A variation of the assignment problem

422

European Journal of Operational Research 68 (1993) 422-426 North-Holland

Theory and Methodology

A variation of the assignment problem * S. Geetha and K.P.K. Nair Faculty of Administration, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 Received January 1991; revised May 1991

Abstract: In this paper, a meaningful variant of the cost minimizing assignment problem with the objective of minimizing the total cost of assignment plus an additional 'supervisory' cost, which depends on the total time of completion of the project is formulated. An efficient method of finding an optimal solution to such a problem is presented with a numerical example to illustrate the same. Keywords: Cost minimizing assignment; Bottleneck assignment; Bicriteria assignment; Trade-off pairs; Algorithm

I. Introduction The cost minimizing assignment problem has been extensively studied in the literature and several polynomial algorithms [3,7,8] are available to solve it. Similarly, efficient algorithms [5,10] are available for solving the time minimizing (bottleneck) assignment problem. In practical situations, the above mentioned problems cannot be viewed as two independent problems, if one is interested in obtaining a solution which minimizes cost as well as time. In view of this, a time-cost trade-off analysis has been suggested by Bakhshi, Bhatia and Puri [2]. In a recent paper by Berman, Einav and Handler [4] an elegant treatment of three different constrained bottleneck problems in networks has been presented. This work includes, among others, a method of

* The work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada awarded to K.P.K. Nair. Correspondence to: S. Geetha, Faculty of Administration, University of New Brunswick, Fredericton, NB, Canada E3B 5A3.

finding all Pareto optimal solutions to the two criteria problem which is equivalent to the timecost trade-off analysis addressed in [2]. In this paper, yet another meaningful objective of minimizing the cost of assignment with an additional 'supervisory' or any other form of cost, which depends on the total time of completion of the project is considered. Properties of an optimal solution to this problem are established and based on these an efficient algorithm is presented. An illustrative numerical example is also included. Also, validation of the algorithm is included. The computational complexity of the algorithm is of O(n 5) in the worst case analysis.

2. Formulation of the problem Consider the cost minimizing assignment problem (CMP),

(el)

Minimize C = ~

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

~_,c,ixij

i=1j=1

(1)

S. Geetha et al. / A variation of the assignmentproblem 3. Properties of an optimal solution to (P)

subject to

n ~_,xij=l i=l ~x~j=l j=l xgj = 0 or 1

for a l l j = l , 2

for a l l i = l ,

. . . . . n,

(2)

2 .... ,n,

(3)

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

(4)

where ciy is the cost of assigning the j-th job to the i-th worker. Let X be the set of all feasible solutions adhering to constraints (2) through (4) above. The corresponding (bottleneck) time minimizing problem (TMP), assuming that all the jobs are started simultaneously, is

The optimal value of CMP gives the minimum cost required to get all the jobs done. The optimum value of TMP gives the minimum completion time to get all the jobs done, assuming that all the jobs are started simultaneously. T i m e - c o s t trade-off analysis gives rise to trade-off relations between time and cost in the above contexts. Now, consider the following bicriteria problem (B). (B)

Minimize

Minimize

t=max{tijlxij=l

subject to

x~X,

}

where tij is the time taken by the i-th worker to complete the j-th job. Several algorithms are available to solve (P1) [3,7,8] and (P2) [5,10] in polynomial time. A more general cost minimizing assignment problem can be described as follows: In addition to the hiring cost of workers to perform the jobs, there is a 'supervisory cost' or some other type of cost associated with the total time of completion of the project. Let F ( > 0) be the 'supervisory cost' per unit of time. Then, the total cost involved in the assignment problem is given by

i=1 j=l

ci, xi, + F{max[ti, I

= 11}

Since tij can be redefined to be the product of time and F, without loss of generality, the problem (P) under consideration can be stated as follows: (P)

Minimize

Q=

~ CijXij i=l j=l +max(tijl xij = l)

subject to

x ~X.

In the next section, certain properties of an optimal solution to (P) are established and these are used in developing the algorithm.

z : [Z 1 : i=1 ~ J=l ~

CijXij'

z2=max(tij]xo=l)] subject to

(P2)

423

x~X.

