formulation of assignment problem

Operations Research - Definition and formulation of Assignment Problem | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research, definition and formulation of assignment problem.

Definition and formulation

Consider the problem of assigning n jobs to n machines (one job to one machine). Let C ij be the cost of assigning i th job to the j th machine and x ij represents the assignment of i th job to the j th machine.

 x ij is missing in any cell means that no assignment is made between the pair of job and machine.( i.e ) x ij = 0.

x ij presents in any cell means that an assignment is made their.In such cases x ij = 1

The assignment model can be written in LPP as follows

Subject to the constrains

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.

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.

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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:

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

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Assignment Problem: Linear Programming

The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.

In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.

It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.

"A mathematician is a device for turning coffee into theorems." -Paul Erdos

Formulation of an assignment problem

Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:

The structure of an assignment problem is identical to that of a transportation problem.

To formulate the assignment problem in mathematical programming terms , we define the activity variables as

for i = 1, 2, ..., n and j = 1, 2, ..., n

In the above table, c ij is the cost of performing jth job by ith worker.

Generalized Form of an Assignment Problem

The optimization model is

Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn

subject to x i1 + x i2 +..........+ x in = 1          i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1          j = 1, 2,......., n

x ij = 0 or 1

In Σ Sigma notation

x ij = 0 or 1 for all i and j

An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.

Assumptions in Assignment Problem

• Number of jobs is equal to the number of machines or persons.
• Each man or machine is assigned only one job.
• Each man or machine is independently capable of handling any job to be done.
• Assigning criteria is clearly specified (minimizing cost or maximizing profit).

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Assignment problem in linear programming : introduction and assignment model.

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Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

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• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
• Linear Programming: Applications, Definitions and Problems

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Assignment problem

1 formulation of the problem, 2 variants of the problem, 3 algorithms for solving the problem, 4 references.

Suppose that there are [math]n[/math] agents and [math]m[/math] tasks, which can be distributed between these agents. Only one task can be assigned to each agent, and each task can be assigned to only one agent. The cost of assignment of the [math]i[/math] -th task to the [math]j[/math] -th agent is [math]c(i, j)[/math] . If [math]c(i, j) = \infty[/math] , then the [math]i[/math] -th task cannot be assigned to the [math]j[/math] -th agent.

The assignment problem : find a feasible set of assignments [math]A = \{ (i_1, j_1), \ldots, (i_k, j_k) \}[/math] , [math]k = \min \{m, n\}[/math] having the maximum total cost:

If [math]m = n[/math] , then we say of the linear assignment problem : each agent is assigned to perform exactly one task, and each task is assigned to exactly one agent.

In the case of unit weights, we have to find a maximum matching in a bipartite graph, and the problem reduces to assigning as much tasks as possible.

• The Hungarian Method [1] [2] [3] for the linear problem. The complexity is [math]O(n^4)[/math] (and can be reduced [4] to [math]O(n^3)[/math] );
• the auction algorithm [5] [6] ;
• the Hopcroft-Karp algorithm [7] for the problem with unit weights. The complexity is [math]O(m \sqrt{n})[/math] .
• ↑ Kuhn, H W. “The Hungarian Method for the Assignment Problem.” Naval Research Logistics Quarterly 2, no. 1 (March 1955): 83–97. doi:10.1002/nav.3800020109.
• ↑ Kuhn, H W. “Variants of the Hungarian Method for Assignment Problems.” Naval Research Logistics Quarterly 3, no. 4 (December 1956): 253–58. doi:10.1002/nav.3800030404.
• ↑ Munkres, James. “Algorithms for the Assignment and Transportation Problems.” Journal of the Society for Industrial and Applied Mathematics 5, no. 1 (March 1957): 32–38. doi:10.1137/0105003.
• ↑ Tomizawa, N. “On Some Techniques Useful for Solution of Transportation Network Problems.” Networks 1, no. 2 (1971): 173–94. doi:10.1002/net.3230010206.
• ↑ Bertsekas, Dimitri P. “Auction Algorithms for Network Flow Problems: a Tutorial Introduction.” Computational Optimization and Applications 1 (1992): 7–66.
• ↑ Zavlanos, Michael M, Leonid Spesivtsev, and George J Pappas. “A Distributed Auction Algorithm for the Assignment Problem,” Proceedings of IEEE CDC'08, 1212–17, IEEE, 2008. doi:10.1109/CDC.2008.4739098.
• ↑ Hopcroft, John E, and Richard M Karp. “An $N^{5/2} $ Algorithm for Maximum Matchings in Bipartite Graphs.” SIAM Journal on Computing 2, no. 4 (1973): 225–31. doi:10.1137/0202019.
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Assignment Problem

