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:
Assignment problem: Hungarian Method Nui Ruppert (Mtk_Nr.: 373224) David Lenh (Mtk_Nr.: 368343) Amir Farshchi Tabrizi (Mtk-Nr.: 372894)
In this OR-Wiki entry we're going to explain the Hungarian method with 3 examples. In the first example you'll find the optimal solution after a few steps with the help of the reduced matrix. The second example illustrates a complex case where you need to proceed all the steps of the algorithm to get to an optimal solution. Finally in the third example we will show how to solve a maximization problem with the Hungarian method.
Inhaltsverzeichnis
- 1 Introduction
- 2 Example 1 – Minimization problem
- 3 Example 2 – Minimazation problem
- 4 Example 3 – Maximization problem
- 6 References
Introduction
The Hungarian method is a combinatorial optimization algorithm which was developed and published by Harold Kuhn in 1955. This method was originally invented for the best assignment of a set of persons to a set of jobs. It is a special case of the transportation problem. The algorithm finds an optimal assignment for a given “n x n” cost matrix. “Assignment problems deal with the question how to assign n items (e.g. jobs) to n machines (or workers) in the best possible way. […] Mathematically an assignment is nothing else than a bijective mapping of a finite set into itself […]” [1]
The assignment constraints are mathematically defined as:
To make clear how to solve an assignment problem with the Hungarian algorithm we will show you the different cases with several examples which can occur .
Example 1 – Minimization problem
In this example we have to assign 4 workers to 4 machines. Each worker causes different costs for the machines. Your goal is to minimize the total cost to the condition that each machine goes to exactly 1 person and each person works at exactly 1 machine. For comprehension: Worker 1 causes a cost of 6 for machine 1 and so on …
To solve the problem we have to perform the following steps:
Step 1 – Subtract the row minimum from each row.
Step 2 – Subtract the column minimum from each column from the reduced matrix.
The idea behind these 2 steps is to simplify the matrix since the solution of the reduced matrix will be exactly the same as of the original matrix.
Step 3 – Assign one “0” to each row & column.
Now that we have simplified the matrix we can assign each worker with the minimal cost to each machine which is represented by a “0”.
- In the first row we have one assignable “0” therefore we assign it to worker 3 .
- In the second row we also only have one assignable “0” therefore we assign it to worker 4 .
- In the third row we have two assignable “0”. We leave it as it is for now.
- In the fourth row we have one assignable “0” therefore we assign it. Consider that we can only assign each worker to each machine hence we can’t allocate any other “0” in the first column.
- Now we go back to the third row which now only has one assignable “0” for worker 2 .
As soon as we can assign each worker to one machine, we have the optimal solution . In this case there is no need to proceed any further steps. Remember also, if we decide on an arbitrary order in which we start allocating the “0”s then we may get into a situation where we have 3 assignments as against the possible 4. If we assign a “0” in the third row to worker 1 we wouldn’t be able to allocate any “0”s in column one and row two.
The rule to assign the “0”:
- If there is an assignable “0”, only 1 assignable “0” in any row or any column, assign it.
- If there are more than 1, leave it and proceed.
This rule would try to give us as many assignments as possible.
Now there are also cases where you won’t get an optimal solution for a reduced matrix after one iteration. The following example will explain it.
Example 2 – Minimazation problem
In this example we have the fastest taxi company that has to assign each taxi to each passenger as fast as possible. The numbers in the matrix represent the time to reach the passenger.
We proceed as in the first example.
Iteration 1:
Now we have to assign the “0”s for every row respectively to the rule that we described earlier in example 1.
- In the first row we have one assignable “0” therefore we assign it and no other allocation in column 2 is possible.
- In the second row we have one assignable “0” therefore we assign it.
- In the third row we have several assignable “0”s. We leave it as it is for now and proceed.
- In the fourth and fifth row we have no assignable “0”s.
Now we proceed with the allocations of the “0”s for each column .
- In the first column we have one assignable “0” therefore we assign it. No other “0”s in row 3 are assignable anymore.
