A NEW MODIFIED APPROACH USING
BEST CANDIDATES METHOD
FOR SOLVING LINEAR ASSIGNMENT
PROBLEMS
ABDALLAH A. HLAYEL¹ , KHULOOD ABU MARIA²
Department of Computer Information System, Faculty of Science and Information Technology,
Al-zaytoonah University of Jordan, P.O. Box 130 Amman 11733 Jordan ¹ hlayel@zuj.edu.jo, ² khulood@zuj.edu.jo
Abstract:
There is an increasing awareness among modern business, engineers, managers, and planners to design and operate their systems even to minimize cost, or to maximum profit (maximum efficiency/business benefits). Accordingly, significant work has been done on business (specially, on manufacturing system) operations for total demand and on the optimal allocation of resources available. Linear Assignment Problems (LAP) is one of the most important optimization problem solving methods (in Operation Research) support this problem. This paper proposes a new modifications on the Best Candidates Method (BCM) and compares the proposed method with other Linear Programming (LP) methods in solving Linear Assignment Problems (LAP). In general, there are many development approaches for LAP to reach the optimal solution through minimize or maximize the objective function. Each problem solving technique (method) has its own time complexity, and solution optimality. Some methods can be used successfully when dealing with small scale problems, while they considered as an inefficient method when solving large scale problems. Performance of different LAP problem solving methods is presented because of their wide used in different area of optimization problems. We introduce our new modifications on BCM in solving LAP problems which has significant improvements in the number of combinations and searching strategy.
Keywords:Operation Research; Linear Programming; Linear Assignment Problems; Optimization Problems; Hungarian Method, Best Candidates Method.
1. Introduction
Operations Research (O.R.) is the discipline of science helping to apply advanced analytical approaches to help make better decisions [Agrawal (2010), Operational Research (2013)].
Operations research gives executives the power to make more effective decisions and build more productive systems, by using techniques such as mathematical modeling to analyze complex situations based on complete data, consideration of all available options, careful predictions of outcomes and estimates of risk, and the latest decision tools and techniques. Operations Research professionals draw upon the latest analytical technologies, including simulation, optimization, or probability and statistics [Winston (2003)] [Operational Research (2013)]. Linear programming (LP) is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions [Purple Math (2013)]. In this context, it refers to a planning process that allocates resources labor, materials, machines, and capital in the best possible optimal way so that costs are minimized or profits are maximized. These resources are known as decision variables. The criterion for selecting the best values of the decision variables is known as objective function. The Limitation of resources availability is known as a constraint set [Purple Math (2013)] [Bazaraa (1990)].
LP is part of a very important area of mathematics called "optimization techniques". This field of study (or at least the applied results of it) is used every day in the organization and allocation of resources. These systems may have hundreds of variables or more [Purple Math (2013)] [Bazaraa (1990)].
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problem increases as the total number of possible combinations depends on the number of agents and tasks (more n! combinations). Hence, the use of this method is not feasible in real world's cases especially for large scale problems which would have a high complexity when implemented this approach.
3.2 Simplex Method
It is an algebraic method for solving LP problems. Where the assignment problem can be formulated as a linear programming problem and be solved using simplex method. However, the formulation of assignment problems can be a tedious job and not efficient especially for large scale problems. It is not difficult to implement, but sometimes it is difficult to explain how the method works. It speeds up the enumeration method by moving step-by-step from one basic feasible solution to another with higher profit until the best is found [Bazaraa (1990)] [Reep (1998)].
3.3 Transportation Method
The assignment problem is a special case of the transportation problem hence transportation method of solution can be used to find optimum allocation. However there are many solution methods that operate on LAP and transportation problems such as [Hlayel (2012)] [Imam (2009)]:
1. Northwest Corner Method. 2. Minimum Cost Method. 3. Genetic Algorithm.
4. Vogel’s Approximation Method. 5. Row Minimum Method.
6. Column Minimum Method. 7. Hungarian Method.
All of these methods are different of their starting basic solution that means not always gets the best solution [Hlayel (2012)] [Hlayel, Alia (2012)]. The Hungarian algorithm is the most used algorithm for solving LP problems, and it can solve LAP.
3.3.1 Hungarian Method
It is used to solve Linear Assignment Problem. It needs to build a matrix for the costs, profit or time of the agents or workers are represented in the rows doing the tasks or jobs that are expressed in the columns. In the case of cost matrix, the element in the i-th row and j-th column represents the cost of assigning the j-th job to the
i-th worker. Hence, Hungarian method based on the principle that if a constant is added to the elements of cost matrix, the optimum solution of the assignment problem is the same as the original problem. Original cost matrix is reduced to another cost matrix by adding a constant value to the elements of rows and columns of cost matrix where the total completion time or total cost of an assignment is zero. This assignment is also referred as the optimum solution since the optimum solution remains unchanged after the reduction. However, the Hungarian Method works as following [Winston (2003)] [Cao (2008)]:
1) Subtract the smallest entry in each row from all the entries of its row. 2) Subtract the smallest entry in each column from all the entries of its column.
3) Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used.
4) Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished. (ii) If the minimum number of covering lines is less than n, an optimal assignment of zeros is not yet possible. In that case, proceed to Step 5.
5) Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to Step 3.
Finally, it is important to signify that the time complexity of the original algorithm was O (n4), and it can be modified to achieve an O (n3) running time.
4. Best Candidates Method with New Modifications
Our method is based on determination of the best candidates then elimination the unwanted one in order to minimize the number of solution combinations to decide the optimal solution [Hlayel (2012)]. However, we can notice that the solution approach using this method as one of LAP methods is divide into two phases. We will describe each phase and clarify the new modifications.
Step1: Prepare the matrix. If the matrix is unbalanced, we balance it and we would not use the added row or column candidates in our solution proccess.
Step2: Determination of the best candidate, it is used for minimization problems (minimum cost) or maximization problem (maximim profit): Elect the best two candidates in each row, if the candidate repeated more than one times elect it also. Check the columns that not have candidates and elect one candidate from them, if the candidate repeated more than one time elect it also.
The second phase, will introduce the following steps:
a. At the end of phase one an index matrix is produced that shows the postion for each candidate. b. Find the direct combinations and calculating the cost for each.
c. Check the unused candidates, by finding the possible candidates for them then calculat the cost for each.
d. Find the optimal solution according to the objective function.
The most important modification in our modified method are:
First, building the index martix that shows and facilitates dealing with combinations.
Second, our method deals with unused elected candidates and take them into acount nervertheless from where we started our serach.
Third, the number of combinations comparing to other bench mark methods is acceptable (small), so the computational time is small.
4.1 Modified-BCM Method Phases and Business Problem Implementation
Problem: Consider the problem of assigning four jobs to four persons . The assignment profit are given as follows in table 1:
Table 1: Person-Job Assignment Profit Matrix.
1 2 3 4
A 3 5 3 6
B 5 7 4 4
C 3 4 5 2
D 4 5 3 5
Phase1: Elect candidates.
Step1: The matrix is balanced, where the number of columns equal to the number of rows as shown in (table 2).
Table 2: Person-Job Assignment Profit Matrix - after Balance.
1 2 3 4
A 3 5 3 6
B 5 7 4 4
C 3 4 5 2
D 4 5 3 5
Step2: Elect the best candidates as shown in table 3.
Table 3: Best Candidates Determination Matrix.
1 2 3 4
A 3 5 3 6
B 5 7 4 4
C 3 4 5 2
Phase2: Obtain the BCM combinations.
The second phase is work as following:
a. Draw the following index matrix that shows the postion for each candidate as clear in table 4.From the table we obtain the soultion set {A2,A4,B1,B2,C2,C3,D2,D4}
Table 4: Best Candidates Combinations Postion Matrix.
1 2 3 4
A - A2 - A4
B B1 B2 - -
C - C2 C3 -
D - D2 - D4
b. Find the direct combinations which are all the candiadates from the solution set, then calculat the cost for each:
Combination1: {B1,A2,C3,D4} where: 5+5+5+5=20.
Combination2: {B1,D2,C3,A4} where: 5+5+5+6=21.
c. Check the unused candidates in the solution set which are {B2,C2}, then find the possible combinations for them , and calculat the cost for each:
Combination3: {B2,C3,D4} then we add to them A1 and become {A1,B2,C3,D4}: 3+7+5+5=20.
Combination4: {B2,C3,A4} then we add to them D1 and become {D1,B2,C3,A4}: 4+7+5+6=22.
Combination5: {B1,C2,A4} then we add to them D3 and become {B1,C2,D3,A4}: 5+4+3+6=18.
Combination6: {B1,C2,D4} then we add tothem A3 and become {B1,C2,A3,D4}: 5+4+3+5=17.
d. Find the optimal solution according to the objective function (maximun of minimum cost):
In our case it is combination number 4 (modified-BCM solution).
5. Results and Aanalysis
There are many methods for solving Linear Assignment Problems (LAP) which try to reach the optimal solution, but they are differ in the following main characteristics:
1) Solution Optimality: is determined to be the best solution from all the feasible solutions. Where by, the optimal solution depends on the right choosing of candidates that form the combination with the optimal solution . Most of methods not always reach the optimal solution because there is no flexibility of choosing the right candidates and the result of obtained combinations for the solution depends on the start choosed candidates, then the methods are force to use the remaining candidate.
2) Time Complexity: is equal to the number of steps that requires to solve an instance of the problem as a function of the size of the input, where its effects on the resources required for the execution of algorithms, and on the space complexity of a problem equal to the memory used by the algorithm.
