multiple optimal solutions assignment problem

Multiple Optimal Solutions: Simplex Method

The optimal solution may not be unique, if the non basic variables have a zero coefficient in the index row (z j -c j ).

This implies that bringing the non basic variable into the basis will neither increase nor decrease the value of the objective function.

Thus, the linear program problem (LPP) has multiple optimal solutions (Alternative optimal solutions) . For example, let us consider the following example of simplex method .

1. Unrestricted Variables 2. Unbounded Solution 3. No Feasible Solution (Infeasible Solution) 4. Multiple Optimum Solutions 5. Degeneracy

Multiple Optimal Solutions Example : LPP

Maximize 2000x 1 + 3000x 2

6x 1 + 9x 2 ≤ 100 2x 1 + x 2 ≤ 20

x 1 , x 2 ≥ 0

After introducing slack variables, the corresponding equations are

6x 1 + 9x 2 + x 3 = 100 2x 1 + x 2 + x 4 = 20 x 1 , x 2 , x 3 , x 4 ≥ 0

Where x 3 and x 4 are slack variables.

Table 1: Simplex Method

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Since z j -c j ≥ 0 for all variables, x 1 = 0, x 2 = 100/9 is an optimum solution of the LPP. The maximum value of the objective function is 100000/3. However, the z j -c j value corresponding to the non basic variable x 1 is zero. This indicates that there is more than one optimal solution of the problem. Thus, by entering x 1 into the basis we may obtain another alternative optimal solution.

However, the value of the objective function will not be improved, although the present solution x 1 = 0, x 2 = 100/9 is shifted to x 1 = 20/3 and x 2 = 20/3.

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Lesson 9. solution of assignment problem.

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Linear Programming optimization with multiple optimal solutions

I am trying to solve the following optimization problem using linear programming (deterministic operations research). According to the book, there are multiple optimal solutions, I don't understand why. I'll show you what I have done.

The problem is:

$max Z=500x_{1}+300x_{2}$

$15x_{1}+5x_{2}\leq 300$

$10x_{1}+6x_{2}\leq 240$

$8x_{1}+12x_{2}\leq 450$

$x_{1},x_{2}\geqslant 0$

I have plotted the lines graphically to get:

The intersection points are:

$(\frac{135}{14},\frac{435}{14})$

$(\frac{5}{2},\frac{215}{6})$

The target function equals to 12000 for two of these points. If this value was the maximum, then I would say that the entire line, edge, between these intersection points is the optimal solution (multiple solutions). However, this is not the maximum. The second intersection I wrote gives a higher value, and therefore is the maximum. So I think there is a single solution.

What am I missing ?

And generally speaking, what is the mathematical justification for having multiple solutions when two points gives the maximum (or minimum)?

• linear-programming
• operations-research

• 1 $\begingroup$ don't you like any of the answers? $\endgroup$ –  LinAlg Commented Jul 31, 2018 at 15:49

3 Answers 3

If you solve the problem graphically you should solve the objective function $Z$ for $x_2$ as well.

$Z=500x_{1}+300x_{2}$

$Z-500x_{1}=300x_{2}$

$\frac{Z}{300}-\frac53x_1=x_2$

Now you set the level equal to zero, which means that $z=0$ and draw the line. This line goes through the origin and has a slope of $-\frac53$. Then you push the line parallel right upward till the objective function touches the last possible point(s) of the feasible solution(s). The graph below shows the process.

All the points on the green line for $\frac52 \leq x_1\leq 15$ are optimal solutions.

All the optimal solutions are on the the line of the second constraint. This result can be confirmed if we have a look on the coefficient of the second constraint and the objective function. The ratios of the coefficients are equal: $\frac{10}6=\frac{500}{300}$. And additionally The second constraint is fullfilled as a equality.

Conclusion: If you see that the slopes of the objective function is equal to one of the constraints then there eventually exists a solution which is a line and not a single point (2 variables).

The second intersection is the one of the red and blue lines. This point does not satisfy the constraint given by the green line.

The optimal solutions are given by the line segment of the green line with its intersection points as end points. This is because the constraint puts the same relative weights on $x_1,x_2$ as the objective function.