Now we are unable to proceed because all the “0”s either been assigned or crossed. The crosses indicate that they are not fit for assignments because assignments are already made.
We realize that we have 3 assignments for this 5x5 matrix. In the earlier example we were able to get 4 assignments for a 4x4 matrix. Now we have to follow another procedure to get the remaining 2 assignments (“0”).
Step 4 – Tick all unassigned rows.
Step 5 – If a row is ticked and has a “0”, then tick the corresponding column (if the column is not yet ticked).
Step 6 – If a column is ticked and has an assignment, then tick the corresponding row (if the row is not yet ticked).
Step 7 - Repeat step 5 and 6 till no more ticking is possible.
In this case there is no more ticking possible and we proceed with the next step.
Step 8 – Draw lines through unticked rows and ticked columns. The number of lines represents the maximum number of assignments possible.
Step 9 – Find out the smallest number which does not have any line passing through it. We call it Theta. Subtract theta from all the numbers that do not have any lines passing through them and add theta to all those numbers that have two lines passing through them. Keep the rest of them the same.
(With this step we create a new “0”)
With the new assignment matrix we start to assign the “0”s after the explained rules. Nevertheless we have 4 assignments against the required 5 for an optimal solution. Therefore we have to repeat step 4 – 9.
Iteration 2:
Step 4 – Tick all unassigned row.
Note: The indices of the ticks show you the order we added them.
Iteration 3:
Iteration 4:
After the fourth iteration we assign the “0”s again and now we have an optimal solution with 5 assignments.
The solution:
- Taxi1 => Passenger1 - duration 12
- Taxi2 => Passenger4 - duration 11
- Taxi3 => Passenger2 - duration 8
- Taxi4 => Passenger3 - duration 14
- Taxi5 => Passenger5 - duration 11
If we define the needed duration as costs, the minimal cost for this problem is 56.
Example 3 – Maximization problem
Furthermore the Hungarian algorithm can also be used for a maximization problem in which case we first have to transform the matrix. For example a company wants to assign different workers to different machines. Each worker is more or less efficient with each machine. The efficiency can be defined as profit. The higher the number, the higher the profit.
As you can see, the maximal profit of the matrix is 13. The simple twist that we do is rather than try to maximize the profit, we’re going to try to minimize the profit that you don’t get. If every value is taken away from 13, then we can minimize the amount of profit lost. We receive the following matrix:
From now on we proceed as usual with the steps to get to an optimal solution.
With the determined optimal solution we can compute the maximal profit:
- Worker1 => Machine2 - 9
- Worker2 => Machine4 - 11
- Worker3 => Machine3 - 13
- Worker4 => Machine1 - 7
Maximal profit is 40.
The optimal solution is found if there is one assigned “0” for each row and each column.
[1] Linear Assignment Problems and Extensions, Rainer E. Burkard, Eranda Cela
[2] Operations Research Skript TU Kaiserslautern, Prof. Dr. Oliver Wendt
[3] The Hungarian method for the assignment problem, H. W. Kuhn, Bryn Mawr College
Fundamental of Operations Research, Lec. 16, Prof. G. Srinivasan
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A Comparative Analysis of Assignment Problem
- Conference paper
- First Online: 06 June 2023
- Cite this conference paper
- Shahriar Tanvir Alam ORCID: orcid.org/0000-0002-0567-3381 5 ,
- Eshfar Sagor 5 ,
- Tanjeel Ahmed 5 ,
- Tabassum Haque 5 ,
- Md Shoaib Mahmud 5 ,
- Salman Ibrahim 5 ,
- Ononya Shahjahan 5 &
- Mubtasim Rubaet 5
Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))
Included in the following conference series:
- International Conference on Big Data Innovation for Sustainable Cognitive Computing
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The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.
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Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11
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Travelling Salesman Problem
This humorously named problem refers to the following situation:
A travelling salesman , named Rover plans to visit each of n cities. He wishes to visit each city once and only once, arriving back to city from where he started. The distance between City i and City j is c ij . What is the shortest tour Rover can take?
