementation science: structuring an implementation problem, upfront evaluation of implementation problems, and a tool for strategic reconfiguration of health services. Challenges facing OR as implementation science include limited evidence or evaluation of impact, limited service user involvement, a lack of managerial awareness, effective communication between research users and OR modellers, and availability of healthcare data. To progress the science, a focus is needed in three key areas: evaluation of OR interventions, transferring the knowledge of OR to health services, and educating OR modellers about the aims and benefits of service user involvement.

Abbreviations

AMU, Acute Medical Unit; CDU, clinical decision-making unit; ED, emergency department; GP, general practitioner; MRI, magnetic resonance imaging; NHS, National Health Service; OR, operational research (UK)/operations research (US); PPI, Patient and Public Involvement; PPT, psychology and psychiatric talking therapies; RIGHT, Research into Global Healthcare Tools; RIL, recovering independent life; SD, system dynamics; TIA, transient ischemic attack; TORCH, Teaching Operational Research for Commissioning in Health

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Acknowledgements

This article presents independent research funded by the National Institute for Health Research (NIHR) Collaboration for Leadership in Applied Health Research and Care (CLAHRC) Wessex. The views expressed in this publication are those of the author(s) and not necessarily those of the National Health Service, the NIHR, or the Department of Health.

Case studies 1 and 3 were funded by NIHR CLAHRC South West Peninsula. Case study 3 used Selective Analytics MapPlace software.

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TM developed the models described in the case studies, conceived the idea for debate, and wrote the paper.

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TM leads the NIHR Collaboration in Leadership in Health Research and CLAHRC Wessex’s methodological hub where he conducts applied health service research in collaboration with the NHS. He is an operational researcher with experience in industry, the public sector, and academic research.

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Monks, T. Operational research as implementation science: definitions, challenges and research priorities. Implementation Sci 11 , 81 (2015). https://doi.org/10.1186/s13012-016-0444-0

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Operational Research in Health-care Settings

Rajesh kunwar.

Department of Community Medicine, TS Misra Medical College, Department of Community Medicine, Prasad Institute of Medical Sciences, Lucknow, Uttar Pradesh, India

V. K. Srivastava

Origin of the term operational research (OR), also known as operations research, can be traced back to World War II when a number of researches carried out during military operations helped British Forces produce better results with lesser expenditure of ammunition. The world soon realised the potential of this kind of research and many disciplines especially management sciences, started applying its principles to achieve better returns on their investments.

Following World War II in 1948, the World Health Organization (WHO) came into existence with research as one of its core functions. It emphasized the need of identifying health-related issues needing research and thereby generation, dissemination, and utilization of the newly acquired knowledge for health promotion.[ 1 ] In 1978, Alma Ata Declaration acknowledged that primary health care was well known globally but, at the same time, also noted that modalities of its implementation were likely be different in different countries depending on their socioeconomic conditions, availability of resources, development of technology, and motivation of the community. A number of issues were yet to be resolved and researched before primary health care was operationalized under local conditions.[ 2 ]

T HE D EFINITION

The kind of research that Alma Ata Declaration recommended for improvement of health-care delivery is essentially OR. Described as “the science of better,” it helps in identifying the alternative service delivery strategy which not only overcomes the problems that limit the program quality, efficiency, and effectiveness but also yields the best outcome.[ 3 ] In its report on “The Third Ten Years of the WHO,” WHO has highlighted the usefulness of OR in improvement of health-care delivery in terms of its efficiency, effectiveness, and wider coverage by testing alternative approaches even in countries with limited national resources.[ 4 ]

OR has been variously defined. Dictionary of Epidemiology defined it as a systematic study of the working of a system with the aim of improvement.[ 5 ] From a health program perspective, OR is defined as the search for strategies and interventions that enhance the quality and effectiveness of the program.[ 6 ] A global meeting held in Geneva in April 2008 to develop the framework of OR, defined the scope of OR in context to public health as “ Any research producing practically usable knowledge (evidence, finding, information, etc.) which can improve program implementation (e.g., effectiveness, efficiency, quality, access, scale up, sustainability) regardless of the type of research (design, methodology, approach), falls within the boundaries of OR .”[ 7 ]

OR, however, is different from clinical or epidemiological research. It addresses a specific problem within a specific program. It examines a system, for example, health-care delivery system, and experiments in the environment specific to the program with alternative strategies to find the most suitable one and has an objective of improvement in the system. On the other hand, clinical or epidemiological research studies individuals and groups of individuals in search of new knowledge. In addition, ethical issues, which form an integral part of all clinical and epidemiological research, have their role poorly defined in OR, more so if it is based on secondary data.

The keyword in all the definitions is improvement, which is to be brought about by means of research in the operation of an ongoing program. Its characteristics include:

• It focuses on a specific problem in an ongoing programme
• It involves research into the problem using principles of epidemiology
• It tests more than one possible solution and provides rational basis, in the absence of complete information, for the best alternative to improve program efficiency
• It requires close interaction between program managers and researchers
• It succeeds only if the research is conducted in the existing environment and study results are implemented in true letter and spirit.