• $\begingroup$ Oh, that's embarrassing, how did I not see that ?!?! Can you kindly explain to me, theoretically, why if two vertices gives the same value, the entire edge is the maximum ? $\endgroup$ –  user2899944 Commented Jul 29, 2018 at 7:19
• $\begingroup$ You can show that for any linear programming problem, the optimum is achieved at a vertex. If we now have two vertices giving us an optimum, it must be the case that the LHS of its constraint is a multiple of the objective function. You should then be able to check that the value you get is the same along the constraint line, hence any feasible point (i.e. the whole line segment between the two vertices) gives an optimum! $\endgroup$ –  ertl Commented Jul 29, 2018 at 7:34

Multiply the second constraint by $50$ to get: $$500x_1+300x_2\le 12000.$$ The value of $Z=500x_1+300x_2$ is $12000$ on every point $(x_1,x_2)$ of the constraint line $500x_1+300x_2=12000$, in particular between $(5/2,215/6)$ and $(15,15)$.

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Dealing with Multiple Optimal Solutions

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• Quirino Paris  

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For many years, LP users have regarded multiple optimal solutions either as an exceptional event or as a nuisance to be avoided. Indeed, in many circles, multiple optimal solutions are a source of embarrassment and, often, the main goal of researchers is to define sufficient conditions for unique solutions.

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Paris, Q. (2016). Dealing with Multiple Optimal Solutions. In: An Economic Interpretation of Linear Programming. Palgrave Macmillan, New York. https://doi.org/10.1057/9781137573926_15

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One of the most well-known combinatorial optimization problems is the assignment problem . Here's an example: suppose a group of workers needs to perform a set of tasks, and for each worker and task, there is a cost for assigning the worker to the task. The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost.

You can visualize this problem by the graph below, in which there are four workers and four tasks. The edges represent all possible ways to assign workers to tasks. The labels on the edges are the costs of assigning workers to tasks.

An assignment corresponds to a subset of the edges, in which each worker has at most one edge leading out, and no two workers have edges leading to the same task. One possible assignment is shown below.

The total cost of the assignment is 70 + 55 + 95 + 45 = 265 .

The next section shows how solve an assignment problem, using both the MIP solver and the CP-SAT solver.

Other tools for solving assignment problems

OR-Tools also provides a couple of other tools for solving assignment problems, which can be faster than the MIP or CP solvers:

• Linear sum assignment solver
• Minimum cost flow solver

However, these tools can only solve simple types of assignment problems. So for general solvers that can handle a wide variety of problems (and are fast enough for most applications), we recommend the MIP and CP-SAT solvers.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

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A new integer model for selecting students at higher education institutions: preparatory classes of engineers as case study.

1. Introduction

2. problem statement, 3. objective and methodology, 3.1. main objective of our work, 3.2. methodology, 3.2.1. organizational structure of higher education institutions.

• Higher education institution: This entity is generally responsible for the management of academic programs and specialty streams. It often develops selection criteria specific to their programs and reviews applications.
• Departments: Academic departments within institutions may be responsible for defining more specific selection criteria related to their specific areas. They may also be involved in the assessment of candidates.
• Selection Committees: Some institutions set up selection committees composed of teaching staff members and experts in the field. These committees evaluate applications and recommend candidates for admission to specialty streams.
• Admission Services: In many institutions, a centralized admissions service processes student applications and coordinates the selection process.
• Academic Advisors: Academic advisors can play an advisory role to students, helping them choose the appropriate specialty streams based on their academic and career goals.
• The Governance Bodies of the institution: In some cases, the governance bodies of the institution, such as boards of directors or management committees, may define general policies concerning selection criteria and specialty streams.

