A Vertex Ranking Procedure for Solving the Linear Fixed-Charge Problem

A Vertex Ranking Procedure for Solving the Linear Fixed-Charge ProblemAuthor(s): Patrick McKeownSource: Operations Research, Vol. 23, No. 6 (Nov. - Dec., 1975), pp. 1183-1191Published by: INFORMSStable URL: http://www.jstor.org/stable/169628 .

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Patrick McKeown 1183

A Vertex Ranking Procedure for Solving the Linear Fixed-Charge Problem

PATRICK McKEOWN State Unsversity of New York, Albany, New York

(Received original January 7, 1974; final, March 24, 1975)

Murty has suggested that the linear fixed-charge problem can be solved by ranking the vertices of the polyhedral constraint set according to their continuous objective values and then adding the fixed charges to determine an optimal solution. Several problems in- volving degeneracy and inequality constraints arise when one at- tempts to implement Murty's method directly. This note describes a method that uses the vertex ranking approach in conjunction with pro- cedures that resolve these difficulties. We also give computational results.

HE linear fixed-charge problem is

min Z=c'x+d'y, Ax=b,

x>P, (P)

Yi3 {0 if Xj=_O' j = 1, * ,n.

If n' equals the number of structural variables and I equals the number of slack and/or surplus variables, then n = n'+1. The A matrix is mXn; c is nX 1; d is nX 1 and nonnegative with the last 1 components equal to zero; and b is a nonnegative m X 1 vector. The cj's are the continuous or 'unit' costs, and the dj's are the fixed or 'setup' costs.

The linear fixed-charge problem was first formulated by Hirsch and Dantzig 41 in 1954. Despite its similarity in form to the standard linear programming problem, P has proven far more difficult to solve. This diffi- culty is due to the discontinuity of the objective function Z caused by the presence of the fixed costs.

The continuous portion of problem P is the associated LP problem.

min Z=C'x Ax=b, (PC)

x> 0,



11 84 Technical Notes

where all dimensions are the same as for problem P. Note that problem P, and problem P share the same feasible region.

Hirsch and Dantzig have proved that an optimal solution of problem P (if one exists) is found at a vertex of T, the feasible region of problem P,. However, no one has found a method analogous to the simplex procedure that will lead to a global solution to problem P. On the contrary, it has been shown that for a nondegenerate problem with all equality constraints, i.e., 1=0, and all positive fixed charges, all vertices are local minima.[5] Hence, other, more powerful tools in addition to the simplex procedure are needed to find a global solution to P.

Previous work on this problem has largely been centered around the fixed charge transportation problem, e.g., Gray[31 and Balinski.111 The only available computational work on the general linear fixed-charge problem is due to Steinberg, [89] who used a search procedure to solve equality constrained general problems.

Murty16] has described a procedure to reach an optimal solution to any linear fixed-charge problem by first ranking the vertices of T in nondecreasing order according to their continuous objective value. The fixed charges for the basic, nondegenerate variables at each such vertex are then added in to determine the optimal solution.

This note focuses on Murty's vertex ranking procedure. Several problems deter its direct implementation. These problems and possible solutions to them are discussed in detail. Finally, we present computational experience with an algorithm based upon improvements and extensions of Murty's procedure and compare it with previous work. This algorithm is very general and, with one minor exception, does not attempt to take ad- vantage of the special structure of any of the three types of test problems attempted. The computer program is written merely as a vehicle to test the extensions and modifications of Murty's procedure.

1. DISCUSSION OF MURTY'S WORK

A practical method of implementing Murty's suggestion is based upon two of his theorems. THEOREM 1. If Sl, S2, *, Sk are the first k vertices of problem P, and are ranked in nondecreasing order according to their objective values ZA then vertex Sk+1 must be adjacent to one of Si, Sk. THEOREM 2. Assume that the first k vertices of problem Pc are ranked in nondecreasing order according to objective value, Sk is the last such vertex, Z is an upper bound on the optimal value to P, Do is a lower bound on the sum of the fixed charges for every solution and Zk> Z-Do. Then no vertex i with value Z > Zk need be ranked for consideration as optimal.

Theorem 1 gives us a method for finding needed vertices of T, and The-




orem 2 gives a stopping criterion on the vertices that must be ranked to determine an optimal solution.

A general vertex ranking procedure for solving P first finds a value of Do that will reduce the number of vertices to be enumerated via the stopping criterion. Then it finds all vertices of P0 satisfying the stopping criterion. We will use the results of Theorem 1 to find these vertices.