If there are n cities, there are (n - 1)! possible ways for his tour. For example, if the number of cities to be visited is 5, then there are 4! different combinations. Such type of problems can be solved by the assignment method.
Let c ij be the distance (or cost or time) between City i to City j and
Use Horizontal Scrollbar to View Full Details
| x = | 1 if a tour includes travelling from city i to city j (for i ≠ j) | |
| 0 otherwise |
The following example will help you in understanding the travelling salesman problem of operation research.
Example: Travelling Salesman Problem
A travelling salesman, named Rolling Stone plans to visit five cities 1, 2, 3, 4 & 5. The travel time (in hours) between these cities is shown below:
| To | |||||
|---|---|---|---|---|---|
| From | 1 | 2 | 3 | 4 | 5 |
| 1 | ∞ | 5 | 8 | 4 | 5 |
| 2 | 5 | ∞ | 7 | 4 | 5 |
| 3 | 8 | 7 | ∞ | 8 | 6 |
| 4 | 4 | 4 | 8 | ∞ | 8 |
| 5 | 5 | 5 | 6 | 8 | ∞ |
How should Mr. Rolling Stone schedule his touring plan in order to minimize the total travel time , if he visits each city once a week?
"As every thread of gold is valuable, so is every minute of time." - John Mason
After applying steps 1 to 3 of the Hungarian method, we get the following assignments.
Use Horizontal Scrollbar to View Full Table.
| To | |||||
|---|---|---|---|---|---|
| From | 1 | 2 | 3 | 4 | 5 |
| 1 | ∞ | 1 | 3 | 1 | |
| 2 | 1 | ∞ | 2 | 1 | |
| 3 | 2 | 1 | ∞ | 2 | |
| 4 | 3 | ∞ | 4 | ||
| 5 | 3 | ∞ | |||
Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.
Select the smallest element from all the uncovered elements. Subtract this smallest element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment. Repeating step 3 on the reduced matrix, we get the following assignments.
| To | |||||
|---|---|---|---|---|---|
| From | 1 | 2 | 3 | 4 | 5 |
| 1 | ∞ | 2 | 1 | ||
| 2 | ∞ | 1 | 1 | ||
| 3 | 1 | ∞ | 2 | ||
| 4 | 3 | ∞ | 5 | ||
| 5 | 4 | ∞ | |||
The above solution suggests that the salesman should go from city 1 to city 4, city 4 to city 2, and then city 2 to 1 (original starting point). The above solution is not a solution to the travelling salesman problem as he visits city 1 twice.
The next best solution can be obtained by bringing the minimum non-zero element, i.e., 1 into the solution. Please note that the value 1 occurs at four places. We will consider all the cases separately until the acceptable solution is obtained. To make the assignment in the cell (2, 3), delete the row & the column containing this cell so that no other assignment can be made in the second row and third column.
Now, make the assignments in the usual manner as shown in the following table.
He starts from city 1 and goes to city 4; from city 4 to city 2; from city 2 to city 3; from city 3 to city 5; from city 5 to city 1.
Substituting values from original table: 4 + 7 + 6+ 4 + 5 = 26 hours.
In this chapter, we focussed on a special type of transportation problem where the objective was to allocate n different facilities to n different tasks. Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as Hungarian Method. If the number of persons is the same as the number of jobs, the assignment problem is said to be balanced. If the number of jobs is different from the number of persons, the assignment problem is said to be unbalanced. Some assignment problems entail maximizing the profit, effectiveness, or layoff of an assignment of persons to tasks or of jobs to machines. The Hungarian Method can also solve such problems. Further, the Hungarian method can also be utilized for solving crew assignment problem and the travelling salesman problem.
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Goal programming Linear programming Simplex Method Transportation Problem
Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods
Table of Contents
Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.
What is the Unbalanced Assignment Problem?
The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.
Formulation of the Unbalanced Assignment Problem
To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.
Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.
Solutions for the Unbalanced Assignment Problem
Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.
In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.
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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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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.
