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Solution structures and sensitivity of special assignment problems
Conzpur. % 0~s. f&s. Vol. Il. No. 4, ~9.397-399 Printed in the U.S.A.
0305-0548/84 L3.00 + .W 0 1984 Pergamon PISS Ltd.
SOLUTION STRUCTURES AND SENSITIVITY OF SPECIAL ASSIGNMENT PROBLEMS H. A. EISELT~ Department of Quantitative Methods, Concordia University, Montreal, Quebec. Canada H4B lR6 and Y. GERCHAK$ Department
of Management
Sciences, University of Waterloo, N2L 3Gl
Waterloo,
Ontario,
Canada
*peti
purpose--We consider assignment problems in which the cost coefficients are weighted averages of scores. we show that in some special cases the optimal solutions of such problems can be determined in advance, and will not change unless certain order relations are altered.
A~~-A~i~ment, Transpo~ation and Bottleneck problems with special cost structures are considered. Under certain order conditions, optimal solutions are shown to possess particularly simple structures. Sensitivity analysis issues are also discussed.
1. INTRODUCTION
While Assi~ment Problems with given parameter values can be solved very e~~ently, determining the range of parameter values for which a given assignment remains optimal may be tedious. The reason is the inherent high degree of degeneracy, due to which a change in optimal basis may leave the optimal extreme point, and hence the assignment, unchanged. It may thus be useful to identify special classes of assignment and related problems whose optimal solutions’ structures can be determined in advance if some order relations among parameters are present. As the optimal solution will then change only if that order is altered, the exact values of parameters are immaterial as long as their relative magnitudes are known. This note focuses on assignment and related problems of the “job-allocation” type. The objective function coefficients cii are obtained as inner products of two sets of k-vectors, as follows Gj
=i~~-l,s~$,a,b, i,j=l,...,
n.
The elements of vector ai may be interpreted as the scores (positive or negative) of person i on k aptitude tests, and vector &, as a relevance-profile of the tests to job j. In the next section we show that if the vectors ai, i = 1,. . . , n and aj,-j = I,. . . , n can be ordered in some simple manner, that order determines the structure of the optimal solution of the assignment problem. Some applications of that result are then mentioned. The following section then extends that result to a Transportation Problem and Bottleneck
THorst A. Eiselt is currently Assistant Professor at Concordia University, Montreal, Canada. He received his Diploma as well as his Doctoral degree from the University of Gdttingen, West Germany. His primary interests are in the fields of Network Flows, Minimax Problems and non-simDlex based approaches to Linear Programming. He coauthored the “Operations Research Handbook” as well ai the book ‘?&rations Research Models: Continuous t%timization” (with G. Pederzoli and C.-L. Sandblom). Dublished bv W. de Gruvter. His research articles h&e appeared in various Operations Research journal; _ $Yigal Gerchak is an Assistant Professor of Management Sciences at the University of Waterloo. He received the B.A. degree in economics and statistics, and the MSc. degree in management sciences from Tel-Aviv University, and the Ph.D. degree from the University of British Columbia in 1980. His current research is in the areas of manpower planning, labor turnover control, criminal justice models, public policy evaluation and applied probability.
397
398
H. A. EISELTand Y. GWCHAK
Assignment and Transportation Problems. We conclude with a discussion of some sensitivity analysis issues in the general job-allocation problem, and provide some intuitive interpretations. 2. ORDERED
JOB-ALLOCATION
ASSIGNMENT
Let eir i = 1, . . . , n and 6,, j = 1, . . . , n be k-vectors such that @I5 a2 zz . . . I @,and &,I&< ‘.. I 8. (elementwise), and let j,, . . ,j, be any permutation of the integers 1, . . . , n. We shall now prove the following Lemma. LEMMA
The sum X1=,a?. 4, attains its maximum when ji = i for all i, and its minimum when ji = n - i + 1 for all i. Proof. Suppose that, in some permutation,
j,,, > j, for some m > I, and consider a transposition of that permutation with j, and j,,, interchanged. Then the sums of inner products corresponding to those permutations will differ only in that the former’s will contam the parttal sum @T-4,+ c-4,,,, while the latter’s will instead contain aF.&, + I-4,. Now, a f
. lj, + a; .6,. - zz;. lin - 0: .6, = (a: - 46,
- ejm)2 0.