3.2.2. Rules of Selection

• Consistency of applications must be respected. Each student must apply to the appropriate streams in a given higher education institution.
• Selection twice is not permitted. Each student shall be assigned at most to a seat in a chosen academic stream in a given higher education institution. In fact, a student chooses a fixed number of streams belonging to a higher education institution and they are assigned at most to one seat from them.
• Collisions shall not be permitted. In a given higher education institution and in a particular academic stream, a seat is assigned at most to one student who has already chosen this stream.
• The maximum number of seats in a particular stream belonging to a higher education institution must be respected. In fact, each stream has its own number of seats that defines the capacity of the stream; then, the number of admitted students in each stream must not exceed the number of available seats.
• The quotas of students having a particular series of bachelor’s degree in each stream must be respected. In fact, in a given higher education institution, each available stream opens access to students with several series of bachelor’s degrees with different quotas.
• Priority is given to the first choices if possible. In fact, each student has the right to choose more than one stream in a given higher education institution and is assigned at most to one from these choices.

3.2.3. Methodology Adopted

4. the mathematical model of student selection problem, 4.1. general features of the model.

• The student who has the right to apply for a seat in an academic stream to pursue their studies. The set of all these students is denoted by S , e.g., S = s t u d e n t # 1 , s t u d e n t # 2 , … , s t u d e n t # S .
• The seat to which an admitted student is assigned to pursue their studies. The set of all available seats is denoted A , e.g., A = s e a t # 1 , s e a t # 2 , … , s e a t # A .
• The academic stream in which a student will pursue their studies. The set of all available streams in the higher education institution is denoted by B , e.g., B = s t r e a m # 1 , s t r e a m # 2 , … , s t r e a m # B .
• The type of Baccalaureate tokens from which each student graduates; the set of all these baccalaureate types is denoted by T , e.g., T = t y p e # 1 , t y p e # 2 , … , t y p e # T .
• The quota of students having a particular type of baccalaureate in a specific stream. The set of all possible quotas in the different streams is denoted by Q , e.g., Q = q u o t a # 1 , q u o t a # 2 , … , q u o t a # Q .
• The number of seats available in an academic stream that represent the capacity of stream. The set of all these numbers is denoted by N , e.g., N = n b r c / c ∈ C .

4.2. Special Features of the Model

• A b = { a ∈ A : a = seat available in the stream b ∈ B }.
• S b = { s ∈ S : s = student applying for the stream b ∈ B to pursue their studies}.
• S t = { s ∈ S : s = student having the series t ∈ T of the baccalaureate}.
• S b , t = S b ∩ S t .
• B s = { b ∈ B : b = a stream chosen by the student s ∈ S } .
• B t = { b ∈ B : b = a stream open for students having the baccalaureate series t ∈ T } .
• T b s a t = { t ∈ T : t = a baccalaureate series required to apply for the stream b ∈ B } whose quota of seats devoted to students with this type of baccalaureate is exhausted.
• T b s a t ¯ = { t ∈ T : t = a baccalaureate series required to applying for the stream b ∈ B } whose quota of seats devoted to students with this type of baccalaureate is not exhausted.
• T b = T b s a t ∪ T b s a t ¯ = { t ∈ T : t = a baccalaureate series required to apply for the stream b ∈ B }.
• N b = { n b / b ∈ B : n b is the number of available seats in the academic stream b }
• Q b , t = { q ∈ Q : q = the quota of seats devoted to bachelors having the series t ∈ T of baccalaureate in the academic stream b ∈ B }.
• P A = { ( s , b ) ∈ S × B : ( s , b ) = a priori assignment in which the student s ∈ S is assigned to the academic stream b ∈ B }.

4.3. Objective Function

• B : the set of all available streams belonging to the higher education institution;
• S b : the set of student applying for the stream b ;
• A b : the set of available seats in the stream b .

4.4. Constraints of the Model

4.4.1. uniqueness constraints.

• This set of constraints ensures that, in a higher education institution, each student shall be assigned at most to one seat in a stream of their choices. This constraint is given by the following equation: ∀ s ∈ S , ∀ b ∈ B s : ∑ a ∈ A b x s , a ≤ 1 . (2) where –   S : the set of all students; –   B s : the set of streams chosen by the student s .
• This set of constraints ensures that, in each stream b , a seat is assigned to one and only one student who has applied for the stream b . This rule is formulated by the following equation: ∀ b ∈ B , ∀ a ∈ A b : ∑ s ∈ S b x s , a ≤ 1 . (3)