• Feb 20, 2024

The assignment problem is a special case of transportation problem.

Suppose there are $n$ jobs to be performed and $n$ persons are available for these jobs. Assume that each person can do each job at a time, though with varying degree of efficiency.

Let $c_{ij}$ be the cost if the $i^{th}$ person is assigned the $j^{th}$ job. Then the problem is to find an assignment so that the total cost of performing all jobs is minimum. (i.e., which job should be assigned to which person with minimum cost). Such problem is called an Assignment Problem (AP).

The tabular form of the assignment problem is as follows

The above table is called the $n\times n$ cost-matrix, where $c_{ij}$ are real numbers.

Thus the objective in the Assignment Problem is to assign a number of jobs to the equal number of persons at a minimum cost or maximum profit. e.g. assigning men to offices, classes to rooms, drivers to trucks, problems to research teams etc.

Mathematical Formulation of AP

Mathematically, the AP can be stated as $$ \begin{equation}\label{eq2.4} \min z = \sum_{i=1}^n\sum_{j=1}^n x_{ij} c_{ij}. \end{equation} $$ subject to $$ \begin{equation}\label{eq2.5} \sum_{j=1}^n x_{ij} =1,; \text{ for } i=1,2,\ldots, n \end{equation} $$

$$ \begin{equation}\label{eq2.6} \sum_{i=1}^n x_{ij} =1,; \text{ for } j=1,2,\ldots,n \end{equation} $$ where $$ \begin{equation*} x_{ij}=\left{ \begin{array}{ll} 1, & \hbox{if $i^{th}$ person is assigned $j^{th}$ job;} \ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$ Constraint \eqref{eq2.5} indicate that one job is done by $i^{th}$ person $i=1,2,\ldots, n$ and constraint \eqref{eq2.6} indicate that one person should be assigned $j^{th}$ job $j=1,2,\ldots, n$.

It may be observed from the above formulation that AP is a special type of Linear programming problem.

Unbalanced Assignment Problem

If the cost matrix of an assignment problem is not a square matrix, the assignment problem is called an Unbalanced Assignment Problem . In such a case, add dummy row or dummy column with zero cost in the cost matrix so as to form a square matrix. Then apply the usual assignment method to find the optimal solution.

Maximal Assignment Problem

Sometimes, the assignment problem deals with the maximization of an objective function instead of minimization.. For example, it may be required to assign persons to jobs in such a way that the expected profit is maximum. In such a case, first convert the problem of maximization to minimization and apply the usual procedure of assignment.

The assignment problem of maximization type can be converted to minimization type by subtracting all the elements of the given profit matrix from the largest element.

Restrictions on Assignment

Sometimes due to technical or other difficulties do not permit the assignment of a particular facility to a particular job. In such a situation, the difficulty can be overcome by assigning a very high cost (say, infinity i.e. $\infty$) to the corresponding cell.

Because of assigning an infinite penalty to such a cell the activity will be automatically excluded from the optimal solution. Then apply the usual procedure to find the optimal assignment.

Linear Assignment Problems and Extensions

Cite this chapter.

• Rainer E. Burkard 4 &
• Eranda Çela 4  

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Assignment problems deal with the question how to assign n items (e.g. jobs) to n machines (or workers) in the best possible way. They consist of two components: the assignment as underlying combinatorial structure and an objective function modeling the ”best way”.