T HE P ROCESS

In health-care settings, an ongoing health program often fails to achieve its expected objective and the program managers are faced with problems factors responsible for which are not apparent. This is the stage where process of OR is initiated. In a standard OR process, planning begins with organization of a research team, which should have a mix of people with different backgrounds such as epidemiology, biostatistics, health managements, etc., The program managers may not be able to carry out the research themselves because of their work responsibility and in all probability, their biased views. However, they need to have a working relationship with the research team to ensure smooth conduct of the research and ownership of the result by all parties.

According to Fisher et al ., OR is a continuous process of problem identification, selection of a suitable strategy/intervention, experimentation of the selected strategy/intervention, dissemination of the findings, and utilization of the information so derived.[ 8 ] However, it may not always be possible to follow a step by step approach in OR since it is carried out in the existing environment, and many of the activities may be taking place simultaneously. The process involves the following steps [ Figure 1 ].

Process of operational research

Identifying problems

Like any other research, it is essential to have a research question as to the first and foremost step for beginning the process of OR. Discussion with program managers and staff, review of project reports and local documentation, discussion with experts in the field and literature search gives an insight into why the problem is occurring and what are possible solutions; and help in the identification of the research question. OR methods are useful for the systematic identification of problems and the search for potential solutions. Structured approaches to identifying options, such as the strategic choice approach or systematic creativity approaches have great potential for use in low-resource settings.[ 9 ]

Choosing interventions

Choosing appropriate interventions is clearly a crucial step. Effectiveness, safety, cost, and equity should all be considered, and researchers should be familiar with standard textbook methods for assessing these. Finding the best combinations and delivery methods is a major research exercise in its own right. Modeling different intervention strategies before rollout is now ubiquitous in many industries but is less common in healthcare.[ 10 ] Modeling work has been done on ways to reduce maternal mortality and in cervical cancer screening in low-resource settings.[ 11 ]

An appropriate intervention design, depending on available time and resources, should have a written protocol spelling out details of steps to be taken during implementation. Only valid and reliable instruments – be it quantitative or qualitative study-should be used; and wherever possible, a pilot study be carried out to further refine the conduct of the intervention. The contribution that OR and management science can make to design and delivery is not restricted to high technology. Oral rehydration therapy is a “low-tech–low-cost–high-impact” innovation, in which OR was used to explore ways it could be administered using readily available ingredients by laypeople, with an escalation pathway to treatment by health-care professionals when necessary.[ 12 ]

Small-scale projects generally need considerable modifications to work on a larger scale. Classic OR techniques such as simulation modeling can be used in locating services, managing the supply chain, and developing the health-care workforce.

Integrating into health systems

After analysis of the result, the information gathered should be disseminated to stakeholders and decision-makers. The modalities of information utilization should have been predecided and included in the research proposal. Successes in global health programs often result from synergistic interactions between individual, community and national actors rather than from any single “magic bullet.” A greater focus is needed on how interventions should be used in a complex behavioral environment, to better capture the dynamics of social networks, and to understand how complex systems can adapt positively to change. This is a task where OR and management science tools can be useful, as demonstrated by systems analysis of programs for cervical cancer prevention[ 13 ] or agent simulation modeling of spread of HIV in villages.[ 14 ]

E VALUATION

One of the greatest challenges for global health is the measurement and evaluation of performance of projects and programs. The WHO defines evaluation as “ the systematic and objective assessment of an ongoing or completed initiative, its design, implementation, and results. The aim is to determine the relevance and fulfillment of objectives, efficiency, effectiveness, impact, and sustainability .”[ 15 ] It may or may not lead to improvement.

Accelerated Child Survival and Development (ACSD) program, an initiative of UNICEF, was implemented in eleven West African countries from 2001 to 2005 with an objective of reducing mortality among under-fives by at least 25% by the end of 2006. Retrospective evaluation of the program was carried out in Benin, Ghana, and Mali by comparing data of ACSD focus districts with those of remainder districts. It showed that the difference in coverage of preventive interventions in ACSD focus areas before and after program implementation was not significant in Benin and Mali. This probably resulted in failure of ACSD program to accelerate survival of under-fives in-focus areas of Benin and Mali as compared to comparison areas. The inputs obtained from the evaluation of the program if translated into policy or national program would have delivered the desired result of ACSD program implementation.[ 16 ] Evaluation, thus, is fundamental to good management and is an essential part of the process of developing effective public policy. It is a complex enterprise, requiring researchers to balance the rigors of their research strategies with the relevance of their work for managers and policymakers.[ 17 ]