2. IMPLEMENTATION OF MURTY'S SUGGESTION

To implement Murty's suggestion in a practical manner, we must solve two problems: (a) finding a suitable value of Do and (b) finding all vertices adjacent to a given vertex, Si.

Determination of a Lower Bound Do

Murty suggested that a value for Do, say Do, be found for a nondegenerate problem by summing the m smallest fixed charges. Note that Do has no relation to feasibility, nor is it necessarily a valid lower bound for degenerate problems. Finally, it is easy to see that by Murty's approach, Do will be equal to zero for a problem with m inequality constraints.

Murty stated that if the highest degree of degeneracy for problem Pc, say n, is known, then a lower bound on the fixed charges can be determined by summing the m-nh smallest fixed charges. However, it has been shown that finding n is as difficult as solving the original problem, [5]

Gray[3" discusses the use of a method for finding lower bounds in cases where an upper bound on each variable uj is known a priori or can be cal- culated. This method uses the problem:

min d'y,

Ax=b,

ujyj- O j= 1,* = n, (Pg) Oy j 1, j=l ,) n,

x ?0.

For fixed-charge transportation problems with supplies { Si } and demands Dj }, the upper bounds can be determined for each xij by calculating uij=

min { S, Dj }. However, if some elements of A are negative for the general linear fixed-charge problem, the calculation of upper bounds will require the solution of a family of linear programming problems.

A method, then, is proposed for finding a value of Do in which the feasibility of various combinations of basic variables is taken into account. This method depends upon the "extreme point optimality principle;" i.e., any optimal solution to problem P will occur at an extreme point of the continuous feasible region, T. Since an optimal solution will occur at an



1 186 Technical Notes

extreme point of T, so will any solution of problem PR that we rank. Hence, it is only necessary to find the feasible basis of problem P0 that has the lowest sum of fixed charges corresponding to the basic, nonzero variables. This is the same as solving problem P with c = 0. Finding an optimal value to such a problem yields the greatest lower bound (glb) on the fixed charges that will be present at any solution of P (i.e., the largest valid value for Do), but requires the solution of a problem that is basically the same as problem P. Hence, finding a value of Do in this fashion is not practical.

Since finding the largest possible Do is as difficult as solving the original problem, we need to find an easier way to determine a feasible basis of problem P, that corresponds to a lower value for Do. We can find such a basis by solving

A0=min EZi1 djyj, ij=n bijyj>,Bt i=l ,m (Pa

yjE{O,l1},j=1, ,n, where

6ij if aij>O, , i 1 if bi>O, 0 if a j < 0, if bi=0,

and aij, bi are, respectively, elements of the constraint matrix and the right- hand side of problem P.

Problem Ps is a set-covering problem that, if solved, would yield the minimum cost combination of columns from P, that could possibly be feasible. This minimum cost, A0, is then a legitimate lower bound on the fixed costs. However, solving Ps may also be as difficult as solving P itself. A relaxed version of Pa, termed Ps', in which yj,.{O, 1 } is replaced by 0<! yj < 1, is a linear programming problem. Hence, the optimal objective value of Pa', AO, can be found by the simplex method. It only remains to show the relation between the value of ho and the lower bound on the fixed charges to problem P. THEOREM 3. Lo, the minimum objective value for problem Ps' is a lower bound on the fixed charges that must be incurred in any solution of problem P.

Proof. Only extreme points of problem P0 need be considered. There- fore, any sum of fixed charges must be due to a combination of nondegenerate, feasible, basic variables of problem P,. The relation between problem P, and Ps' will now be shown.

Let x be a basic feasible solution for Pc, and let the index set of the basic variables be JB. Consider the kth constraint of Pc, i.e., j1 akjxj=

bk> 0. If bk>O, there is at least one jEJB with akj>0; otherwise, x would not satisfy the kth constant. Let y (x) = [y1(x), * *, yn( )I with

yj (x) = 1 if =,>0,




Hence, corresponding to any x, a basic feasible solution of Pc is a y (x) that is feasible in Ps'. Furthermore, d'y (J), the objective value of y (x) in Pa', equals the sum of the fixed charges incurred by x in Pc. Now lvo is the optimal objective value in Pa'. Thus zio cannot exceed the fixed charges incurred in P at any vertex. COROLLARY 1. If the constraint matrix of problem Pc is totally unimodular and nonnegative, then Ao Lo.