Hence any transposition which reverses the natural order decreases the total sum. Thus the total sum is minimized when the j;s are in descending order, and maximized when they are in ascending order.4 Thus if the ranking of individuals by scores on all tests is identical and so is the ranking of jobs by relevance of all tests, it is optimal to assign the best applicant to the job for which (all) tests are most relevant, and so on. If scores are positive, this implies assignment along the main diagonal, and if they represent deficiency points-along the second main diagonal. Note that the weaker hypothesis “cVincreases in i and j” (i.e. ck,2 cg for k 2 i and I >j) is not sufficient for the solution to have the above structure. Since when k = 1 (a single test) the scalars a, and bj can always be ordered, the Lemma identifies the optimal solution of any such problem. This result is known [4]. One well-known application of that logic is in the optimal ordering of sequential files: in order to minimize the expected position in the sequence of the item requested, the items should ideally be arranged in decreasing order of request-probability [5]. Another manifestation of this logic may be in assigning less-senior employees to night-shifts in order to minimize total wages, when night-shift premiums are a function of base-salary, which tends to increase with seniority. It is also used in scheduling theory [2]. Let us now suppose that, for any given job, all tests are of the same relevance, but that the relevance differs among jobs. That is, 8, = xja for all j, where Xj is a constant and B a vector of ones. Then
and denoting Xi_, ai, (the total score of individual i) by yi we get Zy_, YiXj,,for which we know, from the Lemma, that the optimal solution is to assign the individual with highest total score to the job for which tests are most relevant, and so on. 3. EXTENSIONS
TO TRANSPORTATION
AND
BOTTLENECK
PROBLEMS
Any transportation problem can be written as a large assignment problem (by replacing each S-units supply point by S l-unit supply points, and similarly for destinations), where the assignment costs for all combinations generated from a given supply point and destination equal the original unit shipping costs. Hence if unit shipping costs are generated as inner products of ordered vectors, the Lemma implies that the optimal solution of a maximization problem will be obtained by the “NW corner rule”, and that of the minimization problem by the “NE corner rule”.
Solution structures and sensitivity of special assignment problems
399
Consider now the min-max bottleneck assignment problem [l] minimize {maxX,,, ,, {c,}]. Let X be a feasible assignment with objective value Z. This solution can be improved M if Cij2 Z if and only if, upon transforming the costs to C$ = 0 otherwise M > 0, an assignment with .
lforalli,andletck,n_lr+,= ji zero cost can be found [ 31. Consider the assignmen \‘-n-i+ maxi{ci,”_ i + ,}. Then if cij increases in i andj, the cost-transformation must result in c$ = it4 for all i 2 k and j 2 n - k + 1. Hence in the last k columns there are, at most, k - 1 rows having any zero costs, and it is thus impossible to find a complete zero-cost assignment. Thus the assignment ji = n - i + 1 for all i is optimal. It can be similarly shown that, if cii increases in i and j, the optimal solution of the max-min bottleneck assignment problem maximize {mini,j,, {c,}}is also ji = n - i + 1 for all i. It also follows that, under the same hypothesis, the optimal solutions for the min-max as well as max-min bottleneck transportation problems can be obtained by use of the NE corner rule. Note that the weaker hypothesis “I+ increases in i andj” was sufficient in the context of bottleneck problems. 4. SENSITIVITY
ANALYSIS
FOR GENERAL
JOB-ALLOCATION
ASSIGNMENT
We shall assume that a, is individual i’s degree of deficiency at the m-th skill, so the problem is one of minimization. Suppose that such a problem has been solved by the Hungarian method. That is, problem c = (cJ has been transformed to an equivalent problem C = (E,), EV2 0, such that there exists a complete assignment (X$} for which X$ = 1 only if -& =~0. As problems c and C are equivalent, the effect of any change in the former can be analyzed via the effect of a similar change in the latter. Suppose then that person i*‘s deficiency in skill m* has changed, i.e. a:‘,,,.= a,.,,,. - A. Then, for all j, k
ci=
c akbj,,,= ltl=l
c,.~- Ab,.
I
cg
i = i* i # i*’
Let j* be such that X,2,*= 1 (i.e. the job individual i* was previously assigned to). Then it can be easily shown that the optimal solution will not change if A(b,, - b,,.) I Z,.,. for all j. If above is not satisfied, the optimal solution may, and usually will, change. In particular, the above implies that the solution will not change if A > 0 and b,.me2 bj,,,. forallj,orifA
tWe have been told that some related results have been obtained in the doctoral dissertation [6], but were not able to verify that.
