OPERATIONS RESEARCH
Lesson 8. introduction and mathematical formulation.
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Assignment Problem: Meaning, Methods and Variations | Operations Research
After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.
Meaning of Assignment Problem:
An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.
The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.
Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.
Definition of Assignment Problem:
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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:
![mathematical formulation of assignment problem pdf mathematical formulation of assignment problem pdf](https://www.engineeringenotes.com/wp-content/uploads/2017/03/clip_image002_thumb-74.jpg)
Inner Product
Note: The true objective cost function only requires summing entries above the diagonal in the matrix comprised of elements
Since this matrix is symmetric with zeroes on the diagonal, dividing by 2 removes the double count of each element to give the correct cost value. See the Numerical Example section for an example of this note.
Optimization Problem
With all of this information, the QAP can be summarized as:
Computational Complexity
QAP belongs to the classification of problems known as NP-complete, thus being a computationally complex problem. QAP’s NP-completeness was proven by Sahni and Gonzalez in 1976, who states that of all combinatorial optimization problems, QAP is the “hardest of the hard”. [2]
Algorithmic Discussions
While an algorithm that can solve QAP in polynomial time is unlikely to exist, there are three primary methods for acquiring the optimal solution to a QAP problem:
- Dynamic Program
- Cutting Plane
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Branch and Bound Procedures
The third method has been proven to be the most effective in solving QAP, although when n > 15, QAP begins to become virtually unsolvable.
The Branch and Bound method was first proposed by Ailsa Land and Alison Doig in 1960 and is the most commonly used tool for solving NP-hard optimization problems.
A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before one lists all of the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and the branch is eliminated if it cannot produce a better solution than the best one found so far by the algorithm.
Linearizations
The first attempts to solve the QAP eliminated the quadratic term in the objective function of
in order to transform the problem into a (mixed) 0-1 linear program. The objective function is usually linearized by introducing new variables and new linear (and binary) constraints. Then existing methods for (mixed) linear integer programming (MILP) can be applied. The very large number of new variables and constraints, however, usually poses an obstacle for efficiently solving the resulting linear integer programs. MILP formulations provide LP relaxations of the problem which can be used to compute lower bounds.
Numerical Example
Qap with 3 facilities.
Permutation | Cost |
(123) | 91.4 |
99.8 | |
98.4 | |
86.5 | |
103.3 | |
90 |
Applications
Inter-plant transportation problem.
The QAP was first introduced by Koopmans and Beckmann to address how economic decisions could be made to optimize the transportation costs of goods between both manufacturing plants and locations. [1] Factoring in the location of each of the manufacturing plants as well as the volume of goods between locations to maximize revenue is what distinguishes this from other linear programming assignment problems like the Knapsack Problem.
The Backboard Wiring Problem
As the QAP is focused on minimizing the cost of traveling from one location to another, it is an ideal approach to determining the placement of components in many modern electronics. Leon Steinberg proposed a QAP solution to optimize the layout of elements on a blackboard by minimizing the total amount of wiring required. [4]
When defining the problem Steinberg states that we have a set of n elements
as well as a set of r points
In his paper he derives the below formula:
In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation, he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.
The initial placement of components is shown below:
![mathematical formulation of assignment problem pdf mathematical formulation of assignment problem pdf](https://optimization.cbe.cornell.edu/images/thumb/9/95/Initial_wire_length.jpg/412px-Initial_wire_length.jpg)
After the initial placement of elements, it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.
![mathematical formulation of assignment problem pdf mathematical formulation of assignment problem pdf](https://optimization.cbe.cornell.edu/images/thumb/1/1c/Final_wire_length.jpg/377px-Final_wire_length.jpg)
Hospital Layout
Building new hospitals was a common event in 1977 when Alealid N Elshafei wrote his paper on "Hospital Layouts as a Quadratic Assignment Problem". [5] With the high initial cost to construct the hospital and to staff it, it is important to ensure that it is operating as efficiently as possible. Elshafei's paper was commissioned to create an optimization formula to locate clinics within a building in such a way that minimizes the total distance that a patient travels within the hospital throughout the year. When doing a study of a major hospital in Cairo he determined that the Outpatient ward was acting as a bottleneck in the hospital and focused his efforts on optimizing the 17 departments there.
