A Generalized Assignment Problem with Special Ordered ... - CiteSeerX

A Generalized Assignment Problem with Special Ordered Sets: A Polyhedral Approach - Extended Abstract I. R. de Farias, Jr. IBM Corporation, 3200 Windy Hill Rd., Atlanta, GA 30339 (WG09B) E. L. Johnson and G. L. Nemhauser School of Industrial and Systems Engineering Georgia Institute of Technology, Atlanta, GA 30332-0205

1 Introduction Special ordered sets (SOS), introduced by Beale and Tomlin [1], are widely used in branch-andbound algorithms. They were rst implemented in Scicon's UMPIRE, and today nearly all commercial mathematical programming codes contain them. A set of variables fx1; : : :; x g is a special ordered set of type II if x x = 0 whenever ji ? j j  2, i.e. at most two variables in the set can be nonzero, and if two variables are nonzero they must be adjacent in the set. Traditionally, the approach used for solving programming problems with special ordered sets has been branch-and-bound. In general, however, the computational burden of branch-and-bound methods for NP-hard combinatorial optimization problems that do not take into account the structure of the constraints may increase very quickly with only a modest increase in the size of the input. On the other hand, it has been demonstrated that branch-and-cut approaches, which use the polyhedral structure of the problem, frequently improve the performance of branch-and-bound methods substantially. In this paper we present a generalized assignment problem that arises in ber optics cable production scheduling [3] in which special ordered sets of type II appear naturally in the formulation. We study the polyhedral structure of the convex hull of its set of feasible n

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solutions and we present results of our computational experiments on a branch-and-cut scheme for solving this problem using inequalities of a family of facets derived in this paper.

2 Problem Formulation Let M = f1; : : :; mg be a set of non-preemptible jobs to be executed by a machine that processes several jobs simultaneously (multiprocessor). The machine is available for N = f1; : : :; ng consecutive time periods of unit length (shifts) with capacity b > 0 in each time period. The amount of machine capacity used in executing job i is a 2 (0; b], and the duration of each job is the length of a time period, but a job does not necessarily start at the begining of the period. The goal is to minimize scheduling cost. P Denote by x the fraction of time period t during which job i is active. Then 2 x = 1. Also, because the duration of each job is equal to the length of the time period, no job can be active in more than two time periods, and if a job is active in two dierent time periods they must be adjacent. This means that fx 1 ; : : :; x g is a special ordered set of type II 8i 2 M . Our problem can then be formulated as a generalized assignment problem with SOS type II constraints as i

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P P cx 2 2 P 2 a x  b; P 2 x = 1;

min

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M

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M

t

N

t

N

i

it

it

it

x  0;

it

t 2 N; i 2 M; i 2 M; t 2 N;

it

fx ; : : :; x g SOS type II; i 2 M; i1

in

where c is the cost of executing job i completely in time period t. In the polyhedral analysis we will use the formulation (F ) it

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P P dx 2 2 P a x  b; 2 P 2 x  1;

max

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M

i

M

t

N

t

N

i

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it

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x  0; it

t 2 N;

(1)

i 2 M;

(2)

i 2 M; t 2 N;

(3)

fx ; : : :; x g SOS type II; i 2 M i1

(4)

in

with all inequality constraints, that is equivalent to the previous one with an appropriate choice of d , i 2 M , t 2 N . Problem (F ) is at least as hard as a partition problem, and therefore is NP-hard. We denote the LP relaxation of (F ) by LPF and the convex hull of its solution set by PF . PF is full-dimensional, and thus from an inequality point of view it is convenient to work with (F ) since the facets of PF are uniquely determined up to a multiplicative constant. Since the problem P P becomes trivial when 2 a  b, m = 1, or when n < 3, we assume that 2 a > b, m  2 and n  3. Given two indices t1 ; t2 2 N , they are said to be adjacent if either t2 = t1 + 1 or t1 = t2 + 1. A set of distinct indices ft1 ; : : :; t g, 1  l  n, t1 ; : : :; t 2 N , is said to be adjacent if t1 is adjacent to t2 , but not to any other index in the set, t is adjacent to t ?1 , but to no other index in the set, and t is adjacent exactly to t ?1 and to t +1 , 8i 2 f2; : : :; l ? 1g. Note that we may have t1 <





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3 Facets of PF Proposition 1 x  0 is a facet-de ning valid inequality for PF , 8i 2 M and t 2 N .