Standard control trial approaches to evaluation are sometimes feasible and appropriate but often a more flexible systems-oriented approach is required, together with modeling to help assess the effectiveness of preventive interventions.[ 18 ] Decision tree modeling can give rapid insights into the operational effectiveness and cost-effectiveness of procedures[ 19 ] and programs.[ 20 ]

O PERATIONAL R ESEARCH IN H EALTH-CARE S ETTINGS : E XAMPLES

The relevance of OR in health-care settings cannot be overemphasized. It has been successfully used all over the world in various health programs such as family planning, HIV, tuberculosis (TB), and malaria control programs to name a few. Its role in causing improvement in various health programs and the development of policies has been acknowledged globally. Sustained OR efforts of several decades helped in developing the Global strategy for control of TB. India and Malawi provide the most successful example of OR in this field.[ 21 ] In India, it was demonstrated by OR that successful implementation of DOTS strategy throughout the country led to reduction in the prevalence of TB, reduction in fatality due to TB and release of hospital beds occupied by TB patients; and thereby a potential gain to the Indian economy.[ 22 ]

For the treatment of TB, about half of TB patients in India rely on the private sector. In spite of it being a notifiable disease, TB notification from private sector has been a challenge. In 2014, Delhi state, by adopting direct “one to one” sensitization of private practitioners by TB notification committee, was able to accelerate notification of TB cases from the private sector.[ 23 ]

In view of the growing burden of multidrug-resistance TB (MDR-TB), an OR was conducted in the setting of Revised National Tuberculosis Programme on patients with presumptive MDR-TB in North and Central Chennai, in 2014 to determine prediagnosis attrition and pre-treatment attrition, and factors associated with it. Prediagnosis and pretreatment attrition were found 11% and 38%, respectively. The study showed that patients with smear-negative TB were less likely to undergo drug susceptibility testing (DST) and more attention was required to be paid to this group for improving DST.[ 24 ]

One of the most successful examples of OR in India is the experimental study carried out in Gadchiroli district of Maharashtra from 1993 to 1998. In their path-breaking field trial, Bang et al . trained village level workers in neonatal care who subsequently made home visits at scheduled intervals and managed premature birth/low birthweight, birth asphyxia, hypothermia, neonatal sepsis, and breastfeeding problems. This led to a significant reduction in neonatal mortality rates in intervention villages.[ 25 ] Encouraged by the success of this field trial, Home-Based Newborn Care has been adopted by many districts in India to combat neonatal mortality.

In leprosy case detection campaign (LCDC), introduced under National Leprosy Eradication Programme of India in 2016, false-positive diagnosis is a major issue. A study carried out in four districts of Bihar found 30% false-positive cases during LCDC. Using “appreciative inquiry” as a tool, Wagh et al . were able to achieve a decline in false-positive diagnosis.[ 26 ]

OR has been successfully used in hospital settings too. In Latin America, unsafe abortions used to be one of the most common causes of high maternal mortality. Billings and Bensons reviewed ten completed OR projects conducted in public sector hospitals of seven Latin American countries. Their findings indicated that sharp curettage replaced by manual vacuum aspiration for conducting abortion reduced the requirement of resources for postabortion care, reduced cost, and length of hospital stay and reduced maternal mortality.[ 27 ]

C ONCLUSION

Following Alma Ata declaration and Millennium Development Goals, all countries of the world have instituted their own National Health Programmes in a bid to improve health of their countrymen. Although health programs are in place, Governments are committed, guidance from the WHO is available, support from NGOs have been garnered, still many countries have not been able to achieve their desired goals. Operational Research is now being used as a key instrument, especially in resource-poor countries, to tap the untapped information. Administrators are using it as a searchlight for discovering what is still in the dark. It is there to stay. It is high time that the scientific community working in health-care settings gets acquainted with the nuances of OR and uses it more often for improving the outcome of health programs and for making them more efficient and effective.

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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming.   The simplex method of Linear Programming Problems(LPP)   proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point.   Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres.     4 x 5 = 20 routes are possible.   Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs?   The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person?   The objective is minimizing total cost.   This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above.   Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

• To decide the transportation of new materials from various centres to different manufacturing plants.   In the case of multi-plant company this is highly useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.   These two are the uses of transportation model.   The objective is minimizing transportation cost.

Assignment model is used in the following:

• To decide the assignment of jobs to persons/machines, the assignment model is used.
• To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
• To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand.   Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available.   In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

Related Posts:

• Construction of Mathematical Decision Model
• Operations Research approach of problem solving
• Introduction to Crtical Path Analysis
• Waiting Lines and Queuing System in Management Science
• Modeling Techniques in Management Science
• Initial basic feasible solution of a transportation problem
• Procedure for finding an optimum solution for transportation problem
• Introduction to Transportation Problem
• Introduction to Decision Models
• Introduction to Linear Programming (L.P)

One thought on “ Transportation and Assignment Models in Operations Research ”

Exclussive dff. And easy understude

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