0305-0548/84 L3.00 + .W 0 1984 Pergamon PISS Ltd.
SOLUTION STRUCTURES AND SENSITIVITY OF SPECIAL ASSIGNMENT PROBLEMS H. A. EISELT~ Department of Quantitative Methods, Concordia University, Montreal, Quebec. Canada H4B lR6 and Y. GERCHAK$ Department
of Management
Sciences, University of Waterloo, N2L 3Gl
Waterloo,
Ontario,
Canada
*peti
purpose--We consider assignment problems in which the cost coefficients are weighted averages of scores. we show that in some special cases the optimal solutions of such problems can be determined in advance, and will not change unless certain order relations are altered.
A~~-A~i~ment, Transpo~ation and Bottleneck problems with special cost structures are considered. Under certain order conditions, optimal solutions are shown to possess particularly simple structures. Sensitivity analysis issues are also discussed.
1. INTRODUCTION
While Assi~ment Problems with given parameter values can be solved very e~~ently, determining the range of parameter values for which a given assignment remains optimal may be tedious. The reason is the inherent high degree of degeneracy, due to which a change in optimal basis may leave the optimal extreme point, and hence the assignment, unchanged. It may thus be useful to identify special classes of assignment and related problems whose optimal solutions’ structures can be determined in advance if some order relations among parameters are present. As the optimal solution will then change only if that order is altered, the exact values of parameters are immaterial as long as their relative magnitudes are known. This note focuses on assignment and related problems of the “job-allocation” type. The objective function coefficients cii are obtained as inner products of two sets of k-vectors, as follows Gj
=i~~-l,s~$,a,b, i,j=l,...,
n.
The elements of vector ai may be interpreted as the scores (positive or negative) of person i on k aptitude tests, and vector &, as a relevance-profile of the tests to job j. In the next section we show that if the vectors ai, i = 1,. . . , n and aj,-j = I,. . . , n can be ordered in some simple manner, that order determines the structure of the optimal solution of the assignment problem. Some applications of that result are then mentioned. The following section then extends that result to a Transportation Problem and Bottleneck
THorst A. Eiselt is currently Assistant Professor at Concordia University, Montreal, Canada. He received his Diploma as well as his Doctoral degree from the University of Gdttingen, West Germany. His primary interests are in the fields of Network Flows, Minimax Problems and non-simDlex based approaches to Linear Programming. He coauthored the “Operations Research Handbook” as well ai the book ‘?&rations Research Models: Continuous t%timization” (with G. Pederzoli and C.-L. Sandblom). Dublished bv W. de Gruvter. His research articles h&e appeared in various Operations Research journal; _ $Yigal Gerchak is an Assistant Professor of Management Sciences at the University of Waterloo. He received the B.A. degree in economics and statistics, and the MSc. degree in management sciences from Tel-Aviv University, and the Ph.D. degree from the University of British Columbia in 1980. His current research is in the areas of manpower planning, labor turnover control, criminal justice models, public policy evaluation and applied probability.
397
398
H. A. EISELTand Y. GWCHAK
Assignment and Transportation Problems. We conclude with a discussion of some sensitivity analysis issues in the general job-allocation problem, and provide some intuitive interpretations. 2. ORDERED
JOB-ALLOCATION
ASSIGNMENT
Let eir i = 1, . . . , n and 6,, j = 1, . . . , n be k-vectors such that @I5 a2 zz . . . I @,and &,I&< ‘.. I 8. (elementwise), and let j,, . . ,j, be any permutation of the integers 1, . . . , n. We shall now prove the following Lemma. LEMMA
The sum X1=,a?. 4, attains its maximum when ji = i for all i, and its minimum when ji = n - i + 1 for all i. Proof. Suppose that, in some permutation,
j,,, > j, for some m > I, and consider a transposition of that permutation with j, and j,,, interchanged. Then the sums of inner products corresponding to those permutations will differ only in that the former’s will contam the parttal sum @T-4,+ c-4,,,, while the latter’s will instead contain aF.&, + I-4,. Now, a f
. lj, + a; .6,. - zz;. lin - 0: .6, = (a: - 46,
- ejm)2 0.