This research has been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.

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Burkard, R.E., Çela, E. (1999). Linear Assignment Problems and Extensions. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3023-4_2

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Formulation of Assignment problem as integer programming

We need to maintain as quickly as possible a complex system. In particular, we need to replace six of its components $\{P_1,\ldots,P_6\}$ . We have three 3D printers $\{M_1,M_2,M_3\}$ which we can use to fabricate the six components. The following table/matrix states how long it takes (in minutes) the $i$ th printer to print the $j$ th component:

\begin{array}{ccccccc} \hline & P_1& P_2&P_3&P_4&P_5&P_6 \\ \hline M_1 & 23 & 42 & 12 & 32 & 47 & 60\\ M_2 & 25& 37& 13& 37& 51& 64\\ M_3 & 27 &51 &15& 41 &57 &55\\ \hline \end{array}

The complex system will work again only when all the components have been printed. Clearly more components can (and have to be) assigned to single machines and they are made in a sequence, one after the other, and 3D printers can work in parallel. However, you have only two operators and therefore you can only use two machines. How to formulate the problem as a linear (but combinatorial) optimization problem to allocate components to (two out of three) 3D printers so that the maintenance time is minimized.

So far I have tried the following, but I am not quite sure (Please I need help if I was mistaken):

Let $x_{ij}= 1$ if machine $i$ is assigned to component $j$ , $0$ otherwise. The model is \begin{align}\min&\quad23x_{11}+42x_{12}+...+55x_{36}\\\text{s.t.}&\quad x_{11}+x_{12}+x_{13}+x_{14}+x_{15}+x_{16} = 2\\&\quad x_{21}+x_{22}+x_{23}+x_{24}+x_{25}+x_{26} = 2\\&\quad x_{31}+x_{32}+x_{33}+x_{34}+x_{35}+x_{36} = 2\\&\quad x_{11}+x_{21}+x_{31} \ge 1\\&\quad x_{12}+x_{22}+x_{32} \ge 1\\&\quad x_{13}+x_{23}+x_{33} \ge 1\\&\quad x_{14}+x_{24}+x_{34} \ge 1\\&\quad x_{15}+x_{25}+x_{35} \ge 1\\&\quad x_{16}+x_{26}+x_{36} \ge 1\\&\quad x_{ij}\,\,\text{is binary}.\end{align}

• integer-programming
• combinatorial-optimization

• 1 $\begingroup$ "The complex system will work again only when all the components have been printed." Does that mean you'd like the system to end as soon as possible? (Thinking of the objective function here). Besides this, I think this could better be formulated as a scheduling problem on parallel machines, with the additional constraint that only 2 of the 3 machines are used. $\endgroup$ –  dhasson Commented Jul 2, 2020 at 13:37
• $\begingroup$ The Generalized Assignment Problem can serve as inspiration to continue, it's the same you are doing but change the first 3 constraints where every machine is made to print 2 components. First of all, the objective function is total working time of the machines. Instead you want to minimize the makespan (maximum time to finish all jobs), so that system can work again as soon as possible. Second, you may add binary variables $z_i = 1$ if machine $i$ is used, with additional constraints for $\sum_i z_i = 2$ and linking $x$ and $z$. $\endgroup$ –  dhasson Commented Jul 2, 2020 at 15:30
• 1 $\begingroup$ Are you assuming that one machine will never be used, or that operators will move among machines such that all machines might be used at some point, but never more than two running at any given time? $\endgroup$ –  prubin ♦ Commented Jul 2, 2020 at 15:57
• $\begingroup$ @user3752, as dhasson mentioned, it sounds like a parallel machine scheduling problem. Would you see this link? $\endgroup$ –  A.Omidi Commented Jul 3, 2020 at 10:41

Since you mention that you want to operate two out of three machines, it boils down to a problem where you first pick two machines and then perform a standard $R2||C_{\max}$ parallel machine scheduling problem with the two machines that you selected. Such problems are very suitable for dynamic programming/column generation approaches, but your instance is so small that a IP will work fine. And since you ask for an IP, let us consider a straightforward way to model it.