Title: "Cracking the Balanced Assignment Problem in Operations Research!"🔍 Uncover the secrets of the Balanced Assignment Problem in this quick guide to Ope...
Problem 5 A typical assignment problem, presented in the classic manner, is shown in Fig. Here there are five machines to be assigned to five jobs. The numbers in the matrix indicate the cost of doing each job with each machine. Jobs with costs of M are disallowed assignments. The problem is to find the minimum cost matching of machines to jobs.
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 ...
The assignment problem represents a special case of linear programming problem used for allocating resources (mostly workforce) in an optimal way; it is a highly useful tool for operation and project managers for optimizing costs. The lpSolve R package allows us to solve LP assignment problems with just very few lines of code.
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. In an assignment problem , we must find a maximum matching that has the minimum weight in a weighted bipartite graph .
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 ...
This Video will help the student to Understand the Algorithm of Assignment Model.Accordingly, Row Operation / Column Operation and, Row Assignment / Column A...
Playlist of all my Operations Research videos-http://goo.gl/zAtbi4Today I'll tell you how to solve a Minimization type Assignment Problem in just 5 Easy Step...
The assignment problem is a standard topic discussed in operations research textbooks (See for example, Hillier and Lieberman [1] or Winston [2]). A typical presentation requires that n jobs must ...
The Hungarian method is a combinatorial optimization algorithm which was developed and published by Harold Kuhn in 1955. This method was originally invented for the best assignment of a set of persons to a set of jobs. It is a special case of the transportation problem. The algorithm finds an optimal assignment for a given "n x n" cost matrix.
However, solving this task for increasing number of jobs and/or resources calls for computational techniques. This article aims at solving an Assignment Problem using the Gurobi package of Python.
Tables 2, 3, 4, and 5 present the steps required to determine the appropriate job assignment to the machine. Step 1 By taking the minimum element and subtracting it from all the other elements in each row, the new table will be: Table 2 represents the matrix after completing the 1st step. Table 1 Initial table of a.
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
The last phase, interpretation, encompasses making a decision and developing implementation plans. The paragraphs below explain the seven elements of the operations research problem solving process in greater detail. The activities that take place in each element are illustrated through some of the tools or methods commonly used.
e minimisation problem.3. The assignment problem wherein the number of rows is not equal to the number of columns is said t. be an unbalanced problem. Such a problem is handled by introducing dummy row(s) if the number of rows is less than the number of columns and dummy column(s) if the number of columns is le.
Repeating step 3 on the reduced matrix, we get the following assignments. Table. The above solution suggests that the salesman should go from city 1 to city 4, city 4 to city 2, and then city 2 to 1 (original starting point). The above solution is not a solution to the travelling salesman problem as he visits city 1 twice.
Solving Maximisation in an Assignment Problem. The above approach provides a step-by-step process to maximize an assignment problem. Here are the steps in summary: Convert the assignment problem into a matrix. Reduce the matrix by subtracting the minimum value in each row and column. Cover all zeros in the matrix with the minimum number of lines.
Operations research (OR) offers a powerful toolkit for solving optimization problems across diverse fields. These are a curated collection of 25 solved OR problems categorized by key problem types.
The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has ...
Operation Research calculators - Solve linear programming problems of Operations Research, step-by-step online. ... 1.1 Balanced Assignment Problem (Using Hungarian method) 1. A department has five employess with five jobs to be permormed. The time (in hours) each men will take to perform each job is given in the effectiveness matrix. ...
The answer to these questions is "Yes", there is a field in analytics called Operations Research (OR) which uses analytics to solve real-world problems. There are various methods in OR that can be ...
Abstract. This paper is concerned with a target assignment model of a probabilistic and nonlinear nature, but nevertheless one which is closely related to the "personnel-assignment" problem. It is shown here that, despite the apparent nonlinearities, it is possible to devise a linear programming formulation that will ordinarily provide a ...
This paper considers facility location problems in which a firm entering a market seeks to open facilities on a subset of candidate locations so as to maximize its expected market share, assuming that customers choose the available alternative that maximizes a random utility function.