Elshafei identified the following QAP to determine where clinics should be placed:
For the Cairo hospital with 17 clinics, and one receiving and recording room bringing us to a total of 18 facilities. By running the above optimization Elshafei was able to get the total distance per year down to 11,281,887 from a distance of 13,973,298 based on the original hospital layout.
Exam Scheduling System
The scheduling system uses matrices for Exams, Time Slots, and Rooms with the goal of reducing the rate of schedule conflicts. To accomplish this goal, the “examination with the highest cross faculty student is been prioritized in the schedule after which the examination with the highest number of cross-program is considered and finally with the highest number of repeating student, at each stage group with the highest number of student are prioritized.” [6]
- ↑ 1.0 1.1 1.2 Koopmans, T., & Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25(1), 53-76. doi:10.2307/1907742
- ↑ 2.0 2.1 Quadratic Assignment Problem. (2020). Retrieved December 14, 2020, from https://neos-guide.org/content/quadratic-assignment-problem
- ↑ 3.0 3.1 3.2 Burkard, R. E., Çela, E., Pardalos, P. M., & Pitsoulis, L. S. (2013). The Quadratic Assignment Problem. https://www.opt.math.tugraz.at/~cela/papers/qap_bericht.pdf .
- ↑ 4.0 4.1 Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm. SIAM Review . 1961;3(1):37.
- ↑ 5.0 5.1 Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly (1970-1977) . 1977;28(1):167. doi:10.2307/300878
- ↑ 6.0 6.1 Muktar, D., & Ahmad, Z.M. (2014). Examination Scheduling System Based On Quadratic Assignment.
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Electrical Engineering and Systems Science > Systems and Control
Title: optimal assignment and motion control in two-class continuum swarms.
Abstract: We consider optimal swarm control problems where two different classes of agents are present. Continuum idealizations of large-scale swarms are used where the dynamics describe the evolution of the spatially-distributed densities of each agent class. The problem formulation we adopt is motivated by applications where agents of one class are assigned to agents of the other class, which we refer to as demand and resource agents respectively. Assignments have costs related to the distances between mutually assigned agents, and the overall cost of an assignment is quantified by a Wasserstein distance between the densities of the two agent classes. When agents can move, the assignment cost can decrease at the expense of a physical motion cost, and this tradeoff sets up a nonlinear, infinite-dimensional optimal control problem. We show that in one spatial dimension, this problem can be converted to an infinite-dimensional, but decoupled, linear-quadratic (LQ) tracking problem when expressed in terms of the respective quantile functions. Solutions are given in the general one-dimensional case, as well as in the special cases of constant and periodically time-varying demands.
Comments: | 12 pages, 7 figures |
Subjects: | Systems and Control (eess.SY); Optimization and Control (math.OC) |
classes: | 93C20 (Primary), 49K20, 93B17 (Secondary) |
Cite as: | [eess.SY] |
(or [eess.SY] for this version) |
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8.1 Introduction. An assignment problem is a particular case of transportation problem in which a number of operations are to be assigned to an equal number of operators, where each operator performs only one operation. The objective is to minimize overall cost or to maximize the overall profit for a given assignment schedule.
introduction to assignment problem matrix form of assignenmt problem mathematical formulation of an assignment problem difference between transportation problem and assigment problem assigment algorithm (or) hungarian method example of assigment problems question to answer mcq questions with answer k.bharathi,scsvmv. assignment problem 2 / 55
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.
problems, the Transportation and Assignment Problems. Both of these problems can be solved by the simplex algorithm, but the process would result in very large simplex tableaux and numerous simplex iterations. Because of the special characteristics of each problem, however, alternative solution methods requiring signi cantly less mathematical ...
Mathematical Formulation of the Assignment Problem: Note The optimum assignment schedule remains unaltered if we add or subtract a constant from all the elements of the row or column of the assignment cost matrix. Note If for an assignment problem all C ij > 0 then an assignment schedule (x ij) which satisfies ∑ C ij x ij = 0 must be optimal.
9. Do the same for some variants of assignment problems. 10. Give the name of an algorithm that can solve huge assignment problems that are well beyond the scope of Solver. Transportation problems were introduced in Section 3.5 and Section 3.6 did the same for assignment problems.
