Recently, Yadaiah and Haragopal published in the American Journal of Operations Research a new approach to solving the unbalanced assignment problem. They also provide a numerical example which they solve with their approach and get a cost of 1550 which they claim is optimum. This approach might be of interest; however, their approach does not guarantee the optimal solution. In this short paper, we will show that solving this same example from the Yadaiah and Haragopal paper by using a simple textbook formulation to balance the problem and then solve it with the classic Hungarian method of Kuhn yields the true optimal solution with a cost of 1520.

Assignment Problem Hungarian Method Textbook Formulation
1. Introduction

The assignment problem is a standard topic discussed in operations research textbooks (See for example, Hillier and Lieberman  or Winston  ). A typical presentation requires that n jobs must be assigned to n machines such that each machine gets exactly one job assigned to it. If the number of jobs is not equal to the number of machines, then the assignment problem is first balanced. This requires that either dummy (fictitious) jobs or machines are added to the problem so that the number of jobs will equal the number of machines. The typical textbook solution to the balanced assignment problem is then found using Kuhn’s  Hungarian method.

Problems in which there are more jobs than machines and more than one job can be assigned to a machine can easily be handled as a balanced assignment problem with a little modeling effort. The idea is to “make copies” or “clone” the machines. This approach is discussed in Hillier and Lieberman  on page 336 with an example given in Table 8.29 on page 340. Also, problem 28 on page 411 of Winston  illustrates this modeling approach. For example, the numerical problem given in Yadaiah and Haragopal  requires the assignment of eight jobs to five machines such that each machine gets at least one job assigned to it and no machine gets more than two jobs assigned to it. The standard textbook approach would “clone” each of the five machines and add two dummy jobs to create a 10 by 10 balanced assignment problem. In a textbook, this problem would usually be solved with the Hungarian method, but other solution approaches are possible―the formulation is separate from the solution approach.

In Yadaiah and Haragopal  , they use a different approach to solve the unbalanced assignment problem (see their paper for details). If there are n jobs to be assigned to m machines with n strictly greater than m, then they solve a series of k balanced assignment sub-problems each of size m by m where k is the floor (round down) of n/m. The last problem that they solve is a balanced assignment problem of size [n-km] by [n-km]. Instead of using the Hungarian method to solve each sub-problem, they use a Lexi-search approach (See Pandit  and Ramesh  ). Their approach would be an alternative solution methodology to the textbook approach; however, we will show in the next section, using the numerical example from Yadaiah and Haragopal  , that their approach does not guarantee the optimal solution.

2. Yadaiah and Haragopal’s Numerical Example Revisited

Their numerical example is given in Table 1 and their method requires the solution of two sub-problems: a 5 by 5 and a 3 by 3. Please see Yadaiah and Haragopal  for details of their solution approach.

The first sub-problem solved byYadaiah and Haragopal  , is given in Table 2.

Their Lexi-search solution to this sub-problem is the following assignment:

J3 assigned to M1, J4 assigned to M2, J5 assigned to M3, J6 assigned to M5, and J7 assigned to M4 with a cost of 870. The second sub-problem is given in Table 3.

Their Lexi-search solution to this sub-problem is the following assignment:

J1 assigned to M4, J2 assigned to M5, and J8 assigned to M2 with a cost of 680. The final assignment cost is

Cost matrix
JobsMachines
M1M2M3M4M5
J1300290280290210
J2250310290300200
J3180190300190180
J4320180190240170
J5270210190250160
J6190200220190140
J7220300230180160
J8260190260210180
First sub-problem
JobsMachines
M1M2M3M4M5
J3180190300190180
J4320180190240170
J5270210190250160
J6190200220190140
J7220300230180160

870 + 680 = 1550. It can easily be checked using the Hungarian method, that these sub-problems were solved optimally. In their paper,

Yadaiah and Haragopal have a minor typo on page 88, they have 870 + 670 = 1550. It should read 870 + 680 = 1550.

We will now solve the original problem of assigning these eight jobs to five machines such that each machine is used at least once, but not more than twice.

Using the approach suggested in both Hillier and Lieberman  and Winston  , we formulate the balanced 10 by 10 assignment problem given in Table 4.

Solving the balanced assignment problem given in Table 4 using the Hungarian method yields the “reduced” cost matrix given in Table 5.

Second sub-problem
Machines
JobsM2M4M5
J1290290210
J2310300200
J8190210180
Textbook formulation
M1M2M3M4M5M1M2M3M4M5
J1300290280290210300290280290210
J2250310290300200250310290300200
J3180190300190180180190300190180
J4320180190240170320180190240170
J5270210190250160270210190250160
J6190200220190140190200220190140
J7220300230180160220300230180160
J8260190260210180260190260210180
DUM10000000000
DUM20000000000
Final Hungarian method cost matrix
M1M2M3M4M5M1M2M3M4M5
J1403020300403020300
J20604050006040500
J301012010500101201050
J414001060401400106040
J5802006020802006020
J601030000103000
J740120500304012050030
J8700702040700702040
DUM1000050000050
DUM2000050000050

From Table 5, we can read the optimal assignment to be

J1 assigned to M5 at a cost of 210,

J2 assigned to M1 at a cost of 250,

J3 assigned to M1 at a cost of 180,

J4 assigned to M2 at a cost of 210,

J5 assigned to M3 at a cost of 190,

J6 assigned to M5 at a cost of 140,

J7 assigned to M4 at a cost of 180,

J8 assigned to M2 at a cost of 190,

For a total minimum cost of 1520 (not 1550).

Both of these assignments use each machine at least once and no more than twice, but the standard textbook formulation solved with the Hungarian method gets the guaranteed optimal solution of 1520―maybe simpler is better. The flaw with their approach is that their assignment is not optimal because their decomposition into sub- problems does not guarantee an optimal solution to the original problem as illustrated by this example.

Cite this paper

Nathan Betts,Francis J. Vasko, (2016) Solving the Unbalanced Assignment Problem: Simpler Is Better. American Journal of Operations Research,06,296-299. doi: 10.4236/ajor.2016.64028

ReferencesHillier, F.S. and Lieberman, G.J. (2010) Introduction to Operations Research. 9th Edition, McGraw-Hill, New York.Winston, W.L. (2004) Operations Research: Applications and Algorithms. Thomson, Belmont.Kuhn, H.W. (1955) The Hungarian Method for the Assignment Problem. Naval Research Logistics Quarterly, 5, 83- 97. http://dx.doi.org/10.1002/nav.3800020109Yadaiah, V. and Haragopal, V.V. (2016) A New Approach of Solving Single Objective Unbalanced Assignment Problem. American Journal of Operations Research, 6, 81-89. http://dx.doi.org/10.4236/ajor.2016.61011Pandit, S.N.N. (1963) Some Quantitative Combinatorial Search Problems. PhD Thesis, IIT, Khargpur.Ramesh, M. (1997) Lexi-Search Approach to Some Combinatorial Programming Problem. PhD Thesis, University of Hyderabad, Hyderabad.