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Proposition 2 P x  1 is a facet-de ning valid inequality for PF , 8i 2 M .

2

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t=1

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The inequalities in the previous propositions are called trivial inequalities since they are valid for LPF . We next present inequalities that are facets of PF but are not valid for LPF . These 3

inequalities can be used as cuts in a branch-and-cut scheme for solving (F ). We also give a full inequality description of PF for a particular case, and we show that the inequalities of the three families cut o all infeasible vertices of LPF . The inequalities are based on the concept of l-covers de ned below.

De nition Given a positive integer l, a set I  M is said to be an l-cover when P 2 a > lb. P An l-cover is minimal if 2 ? a  lb 8r 2 I . Given an l-cover I and i 2 I , we de ne a = P 2 maxf0; lb ? 2 ? a g. i

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(l)

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Theorem 1 Let I  M be an l-cover, 1  l  n ? 2, and suppose that 9 i0 2 I with a 0 > 0. Let T = ft ; : : :; t g  N be a set of adjacent indices, and de ne T 0 = ft 2 N : jt ? t0 j  2 8t0 2 T g. When T 0 = 6 ;, the following family of inequalities is valid and facet-de ning: (l)

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Xa Xx + Xa X x (l)

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 lb:

(5)

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Corollary 1 The inequalities (1) are facet-de ning i P 2 i

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a  b 8j 2 M . i

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Theorem 2 When M is a minimal 1-cover, (5) for l = 1 together with (2) and (3) give PF . 2

Corollary 2 When m = 2, (5) for l = 1, (2), and (3) give PF . 2 When M is not a 1-cover, LPF may contain other vertices that are not cut o by (5). We present next another family of facets that cut o some of these vertices of LPF . In the theorem that follows we de ne a pair of adjacent indices t1 ; t2 2 N , and we postulate the existence of a 4

third index t3 2 N ? t1 that is adjacent to t2 . Note that this implies that 2  t2  n ? 1. When 2  t1  n ? 1, t1 also is adjacent to another index of N besides t2 . We denote this index by t0 .

Theorem 3 Let I  M be a 2-cover, and suppose 9 i0 2 I with a 0 > 0. Let t ; t 2 N be two adjacent indices, and T 0 = ft 2 N : jt ? t j  2 and jt ? t j  2g. Assume 9 t 2 N ? t that is (2)

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adjacent to t2 . De ne

a~ = maxf0; aa 0 (b ?

X

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2 ?f 0 g I

2

3

1

a )g; i 2 I ? i0: r

i;i

The following family of inequalities is valid and facet-de ning for PF . 1) If 9 t0 2 N ? t2 that is adjacent to t1 :

X

a (x 1 + x 2 ) + a 0 x 0 1 + a(2) 0 (x 0 2 + x 0 3 ) + i

2?

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i0

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Xa X x (2)

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2) If 6 9 t0 2 N ? t2 that is adjacent to t1 :

X 2 ?0

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T0

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a (x 1 + x 2 ) + a 0 x 0 1 + a(2) 0 (x 0 2 + x 0 3 ) +

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+

X I

(2) i

2

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2?

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Xa X x i

a~ x 0  2b

T0

i0

(6)

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 2b:

(7)

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Another family of facet-de ning valid inequalities for PF , which can be seen to be dierent from the family in theorem 3, is given below.