For a formulation, consider the following decision variables: $$y_j = \left\{ \begin{array}{ll} 1 & \mbox{ if } M_j \mbox{ is being operated } \\ 0 & \mbox{otherwise} \end{array} \right.$$ and $$x_{ij} = \left\{ \begin{array}{ll} 1 & \mbox{ if } P_i \mbox{ is made on } M_j \\ 0 & \mbox{otherwise} \end{array}\right.$$ Also, let's assume that $a_{ij}$ is the time needed to produce $P_i$ on $M_j$ .

Now the following IP can be formulated: $$\begin{array}{llll} \min & z \\ \mbox{s.t.} & \sum_{j} y_j & \leq K \\ & \sum_{j} x_{ij} & = 1 & \forall i \\ & x_{ij} & \leq y_j & \forall i \forall j \\ & \sum_i a_{ij} x_{ij} & \leq z & \forall j \\ & x_{ij} \in \{0,1\} & & \forall i \forall j \\ & y_j \in \{0,1\} & & \forall j \\ & z \in \mathbb{R} \end{array}$$ Where the objective variable $z$ represents the makespan, the first constraint states that at most $K$ machines can be used (in your instance, $K=2$ , i.e. $K$ is the number of operators), the second constraint states that each $P_i$ must be executed on exactly one $M_j$ , the third constraint states that a $P_i$ can only be produced on a $M_j$ if that $M_j$ is being operated and the fourth constraint states that the makespan $z$ should be at least the time spent on each individual machine.

If you want to allow operators to switch between machines, the problem becomes more complicated, as you need to make sure that both an operator and a machine is available when you produce an item; you probably need to keep track of both the operator and the machine resources in your model and define constraints to avoid conflicts for which you may or may not need to determine an order as well.

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OPERATIONS RESEARCH

Lesson 8. introduction and mathematical formulation.

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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Quadratic assignment problem

Author: Thomas Kueny, Eric Miller, Natasha Rice, Joseph Szczerba, David Wittmann (SysEn 5800 Fall 2020)

• 1 Introduction
• 2.1 Koopmans-Beckman Mathematical Formulation
• 2.2.1 Parameters
• 2.3.1 Optimization Problem
• 2.4 Computational Complexity
• 2.5 Algorithmic Discussions
• 2.6 Branch and Bound Procedures
• 2.7 Linearizations
• 3.1 QAP with 3 Facilities
• 4.1 Inter-plant Transportation Problem
• 4.2 The Backboard Wiring Problem
• 4.3 Hospital Layout
• 4.4 Exam Scheduling System
• 5 Conclusion
• 6 References

Introduction

The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957 [1] , 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. It has been used to optimize backboards, inter-plant transportation, hospital transportation, exam scheduling, along with many other applications not described within this page.

Theory, Methodology, and/or Algorithmic Discussions

Koopmans-beckman mathematical formulation.

Economists Koopmans and Beckman began their investigation of the QAP to ascertain the optimal method of locating important economic resources in a given area. The Koopmans-Beckman formulation of the QAP aims to achieve the objective of assigning facilities to locations in order to minimize the overall cost. Below is the Koopmans-Beckman formulation of the QAP as described by neos-guide.org.

Quadratic Assignment Problem Formulation

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

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.

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:

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.

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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Leveraging Computational Thinking in the Era of Generative AI

What is the value of computational thinking in the era of GenAI?

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Computational thinking was first introduced by Papert (1980), and a quarter of a century later, was illuminated and elaborated by Wing (2006). In August 2022, the concept of computational thinking was revisited by Mike et al., this time from the data science perspective (Mike et al., 2022). A few months later, in November 2022, OpenAI launched ChatGPT, which was the first signal of a new storm—Generative AI, or GenAI—that has changed many domains of our lives since then. Specifically, in the context of our discussion, GenAI applications re-raised the discussion about the necessity of computational thinking in general; and in particular, in the context of CS and data science as well as CS education.