A solution ~ ~ X is efficient in the above bicriteria problem if and only if for all x ~ X , z(x) <<. z(~) implies z ( x ) = z(~). For x ~ X , the objective value in problem (P) is

Q ( x ) = z , ( x ) + z2(x ). Therefore, we have the following lemma, whose proof by contradiction is obvious.

Lemma 1. An optimal solution of problem (P) is attained at an efficient solution of the bicriteria problem (B). As seen from [2,4], there are at most n 2 efficient solutions in problem (B) and these can be enumerated and the objective value Q can be determined. Because of Lemma 1, the one that results in minimum Q is optimal to problem (P). In the worst case, the computational complexity of this scheme is of O(n5). Now consider the mapping of all feasible solutions X of problem (B) through the two objective functions in the bicriteria space. Let Z be the set of all points so mapped in the bicriteria space. A point ~ ~ Z is efficient in Z if and only if for all z ~ Z, z < ~, implies ~. Clearly, given an efficient solution of problem (B) there exists a corresponding efficient point in Z. Let H be the convex hull of the set of points Z. A point kt ~ H is efficient if and only if for all h ~ H, h < h implies h = h. Clearly, the efficient points in this case form a piecewise linear convex frontier and the extreme points on this frontier

424

S.

Geetha et al. / A variation of the assignment problem

are called efficient extreme points of H. Thus, each efficient extreme point of H is indeed an efficient point of Z, but the converse is not necessarily true. These considerations lead to the following lemma, whose proof is straightforward.

Lemma 2. An optimal solution to problem (P) maps into an efficient extreme point of the convex hull H. Though Lemma 2 provides a stronger characterization of an optimal solution to problem (P), its utility in structuring the algorithm is rather limited. This is because of the fact that when an efficient point z ~ Z is obtained, it is not easy to figure out whether or not it is an extreme point of the convex hull H, until we have identified all points of Z. For the very same reason resulting from the nonlinearity of the second objective, the method of Aneja and Nair [1] cannot be modified for solving (P). However, Lemma 2 shows that an optimal solution maps into an efficient extreme point of H and furthermore, it is easy to see that the iso-value (Q) curves in the bicriteria space are straight lines of the form Q = z 1 + z 2. These aspects together with an easily obtainable lower bound on z 2 would be used in the development of the algorithm; however, the worst case complexity will remain unchanged, although on the average, the performance would be better.

plying a binary search scheme. A readily available lower bound, though not necessarily attainable, is the n-th element of t~i's arranged in non-decreasing order and let this be denoted by b.

Algorithm. Step 0 (Initialization): Set iteration number k = 0. Solve the problem minimize

zI = ~

x~X

i-1

~ c,ixij j=l

and obtain x k, zf, z~ and Qk = z ~ + z2~. Set x * = x ~, Q* = Q~ and go to Step 1. Step 1. Revise cij's as follows:

Cij

if tij <

o~

if tij >/Zk2 .

Cij =

Z 2, k

Set k = k + l a n d g o t o S t e p 2 . Step 2. Solve the problem minimize

zI= ~

x~X

i-1

~ cijxij j=l

Cij'S and obtain x k, z~, z2k and Qk = z f + zk2. If Zlk > / Q * - b , terminate; in this case, x * is an optimal solution and Q* is the optimal value of (P). If Q k < Q , , update Q* and x* by setting x * = x ~ and Q* = Qk. Go to Step 1 with z~. with revised

4. A method of solving problem (P) We assume that cost minimizing problem (P1) has finite optimum for otherwise problem (P) will result in unbounded optimum. The algorithm given below generates a sequence of feasible solutions to problem (B) with strictly decreasing z 2 values. In each case, the objective value Q in problem (P), given by Q = z 1 + z 2 is evaluated and the one that gives minimum of Q is optimal in problem (P). Knowledge of a lower bound for

z z = m a x ( t i j l x i j = 1) would be helpful in developing an algorithm. Minimum z 2 given by the bottleneck time minimizing problem is the attainable lower bound; however, it requires efforts of O[(log n2)n3], ap-

Validation of the algorithm. Clearly, z 1, z 2. . . . generated by the algorithm are mappings of feasible solutions to problem (B). The sequence z~, Zlz, ... is non-decreasing while z 12, z22, ... is strictly decreasing. Given the current Q* = z~* + z~ at any iteration, for all x such that Zz(X) >/z~, Q(x) >~Q* and this can be seen easily by induction. The termination criteria ensures that for x such that z2(x) < z~, Q(x) >1Q* and this follows from the fact that z~, Zl2 ... is non-decreasing and b is a lower bound on z 2. Thus, upon termination x * is optimal with value Q* and furthermore, it is an efficient extreme point of the convex hull H. The complexity of the algorithm as stated earlier is of O(nS), in the worst case. On average, the performance may be better depending on the strength of the bound b.