Theorem 4 Let I  M be an l-cover, 3  l  n?1 and suppose that 9i 2 I with a > 0. Consider P a partition of I into I and I with 2 1 a > (l ? 1)b and a > 0 8i 2 I . Let T = ft ; : : :; t g  N be a set of adjacent indices, and T 0 = ft 2 N : jt ? t0 j  2 8t0 2 T g. Assume 9 t 2 T ? t ? that (l)

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(l)

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2

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l+1

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is adjacent to t . Then, l

X a Xx + X a X 21

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2

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T

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22

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2 ?

t

T

tl

x + it

Xa 22

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(l)

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(x + x

is a family of facet-de ning valid inequalities for PF .

5

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+1 ) +

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Xa X x (l)

2

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2

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T0

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 lb

(8)

2

Theorem 5 Let x be a vertex of LPF that does not satisfy the SOS constraints (4). Then there are inequalities among (5), (6),(7), and (8) that are violated by x.

4 Computational Experience We now report computational experience using a branch-and-cut scheme to solve (F ). We use as cuts the inequalities (5) for l = 1, and we compare our results with the traditional branch-andbound approach on 55 randomly generated test problems.

4.1 Test Problems The complexity of (F ) grows exponentially with the number of special ordered sets, even when all sets have 3 variables. We then generated 55 test problems randomly, each with a dierent seed, and we xed the number of variables in the sets while increasing the number of special ordered sets. In all test problems each special ordered set had 20 variables, and the number of sets ranged from 100 to 200 with intervals of 10. For a given problem size we tested 5 dierent instances. The objective function coecients were randomly generated between 1 and 100, the knapsack coecients were randomly generated between 1 and 20, and b was equal to 20 in every problem. Note that the number of variables in the model is equal to the number of special ordered sets times the number P of variables in each set. Also the number of convexity constraints 2 x  1 is equal to the number of special ordered sets, while the number of knapsack constraints is always equal to 20. Thus the size of the problems ranged from 120 constraints (not including nonnegativity) and 2,000 variables to 220 constraints and 4,000 variables. t

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4.2 Node Selection, Branching and Separation Strategies We tested two dierent node selection strategies, depth- rst-search and best bound, and we found that best bound performed better by far. Therefore, computational results reported use best bound. The choice of branching set is done as follows. When testing the feasibility of the LP relaxation solution at a given node we test the feasibility of each set fx 1; : : :; x g begining with i = 1 until an i

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infeasible set is found. Let i be the smallest value of i for which at least two variables not adjacent were found to be positive. Let k1 2 N be the smallest index of a variable in the set corresponding to i that appears in a SOS2 violation. We then choose fx  1 ; : : :; x  g to be the branching set, P 1 x  = 0 and P = +2 x  = 0 to the branches. and we add the constraints =1 1 The separation problems for the families of inequalities (5) are NP-Complete. We use a separaP tion heuristic based on the following observation. Let x~ 2 LPF and 2 a x~ = b, I  M; t 2 N . If x~ 0 > 0 and x~ 0 0 > 0, i0 2 I and jt0 ? tj  2, x~ is cut o by (5) whenever I is a 1-cover and a(1) 0 > 0. We then use the following strategy for separation. Let i1 ; : : :; i 2 M , with i1 <


< i , be such that the corresponding set contains a SOS2 violation. Starting with i = i1 until i = i or a cut is found, let t1 ; : : :; t 2 N , with t1 


 t , be the indices of the variables in the set corresponding to i that appear in a SOS2 violation. Suppose i is the index being currently considered. Starting with t = t1 until t = t or a cut is found, we search for a t for which P 2 a x~ = b, where I = fi 2 M : x~ > 0g, I is a 1-cover and a(1) > 0. The inequality (5) P a x + P a(1) P 0 x  b cuts o x~. 2 2 2 When an inequality (5) is found, it is added to the constraint set of the current node, and another LP-relaxation is solved. We repeat this process until we nd a feasible solution, in which case the current node is fathomed, or the optimal value of the new LP-relaxation is smaller than the value of the best solution found so far, in which case again the current node is fathomed, or we cannot nd a cutting plane, in which case we branch. i