Indeed, is computational thinking relevant in the era of GenAI? After all, if the technology does the programming work and the data analysis, what is the added value of computational thinking? Herein, we argue that not only does the importance of computational thinking increase, but moreover, as everyone uses GenAI, computational thinking becomes an essential skill for all, as it is manifests in a new form—prompt engineering—that all users of GenAI are required to master.

Computational Thinking

Computational thinking is shaped in terms of the questions “What can humans do better than computers? And what can computers do better than humans?” (Wing, 2006), which a flood of papers have sought to answer and define which human cognitive and social skills are required for problem-solving processes in general, and for solving computational problems in particular. Today, the following skills are recognized as key components of computational thinking: problem formulation; problem decomposition; organization and logical analysis of data; data representation using models and simulations; abstraction; suggestion and assessment of multiple solutions to a given problem; implementation of the chosen solution; and generalization (e.g., Wing, 2010). Computational thinking skills are recognized today as important not only in the context of CS, but also as important skills for everyone in the 21 st century, and can be applied in various contexts (Günbatar, 2019; Wing, 2017).

Prompt Engineering

Prompt engineering refers to the “systematic practice of constructing prompts to improve the generated output of a generative model” (Dang et al., 2022). Prompt engineering is a new field of research, asking what the best practices are for shaping prompts for GenAI tasks in general, and for programming tasks in particular (Oppenlaender et al., 2023). From an educational perspective, researchers ask how to teach prompt engineering, and how to evaluate the results of such teaching and learning processes (Denny et al., 2023, 2024). Currently, for the majority of end users, writing effective prompts is largely a trial-and-error process (Dang et al., 2022). Massive online open courses (MOOCs) suppliers, such as Coursera, now offer courses in prompt engineering ( https://www.coursera.org/learn/prompt-engineering-for-web-developers) that include topics such as breaking tasks into smaller steps, prompt iterativity, generating code with pseudocode, and test and edge cases. Figure 1 presents ChatGPT’s advice for the best practices for prompting.

Figure 1. Best practices for prompting: ChatGPT’s answer

Interactions between Computational Thinking and Prompt Engineering in Computer Science Education

GenAI apps are becoming an inevitable technology in the field of CS education (Armoni and Hazzan, in press). Students use them to learn, debug, and solve short coding exercises, wherein these apps excel. We suggest GenAI apps facilitate new opportunities for learning and teaching (Denny et al., 2022; Yilmaz and Karaoglan Yilmaz, 2023), and that therein, the importance of computational thinking increases due to its strong interaction with prompt engineering, as illustrated in Figure 2.

It has been found that learning how to prompt enhances the learning of several computational thinking skills. Specifically, students who use GenAI should describe clearly the steps needed to solve a problem (Denny et al., 2023), focus on the algorithm to determine its subprograms, and combine the sub-solutions into the entire solution (Yilmaz and Karaoglan Yilmaz, 2023). Efficient prompting can accordingly be viewed as writing pseudocode (Denny et al., 2022). After all, shaping prompts to produce specific results is a problem-solving process, and hence requires computational thinking.

To test the degree of resemblance between computational thinking and prompt engineering, we replaced “computational thinking” with “prompt engineering” in Wing’s paper (Wing, 2006), with the following results:

• Prompt engineering “is interpreting code as data and data as code.” (p. 33) as the prompt data should be treated as code, since efficient prompting is writing pseudocode; on the other hand, the prompt pseudocode is data for the LLM.
• Prompt engineering “is using abstraction and decomposition when attacking a large complex task.” (p. 33) as to implement large complex systems, the task should be broken down into small sub-tasks, each of them then articulated by a prompt.
• Prompt engineering “is having the confidence [that] we can safely use, modify, and influence a large complex system without understanding its every detail.” (p. 33), as using pseudocode, instead of a specific programming language, enables to perform [sic] the task without the need to master the details of a specific language.
• Finally, prompt engineering “is search, search, and more search, resulting in a list of Web pages, a strategy for winning a game, or a counterexample.” (p. 34) as efficient prompting involves experimenting with different prompts in learning processes and with iterative refinements.