S. Geetha et al. / A variation of the assignment problem

425

Table 1 The c i j and tij v a l u e s

i

j 1

2

3

4

5

6

1

c# = 6 tij = 4 6 6 11 2 9 12 4 9 3 17

3 20 4 18 7 8 10 13 6 8 5 13

5 9 6 8 4 20 8 14 7 7 11 3

8 3 5 7 8 7 6 6 9 14 10 4

10 8 9 17 3 15 10 9 8 5 12 13

6 9 8 8 2 7 4 10 7 9 8 7

2 3 4 5 6

minimizing the cost of assignment with an additional 'supervisory' cost, which depends on the total time of completion of the project is formulated and a method of finding an optimal solution for the problem is developed. It can be seen that in the case of a transportation problem also, a similar model can be developed and the algorithm given in Section 4 can be modified easily to solve such a problem. Also, extensions of the above to 3-dimensional assignment [9] and transportation [6] problems are apparent.

5. Numerical example Consider Problem (P) with Cij and tij values as given in Table 1. Arranging the t# values in non-decreasing order, gives the 6th element in that order as 5 and therefore, b = 5. Table 2 gives the solution stages. The optimal value of the problem under consideration is 41 and an optimal solution giving this value is X13 , X24 , X35 , X46 ~ X51 ~ X62 ~

1.

6. Conclusion

Acknowledgement

The assignment problem has two aspects, namely, the cost and the time. In this paper, a meaningful variant of CMP with an objective of

The authors wish to thank Dr. S.N. Kabadi for his valuable suggestions and discussion on this work.

Table 2 Solution stages Iteration k

z~

xk

0 1

25 26

X12,X24,X35,X46,X53,X61 x 13,x24, x35,x46,x52,x61

= =

1 1

z~

Qk

Q,

Q* - b

Remarks

20 17

45 43

45 43

45 - 5

zkl < Q * - b , Q k

r


Record new Q* 2

26

x13,x24,x35,Xa6,X51,x62

=

1

15

41

41

43 - 5

z~ < Q * - b , Q k


Record new Q* 3

31

x13,x24,x36,Xa5,X51,x62

=

1

13

44

41

41 - 5

4

39

x13,x24,x32,Xa5,X51,x66

=

1

9

48

41

41-5

z~ < Q* - b,Q k > Q* Do not record

Zkl > Q * - b Terminate

426

S. Geetha et al. / A variation of the assignment problem

References [6] [1] Aneja, Y.P., and Nair, K.P.K., "Bicriteria transportation problem", Management Science 25 (1979) 73-78. [2] Bakshi, H.C., Bhatia, H.L., and Puri, M.C., "Trade-off relations in assignment problems", Cahiers du Centre d'Etudes de Recherche Opdrationnelle 21 (1979) 367-374. [3] Barr, R.S., Glover, F., and Klingdam, D., "The alternating basis algorithm for assignment problems", Mathematical Programming 13 (1977) 1-13. [4] Berman, O., Einav, D., and Handler, G., "The constrained bottleneck problem in networks," Operations Research 38 (1990) 178-181. [5] Garfinkel, R.S., "An improved algorithm for bottleneck

[7]

[8]

[9] [10]

assignment problem", Operations Research 18 (1971) 1717-1751. Haley, K.B., "The solid transportation problem", Operations Research 10 (1962) 448-463. Hung, M.S., and Rom, W.O., "Solving the assignment problem by relaxation", Operations Research 28 (1980) 969-982. Kuhn, H.W., "The Hungarian method for the assignment problem", Naval Research Logistics Quarterly 2 (1955) 83-97. PierskaUa, W.P., "The multidimensional assignment problem", Operations Research 16 (1968) 422-430. Ravindran, A., and Ramaswamy, V., "On the bottleneck assignment problem", Journal of Optimization Theory and Applications 21 (1977) 451-458.