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4.3 Computational Results We used an IBM RS6000/550 to run our test problems and MINTO 3.0 as branch-and-bound algorithm and CPLEX 4.0 as LP solver. We limited the size of the branching tree to 50,000 nodes. Of the 55 problems tested, the usual branch-and-bound approach could not nd a feasible solution for 19 problems, and it could not nd an optimal solution or prove optimality for 5 problems. With the branch-and-cut approach, however, all 55 problems were solved to proven optimality. 7

Table 1 has for each problem size the average CPU time in seconds with and without cut generation to obtain a proven optimal solution and the rst feasible solution. Table 2 has for each problem size the average number of nodes processed with and without cut generation, as well as the number of cuts generated, to obtain a proven optimal solution and the rst solution. For those problems that were not solved to optimality we included in the averages of table 2 the number of nodes processed (50,000) when the algorithm was halted, as well as the computational time spent so far in the time averages of table 1. The same was done for the averages relative to the rst solution for those problems for which branch-and-bound failed to deliver a feasible soluiton. Table 1: Time with and without cuts

SETS CONS VARS NO CUTS CUTS NO CUTS CUTS 1st SOL 1st SOL 100 110 120 130 140 150 160 170 180 190 200 Total

120 130 140 150 160 170 180 190 200 210 220

2,000 2,200 2,400 2,600 2,800 3,000 3,200 3,400 3,600 3,800 4,000

3,263 3,530 3,474 3,993 4,473 2,116 4,572 3,508 3,486 5,685 3,012 41,112

718 4,000 498 1,102 1,706 318 1,448 486 465 2,281 1,082 14,104

3,049 3,277 3,117 4,978 4,443 1,780 3,825 3,226 3,225 5,490 2,728 39,168

560 2,784 400 772 1,015 295 1,042 453 240 1,723 989 10,273

From tables 1 and 2 we can see that the above strategy reduced the computational time and the number of nodes processed in the test problems considerably, even though only one family of 8

Table 2: Nodes and cuts

OPTIMALITY 1st SOLUTION SETS NODES NODES CUTS NODES NODES CUTS NO CUTS W/ CUTS NO CUTS W/ CUTS 100 110 120 130 140 150 160 170 180 190 200 Total

33,798 30,582 33,129 36,055 34,978 19,745 30,962 25,233 22,359 36,950 20,253 324,044

5,481 9,421 18,542 32,830 3,408 5,331 8,031 12,131 8,009 13,750 2,272 4,127 6,251 11,020 2,375 3,966 2,124 3,353 9,107 14,012 4,231 7,491 69,831 117,432

29,830 28,751 27,786 34,041 34,785 18,100 27,082 22,600 20,646 34.499 17,750 295,870

3,477 13,083 2,068 4,861 4,602 2,175 4,468 1,954 478 5,119 3,749 46,034

9,196 26,113 4,133 9,719 9,202 3,886 6,474 2,739 1,825 10,234 7,212 90,733

inequalities was used and the separation strategy was rather simple.

5 Further Research In this paper a polyhedral approach is used to tackle a problem with non-linear constraints and continuous variables. This approach has been successfully applied to 0-1 linear problems for over 25 years now. However, to the best of our knowledge, many relevant questions relative to continuous models have not been addressed yet, such as how to obtain facets in general, or how to use them in branch-and-cut schemes. Some of these questions have been studied in [2] and are the subject of other forthcoming papers. 9

References [1] Beale,E.M.L. and J.A. Tomlin, \Special Facilities in a General Mathematical Programming System for Non-Convex Problems Using Ordered Sets of Variables," in Lawrence, J., ed., Proceedings of the Fifth International Conference on Operations Research , Tavistock Publications, London 1970, 447-454. [2] de Farias,I.R., \A Polyhedral Approach to Combinatorial Complementarity Problems," Ph.D. Thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 1995. [3] Gue,K.R., G.L. Nemhauser, and M. Padron \Production Scheduling in Multiprocessor Flowshops," LEC, Georgia Institute of Technology, 1994.

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