Undergraduate CS Students’ Expressions of the Interrelationships between Computational Thinking and Prompt Engineering

Yael Erez, co-author of this blog, teaches an Introduction to Computer Science (CS1) course to CS undergraduate students. In the Fall 2023 semester, several tasks that involved problem-solving with the help of GenAI apps, such as ChatGPT, were assigned to the students. Two of these assignments are detailed in brief in Table 1.

Data collected using these assignments demonstrate the increased importance of computational thinking for prompt engineering:

• When the students fix the incorrect cyclic shift with the help of ChatGPT (Assignment #1), not only does assessment of the app solutions enhance their computational thinking skills, such as critical thinking and program comprehension, but the computational thinking skill of problem formulation is required for prompt engineering.
• When the students use the app to implement the contact list (Assignment #2), not only do the complexity and length of the implementation compel them, as part of the prompt engineering task, to decompose the problem into sub-problems, thereby enhancing their learning of the computational thinking skill of decomposition, but computational thinking skills such as successive refinement are required for prompt engineering.

While the above examples do not require the students to write code on their own, suggesting that GenAI apps may eliminate the need to master a specific language syntax, these tasks underscore the importance of students’ computational thinking, as the following data illustrates.

After each assignment submission, the students were asked to answer a questionnaire containing an open question: “Write down three new ideas that you learned from solving the question.” Many of the students’ answers to this question referred to the skills needed for solving programming problems with ChatGPT, demonstrating the strong interrelationships between computational thinking and prompt engineering.

One answer to the above-mentioned question was “Formulating the problem explicitly,” and another similar answer was “Accuracy in formulating requests,” indicating the importance of the computational thinking skill of problem formulation, and the importance of clarity and specificity for prompt engineering. Other students answered, “This time, unlike previous times, I sent it a few details each time and was able to lead it to solving the problem faster,” and “Reducing the problem to small tasks,” describing the problem decomposition phase of computational thinking and the iterative refinements of prompt engineering. Another student referred to system design and implementation, describing the process of “First understand the problem, and then direct the tools to solve the problem.” The computational thinking skills of critical thinking and solution evaluation required for prompt engineering were cited by another student, who wrote, “I knew what ChatGPT’s ability is, it doesn’t always answer correctly.”

Effective prompt engineering requires computational thinking skills such as problem formulation, problem decomposition, data organization and analysis, and suggestion and assessment of multiple solutions to a given problem. Accordingly, not only does experiencing prompt engineering improve our computational thinking skills, but computational thinking is required for prompt engineering.

These interactions between prompt engineering and computational thinking underscore the importance of computational thinking in the era of GenAI. While GenAI apps may facilitate programming tasks, even for complex systems, thereby eliminating the need to master a specific language syntax, they still lack computational thinking. Therefore, humans’ computational thinking is required in the GenAI era.

While we introduced the interactions between computational thinking and prompt engineering mainly in the context of CS, GenAI tools are available to all, and are widely used in fields other than CS, and by humans who are not computer scientists. For example, Koby Mike, co-author of this blog, teaches data science to social science graduate students. In these courses, following the incorporation of ChatGPT and prompt engineering in the course, while the volume of Python programming in the curriculum dramatically decreased, the volume of computational thinking expanded. This finding is aligned with Wing’s statement “Computational thinking is a fundamental skill for everyone, not just for computer scientists.” Continuing the exercise of replacing “computational thinking” with “prompt engineering” in Wing’s paper, evoking a discussion about computational thinking (Wing, 2006), the wide use of GenAI tools suggests that prompt engineering is a fundamental skill for everyone, not just for computer scientists.