Step 3. Cover all the zeros of the matrix with the minimum number of horizontal or vertical lines. Step 4. Since the minimal number of lines is 3, an optimal assignment of zeros is possible and we are finished. Since the total cost for this assignment is 0, it must be. Step 3.
mathematical formulation are discussed and an example illustrating the design considerations is presented. Two major aspects in mathematical formulation involve the defining ... equations was dum to our understanding of the basic assignment problem,.7-and thus we were able to Simplify Lhe Conceptual problems associated with (he constralnts of ...
Transportation and Assignment problem and their optimal solutions. Inventory Control. LECTURE NOTES ON LINEAR PROGRAMMING Pre-requisites: Matrices and Vectors CHAPTER I Mathematical formulation of Linear Programming Problem Let us consider two real life situations to understand what we mean by a programming problem.
INTRODUCTION AND MATHEMATICAL FORMULATION. 8.1 Introduction. In earlier module, transportation problem and the technique of solving such a problem was discussed. In this lesson, the Assignment Problem, which is a special type of transportation problem, is introduced. Here the objective is to minimize the cost or time of completion of a number ...
This book provides a comprehensive treatment of assignment problems from their conceptual beginnings in the 1920s through present-day theoretical, algorithmic, and practical developments. The revised reprint provides details on a recent discovery related to one of Jacobi's results, new material on inverse assignment problems and quadratic ...
e BiQuadratic Assignment Problemgeneralization of the QAP is the BiQuadratic Assignment Problem, denoted BiQAP, which is essentially a quartic assignment problem with cost coefficients formed by the product. of two four-dimensional arrays. More specifica. ly, consider two n4 × n4. rrays= (fijkl) and D = (dmpst).
history of sophisticated mathematical techniques, many of which built on linear programming for generating a global view of large, complex optimization problems [5]. 2. Mathemtical LP Model for assignment problem Some linear programming models for the assignment problem is presented .It is assumed that the cost (or time) for every
the linear sum assignment problem (LSAP). They range from primal-dual combinatorial algorithms, to simplex-like methods, cost operation algorithms, fore. t algorithms, and relaxation ap-proaches. The worst-case complexity of the best sequential algorithms for the LSAP is.
Linear programming (LP) can be defined as' a mathematical method of determining an optimum assignment of interdependent activities, given the availability of resources'. LP models may be constructed for ... 2.2 Formulation of Assignment Problem[34] The decision problem becomes complicated when a number of resources are required to be ...
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 ...
Outline Problem background Mathematical formulation Algorithms Assignment Traveling Salesman Problem (TSP) One of the most studied problems in the area of optimization. The name is a mystery, but gives a clear connection to the applications of the problem. 1832, handbook Der Handlungsreisende for traveling salesmen. Stated as a mathematical problem in the 1930's:
Abstract: Assignment problem is an important problem in mathematics and is also discuss in real physical world. In this paper we attempt to introduce a new proposed approach for solving assignment problem with algorithm and solution steps. We examine a numerical example by using new method and compute by existing two methods. Also
tational burden, this method is not suitable. For solving assignment problem, we will discuss an efficient method which was developed by a Hungarian Mathematician D. Konig. 2.1.1 Mathematical Formulation of Assignment Problem Consider the problem of assignment of a company which has n machines of different
Introduction. The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957, is a mathematical optimization module created to describe the location of invisible economic activities.An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics.
models the source is connected to one or more of destination. The most common. method to solve assignment models is the Hungarian metho d. In this paper. introduced another method to solve ...
The optimum assignment schedule remains unaltered if we add or subtract a constant from all the elements of the row or column of the assignment cost matrix. Note. If for an assignment problem all Cij > 0 then an assignment schedule (xij) which satisfies ∑ Cij xij = 0 must be optimal. Consider the problem of assigning n jobs to n machines (one ...
A mathematical formulation of the problem. Let x and y represent the number of cabinets of kind 1 and 2 that he must produce. Non-negative constraints are non-negative limitations. The company is allowed to invest 540 hours of labour and must build up to 50 cabinets. Hence, 15x + 9y <= 540. x + y <= 50.
The problem formulation we adopt is motivated by applications where agents of one class are assigned to agents of the other class, which we refer to as demand and resource agents respectively. Assignments have costs related to the distances between mutually assigned agents, and the overall cost of an assignment is quantified by a Wasserstein ...