4.4.2. Capacity Constraints

• In a specified stream, the number of admitted students is limited. So, the number of assigned students to a stream shall not exceed the number of available seats in this stream. ∀ b ∈ B : ∑ s ∈ S b ∑ a ∈ A b x s , a ≤ n b . (4) where –   n b : the number of available seats in the stream b .
• For each stream b ∈ B , the number of students having a particular series of baccalaureate and assigned to seats in the stream b shall not exceed the quota of seats devoted to students having this series of baccalaureate in the stream b , described as follows: ∀ b ∈ B , ∀ t ∈ T b : ∑ s ∈ S b , t ∑ a ∈ A b x s , a ≤ q b , t . (5) where –   T b : the set of all baccalaureate series required to applying for the stream b ; –   S b , t : the set of students applying for the stream b ∈ B and having the series t of baccalaureate; –   q b , t : the quota of seats devoted to bachelors having the baccalaureate series t in the stream b .
• In several cases, some students hold some types of baccalaureates in some streams that cannot exceed their quotas of seats, which can generate some unoccupied seats in such streams. In order to maximize the use of the academic resources, we redistribute these unoccupied seats among the other students holding other types of baccalaureate whose quotas of seats are exhausted. This property can be added by replacing the constraints ( 5 ) with the following constraints ( 6 ): ∀ b ∈ B , ∀ t ∈ T b : ∑ s ∈ S b , t ∑ a ∈ A b x s , a ≤ q b , t + p b , t . (6) where p b , t is calculated using the following method: –   For b ∈ B and t ∈ T b s a t ¯ , we calculate the number of unoccupied seats in the stream b devoted to students holding the baccalaureate type t : r b , t = q b , t − S b , t –   Then, we calculate the total number of unoccupied seats in the stream b : r b = ∑ t ∈ T b s a t ¯ r b , t –   Therefore, we redistribute this number of unoccupied seats among students holding the type of baccalaureate t ∈ T b s a t using the following equation: h b , t = r b T b s a t where x refers to the integer part of x . –   The integer division generates a reminder denoted r e m b , which is also redistributed among students holding the baccalaureate type t ∈ T b s a t using the following equation: p b , t = h b , t + 1 t i ∈ { t i ∈ T b s a t : i = 1 , … , r e m b } h b , t t i ∈ { t i ∈ T b s a t : i = r e m b + 1 , … , | T b s a t | }

4.4.3. Consecutiveness Constraints

• Consecutiveness of student choices: For every student, priority to first choices must be considered. In fact, each student wishes satisfy their first choice before the second, etc. This rule can be formulated by the following equation: ∀ s ∈ S , ∀ b i ∈ B s ( i = 1 , … , B s − 1 ) : ∑ a ∈ A b i + 1 x s , a ≤ ∑ a ∈ A b i x s , a (7)
• Consecutiveness of competent students: This set of constraints ensures that priority is given to competent students. In this sense, students who have the higher weight coefficient will be assigned to a seat from their chosen streams. The following constraint formulates this rule: ∀ s , r ∈ S , ∀ b ∈ B s ∩ B r ( w s , b < w r , b ) : ∑ a ∈ A b x s , a ≤ ∑ a ∈ A b x r , a + ∑ b ′ ∈ B s ∩ B r ∖ b ∑ a ∈ A b ′ x r , a (8) where –   w s , b : the weight coefficient of the student s . In this equation, the term ∑ b ′ ∈ B s ∩ B r ∖ b ∑ a ∈ A b ′ x r , a takes two possible values: It takes the 1 value if the student r is assigned to one of the streams b and b ′ —in this case, the student s can be assigned to a seat in the stream b ; otherwise, it takes the 0 value if the student r is not assigned to any stream—in this case, and given that w s , b < w r , b , the student r should not assigned to any stream.

4.4.4. Pre-Assignment Constraints

• This set of constraints ensures that certain students will be assigned to a seat in a stream of their choices and could be used either for the pre-allocation of students or for better handling and reducing complexity and computational difficulties. These constraints are formulated by the following equation: ∀ ( s , b ) ∈ P A : ∑ a ∈ A b x s , a = 1 . (9) where –   P A : the set of a priori assignments to the list of students s ∈ S b .