When prompting efficiently, computational thinking is applied and improved, even for everyday tasks such as planning a trip, solving riddles, or knowledge expansion. If Wing wrote her viewpoint today, it would likely include the following: Computational thinking is required for having effective conversations with LLMs to solve problems efficiently and quickly. Hence Wing’s message to “make computational thinking commonplace” (Wing, 2006) has become a reality.

Armoni, Y. and Hazzan, O. (in press). The Inevitability of AI Technology in Education: Futurism Perspectives on Education in 10-20 years .

Dang, H., Mecke, L., Lehmann, F., Goller, S., and Buschek, D. (2022). How to Prompt? Opportunities and Challenges of Zero- and Few-Shot Learning for Human-AI Interaction in Creative Applications of Generative Models .

Denny, P., Becker, B. A., Leinonen, J., and Prather, J. (2023). Chat Overflow: Artificially Intelligent Models for Computing Education – renAIssance or apocAIypse? Proceedings of the 2023 Conference on Innovation and Technology in Computer Science Education V. 1 , 3–4. https://doi.org/10.1145/3587102.3588773

Denny, P., Kumar, V., and Giacaman, N. (2022). Conversing with Copilot: Exploring Prompt Engineering for Solving CS1 Problems Using Natural Language (arXiv:2210.15157). arXiv. http://arxiv.org/abs/2210.15157

Denny, P., Leinonen, J., Prather, J., Luxton-Reilly, A., Amarouche, T., Becker, B. A., and Reeves, B. N. (2024). Prompt Problems: A New Programming Exercise for the Generative AI Era. Proceedings of the 55 th ACM Technical Symposium on Computer Science Education V. 1 , 296–302. https://doi.org/10.1145/3626252.3630909

Günbatar, M. S. (2019). Computational thinking within the context of professional life: Change in CT skill from the viewpoint of teachers. Education and Information Technologies , 24 (5), 2629–2652. https://doi.org/10.1007/s10639-019-09919-x

Mike, K., Ragonis, N., Rosenberg-Kima, R. B., and Hazzan, O. (2022). Computational Thinking in the Era of Data Science. Communications , 65 (8), 33–35. https://doi.org/10.1145/3545109

Oppenlaender, J., Linder, R., and Silvennoinen, J. (2023). Prompting AI Art: An Investigation into the Creative Skill of Prompt Engineering .

Papert, S. (1980). Mindstorms : Children, computers, and powerful ideas . Basic Books.

Wing, J. M. (2006). Computational Thinking. Communications , 49 (3), 33–35. https://doi.org/10.1145/1118178.1118215

Wing, J. M. (2010). Computational Thinking: What and Why? http://www.cs.cmu.edu/~CompThink/resources/TheLinkWing.pdf

Yilmaz, R., and Karaoglan Yilmaz, F. G. (2023). The Effect of Generative Artificial Intelligence (AI)-based Tool Use on Students’ Computational Thinking Skills, Programming Self-Efficacy and Motivation. Computers and Education: Artificial Intelligence , 4 , 100147. https://doi.org/10.1016/j.caeai.2023.100147

Yael Erez is a lecturer in the Department of Computer Science of the Technion—Israel Institute of Technology, and a faculty member in the Department of Electrical Engineering of the Braude College of Engineering. She has BSc and MSc degrees in Electrical Engineering from the Technion, and is currently a Ph.D. student in the Technion’s Faculty of Education in Science and Technology.

Koby Mike is a postdoctoral fellow in the Gender Studies program at Bar-Ilan University; he earned BSc and MSc degrees in Electrical Engineering from Tel Aviv University, and a Ph.D. in Education in Science and Technology from the Technion.

Orit Hazzan is a professor at the Technion’s Faculty of Education in Science and Technology. Her research focuses on computer science, software engineering, and data science education; for additional details, see https://orithazzan.net.technion.ac.il/ . 

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