3) Size Scalibility: is the ability of a system, network, or process to handle a growing amount of work in a capable manner or its ability to be enlarged accommodate that growth with the same time complexity.
a. The enumeration method can always reach the optimal solution , but with high complexity espicially for large scal problem.
b. Hungarian method one of the best and famous algorithms used on solving LAP , but it not alwayes reach the optmal solution.
c. The modified-BCM can always obtained the optimal solution because of its flexibility to the right chossing of candidates in each solution step, also has less time complexity and scalable for large size problem.
Table 5: Comparisons Between Linear Programming Methods.
6. Conclusion
In our paper we introduce a new LAP method called modified-BCM. We describe the new modifications, and clarify the advantages of the modified-BCM method. We highlight on the new modifications and how we solve LAP problems using our approach. The study also gives the reader basic concepts on operations research, Linear Programming, and shows the comparison study between the LAP solution methods performance's. As the comparison result, the modified-BCM is the most efficient and accurate comparing to other currently used methods, and it can be easly used in different areas and applications in optimization problems.
References
[1] Al Rahedi, Naef T.; and Atoum, Jalal, (2009): Solving the Traveling Salesman Problem Using New Operators in Genetic Algorithms.
Am. J. Applied Sci., 6: 1586-1590. DOI: 10.3844/ajassp.2009.1586.1590.
[2] Agrawal, Sangeeta; Subramanian, KR; and Kapoor, S. (2010): Operations Research - Contemporary Role in Managerial Decision
Making, IJRRAS 3 (2), May.
[3] Bazaraa ,Mokhtar, S.; Jarvis ,John, J. ; and Sherali, Hanif, D., P. (1990): Linear Programming and Network Flows. John Wiley &
Sons, Second Edition.
[4] Binitha, S et al.; Sathya, S., Siva. (2012):A Genetic Algorithm Approach for solving the Trim Loss Optimization Problem in Paper
Manufacturing industries, International Journal of Engineering Science and Technology (IJEST), ISSN: 0975-5462, Vol. 4, No.05, May 2012.
[5] Cara, Behin. (2012): https://www.deaos.com/Help.aspx?name=Linear%20Programming.
[6] Cao, Yi. (2008): Hungarian Algorithm for Linear Assignment Problems 10 Jul.
[7] Dell’Amico, Burkard ; Martello, S. (2009):Assignment Problems. SIAM. ISBN 978-0-898716-63-4.
[8] Ebrahimipoor, A.R.; Alimohamadi, A.; Alesheikh, A.A.; Aghighi, H. (2009): Routing of water pipeline using gis and genetic algorithm.
J. Applied Sci., 9: 4137-4145. en.wikipedia.org/wiki/Optimization_(mathematics)
[9] Hlayel ,Abdallah; Alia, Mohammad. (2012): Solving Transportation Problems Using the Best Candidates Method, DOI:
10.5121/cseij.2503. 23.
[10] Hlayel, Abdallah .(21012): The Best Candidates Method for Solving Optimization Problems. jcssp.2012.711.715.
[11] Imam, Taghrid; Elsharary , Gaber .(2009) : Solving Transportation Problem Using Object-Oriented Model, IJCSNS, VOL.9 No.2,
February.
[12] James, Thomas. (2013): Five Areas of Application for Linear Programming Techniques | eHow.com
http://www.ehow.com/facts_7789072_five-application-linear-programming-techniques.html#ixzz2R6RUOj1Z
[13] Khandelwal, Anju. (2011): Optimal Execution Cost of Distributed System: Through Clustering, International Journal of Engineering
Science and Technology (IJEST), ISSN: 0975-5462, Vol. 3 No. 3 March 2011.
[14] Operational Research: Science of Better (2013):http://www.scienceofbetter.org/what/index.htm.
[15] Purple Math (2013):http://www.purplemath.com/modules/linprog.htm
[16] Reddy, S., Narayana; Varaprasad,V.; and Veeranna ,V. (2012): Optimization of Multi-Objective Facilitity Layout Using
Non-Traditional Optimization Technique, International Journal of Engineering Science and Technology (IJEST), ISSN : 0975-5462,Vol. 4 No.02 February 2012.
[17] Reep, J, Leavngood, S. (1998): Using the Simplex Method to Solve Linear Programming Maximization Problems, Operations
Research, EM 8720-E, October.
[18] Samimi , Amir ; Aashtiani ,hedayat, Z. ; and Mohamm adian ,Abolfazl, (Kouros), (2009): A Short-term Management strategy for
Improving transit network efficiency. Am. J. Applied Sci., 6: 241-246. DOI: 10.3844/ajassp.2009.241.246.
[19] Winston, Wayne L. P. (2003): Operations Research: Applications and Algorithms. Duxbury Press, fourth edition.
[20] Zapee, C.; Webster, W.; and Horowitz, I. (1993): Using Linear Programming to Determine Post-Facto Consistency in Performance
Evaluations of Major League Baseball Players,” Interfaces, November– December, PP. 107–13.
Characteristics Time complexity Size Scalability Solution Optimality Methods
Enumeration High Complexity Not Scalable Optimal
Hungarian O(n3) Not Scalable Not always