4.5. Determination of Weight Coefficients

5. real case study, 5.1. presentation of the moroccan preparatory classes.

• Appropriate a work methodology based on organization, investigation, personal initiative, and perseverance.
• Develop autonomy and skills for thinking and reasoning as well as communication.
• Adapt to different problem situations.
• Gradually familiarize themselves with situations requiring sustained effort that may arise during their subsequent training and during execution of their professions.

5.2. Ways of Access to Preparatory Classes

• For holders of a Moroccan baccalaureate, access to preparatory classes is regulated by a ministerial letter. It specifies the procedures and steps to be followed for the application as well as all the selection and registration procedures.
• For students of Moroccan nationality and holding a foreign baccalaureate, application files are assessed at the central service level by a special committee. About twenty places, all courses combined, are reserved each year for this category of candidates.
• For students of foreign nationality with a foreign baccalaureate, the Moroccan Agency for International Cooperation (AMCI) centralizes application files. It also manages the quota reserved for this category of candidates. Around one hundred places are reserved for these candidates across the study streams.

5.3. Statement of the Student Selection Problem at Preparatory Classes

5.4. organizational structure of the preparatory classes.

• The number of repeated years in the baccalaureate phase;
• Age, specialty, and the total mark of the student;
• A mark awarded by the class council;
• A qualifying average of subjects, calculated in terms of the educational streams chosen.

5.5. Data Description

• Streams available in this center;
• Type of Baccalaureate series required to access each stream;
• Quota of seats devoted to holders of each Baccalaureate series;
• Number of seats available in each stream in this center.
• N 1 = 0 , i f   t h e   c o r r e s p o n d i n g   s t u d e n t   h a s   r e p e a t e d t h e   s e c o n d   y e a r   o f   B a c c a l a u r e a t e   c y c l e . 5 , i f   t h e   c o r r e s p o n d i n g   s t u d e n t   h a s   r e p e a t e d t h e   f i r s t   y e a r   o f   B a c c a l a u r e a t e   c y c l e . 10 , o t h e r w i s e .
• N 2 = M 1 + 2 ∗ M 2 3 where –   M 1 : The passing rate achieved in the first year of the Baccalaureate cycle; –   M 2 : The passing rate achieved in the second year of the Baccalaureate cycle.
• N 3 : The qualifying average of subjects for each stream. For example, for the stream MPSI, the parameter N 3 is calculated using the equation N 3 = 4 M + 3 P h y + 0 , 5 L V 2 + 1 F r + 0 , 5 A r 9 where M , P h y , L v 2 , F r , and A r are the marks attained in the final exam of the second year, namely, in the subjects considered for accessing the MPSI stream: –   M : the math mark; –   P h y : the physics mark; –   A r : the Arabic mark; –   F r : the french mark; –   L V 2 : the English mark.
• N 4 : a grade attributable by the class council (rated on 25).
• N 5 = B s − o r d b B s : a grade that takes into account the range of the stream chosen, where B s is the number of possible choices.
• N 6 = t m a x − t s t m a x − t m i n : a grade that takes into account the exact date of students candidature, where –   t m a x : deadline for submission of candidature on the web site; –   t m i n : starting date for submission of candidature on the web site; –   t s : exact submission date of candidature of student s on the web site.

6. Results and Discussion

6.1. computational environment, 6.2. results presentation, 6.3. comparative study, 7. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Majdoub, S.; Loqman, C.; Boumhidi, J. A New Integer Model for Selecting Students at Higher Education Institutions: Preparatory Classes of Engineers as Case Study. Information 2024 , 15 , 529. https://doi.org/10.3390/info15090529

Majdoub S, Loqman C, Boumhidi J. A New Integer Model for Selecting Students at Higher Education Institutions: Preparatory Classes of Engineers as Case Study. Information . 2024; 15(9):529. https://doi.org/10.3390/info15090529

Majdoub, Soufyane, Chakir Loqman, and Jaouad Boumhidi. 2024. "A New Integer Model for Selecting Students at Higher Education Institutions: Preparatory Classes of Engineers as Case Study" Information 15, no. 9: 529. https://doi.org/10.3390/info15090529

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