Hence any transposition which reverses the natural order decreases the total sum. Thus the total sum is minimized when the j;s are in descending order, and maximized when they are in ascending order.4 Thus if the ranking of individuals by scores on all tests is identical and so is the ranking of jobs by relevance of all tests, it is optimal to assign the best applicant to the job for which (all) tests are most relevant, and so on. If scores are positive, this implies assignment along the main diagonal, and if they represent deficiency points-along the second main diagonal. Note that the weaker hypothesis “cVincreases in i and j” (i.e. ck,2 cg for k 2 i and I >j) is not sufficient for the solution to have the above structure. Since when k = 1 (a single test) the scalars a, and bj can always be ordered, the Lemma identifies the optimal solution of any such problem. This result is known [4]. One well-known application of that logic is in the optimal ordering of sequential files: in order to minimize the expected position in the sequence of the item requested, the items should ideally be arranged in decreasing order of request-probability [5]. Another manifestation of this logic may be in assigning less-senior employees to night-shifts in order to minimize total wages, when night-shift premiums are a function of base-salary, which tends to increase with seniority. It is also used in scheduling theory [2]. Let us now suppose that, for any given job, all tests are of the same relevance, but that the relevance differs among jobs. That is, 8, = xja for all j, where Xj is a constant and B a vector of ones. Then
and denoting Xi_, ai, (the total score of individual i) by yi we get Zy_, YiXj,,for which we know, from the Lemma, that the optimal solution is to assign the individual with highest total score to the job for which tests are most relevant, and so on. 3. EXTENSIONS
TO TRANSPORTATION
AND
BOTTLENECK
PROBLEMS
Any transportation problem can be written as a large assignment problem (by replacing each S-units supply point by S l-unit supply points, and similarly for destinations), where the assignment costs for all combinations generated from a given supply point and destination equal the original unit shipping costs. Hence if unit shipping costs are generated as inner products of ordered vectors, the Lemma implies that the optimal solution of a maximization problem will be obtained by the “NW corner rule”, and that of the minimization problem by the “NE corner rule”.
Solution structures and sensitivity of special assignment problems
399
Consider now the min-max bottleneck assignment problem [l] minimize {maxX,,, ,, {c,}]. Let X be a feasible assignment with objective value Z. This solution can be improved M if Cij2 Z if and only if, upon transforming the costs to C$ = 0 otherwise M > 0, an assignment with .
lforalli,andletck,n_lr+,= ji zero cost can be found [ 31. Consider the assignmen \‘-n-i+ maxi{ci,”_ i + ,}. Then if cij increases in i andj, the cost-transformation must result in c$ = it4 for all i 2 k and j 2 n - k + 1. Hence in the last k columns there are, at most, k - 1 rows having any zero costs, and it is thus impossible to find a complete zero-cost assignment. Thus the assignment ji = n - i + 1 for all i is optimal. It can be similarly shown that, if cii increases in i and j, the optimal solution of the max-min bottleneck assignment problem maximize {mini,j,, {c,}}is also ji = n - i + 1 for all i. It also follows that, under the same hypothesis, the optimal solutions for the min-max as well as max-min bottleneck transportation problems can be obtained by use of the NE corner rule. Note that the weaker hypothesis “I+ increases in i andj” was sufficient in the context of bottleneck problems. 4. SENSITIVITY
ANALYSIS
FOR GENERAL
JOB-ALLOCATION
ASSIGNMENT
We shall assume that a, is individual i’s degree of deficiency at the m-th skill, so the problem is one of minimization. Suppose that such a problem has been solved by the Hungarian method. That is, problem c = (cJ has been transformed to an equivalent problem C = (E,), EV2 0, such that there exists a complete assignment (X$} for which X$ = 1 only if -& =~0. As problems c and C are equivalent, the effect of any change in the former can be analyzed via the effect of a similar change in the latter. Suppose then that person i*‘s deficiency in skill m* has changed, i.e. a:‘,,,.= a,.,,,. - A. Then, for all j, k
ci=
c akbj,,,= ltl=l
c,.~- Ab,.
I
cg
i = i* i # i*’
Let j* be such that X,2,*= 1 (i.e. the job individual i* was previously assigned to). Then it can be easily shown that the optimal solution will not change if A(b,, - b,,.) I Z,.,. for all j. If above is not satisfied, the optimal solution may, and usually will, change. In particular, the above implies that the solution will not change if A > 0 and b,.me2 bj,,,. forallj,orifA
tWe have been told that some related results have been obtained in the doctoral dissertation [6], but were not able to verify that.