Proof. Any nonnegative, totally unimodular matrix consists entirely of ones and zeros. Hence, the constraint matrix of problem Ps is identical to that of problem Pc. Since the constraint matrix of problem Pa is totally unimodular, the LP solution to problem Ps' will be identical to the integer programming solution. Hence, Ao = Lio.

For a fixed-charge transportation problem, P0 will have a nonnegative, totally unimodular constraint matrix. For such a problem integer values of yj will be obtained. However, a lower bound for the fixed charge transportation problem will be improved if problem (Ta) is solved:

min AT = EdjyijX

Ej_1 y,j>, (TS)** M

Ei=1 yij>,Yjy, j-1*** NY

yij ?0 for all i, j,

where we have arranged the supplies Si and the demands Dj so that S1> S2>?-._SM, and S>?D,?D2?**...? DN. Then oi=min{l} such that D1+D2+ --+D>Si andj=lforj=1, * *,N.

Problem Ta takes into account the need for row feasibility of the transportation problem and yields values of Do that are greater than or equal to those found from Ps'. At the same time, Ts uses the principle proved in Theorem 3.

Finding Adjacent Extreme Points

As discussed earlier, the algorithm must be able to find the vertices adjacent to any vertex of the constraint set of problem Pc. When the given extreme point is nondegenerate, the problem of finding adjacent extreme points can be handled through the simplex tableau; i.e., one-for-one ex- changes of basic and nonbasic variables can be made by pivoting. If there are m basic variables and n total variables, we will thus find all n-m of the extreme points adjacent to a given nondegenerate extreme point.

In the degenerate case, the situation is complicated by the fact that there are multiple bases that represent a given degenerate extreme point. Hence, the procedure described above would work only if applied to all of the basic representations of a degenerate extreme point. This is exactly the method that Murty suggests.



1188 Technical Notes

We used the Chernikova-Rubin algorithm"7' to determine adjacent extreme points of a degenerate extreme point. This algorithm looks at the cone of edges emanating from the extreme point under consideration. In the case of a nondegenerate extreme point, these edges are represented by the m-n nonbasic columns of the standard simplex tableau. However, when the extreme point is degenerate, the Chernikova-Rubin algorithm must be used. This algorithm is an adaptation of the original Chernikova algorithm"2' that enables us to find all vertices adjacent to a degenerate vertex.

3. COMPUTATIONAL RESULTS

We have developed an algorithm that will solve general linear fixed-charge problems. Our algorithm will also solve transportation or warehouse-loca- tion problems, but in most cases not as efficiently as algorithms specially designed for that purpose. In addition to general linear fixed-charge problems, we tested some transportation problems, primarily to investigate the ability of this procedure to handle degenerate problems. In these cases, row feasibility was required in calculating a lower bound on fixed charges.

We tested general linear fixed-charge problems of two types. Both types were based on 240 5 X 10 problems randomly generated by Stein- berg.!8' The first type has -20<aij<20, 1?< bi<999, 1_c;<20 and 1< dj< 999, with 50 percent density and all equality constraints. Larger problems were formed by placing the 5 X 10 problems on the diagonal with zeros elsewhere. We tested nine 5 X 10 problems, five 10 X 20 problems, and one 15X30 problem, using a FORTRAN IV code on a UNIVAC 1108. These are problems 1-15 in Table I.

The second type of general linear fixed-charge problems was similar to the first, except that 0? aij5 20, 1 _ dj_ 499, and it had greater-than constraints. This type of problem obviously has m more variables than a comparable Type I problem, e.g., a 5X10 becomes a 5X15. All of the Type I problems except the 15X30 were changed to Type II problems and tested. These are problems 10-29 in Table I.

As discussed earlier, transportation problems were used to test the degeneracy aspects of the algorithm. In particular, we used four problems from GrayYl] A fifth problem (33 and 33A) was a smaller version of one of these problems (34). Each problem, if successfully solved in its original form, was changed to become degenerate (or nondegenerate). These problems are shown in Table II as 30-34 and 30A-33A.

Table I gives the average time to solution for the 29 problems solved and the fastest and slowest times for each type of problem. It also shows the average number of vertices investigated during the algorithm and comparison times for Steinberg's branch-and-bound procedure (IBM 7072). Table




TABLE I GENERAL LINEAR FIXED-CHARGE TEST PROBLEMS

Summary of Results

Problem size Constraint Ave. sol. Average Prev. ave.

type time (secs) Fast/slow vertices sol. time Number (mXn)

1-9 5X10 Equal to 0.496 0.321/0.620 20.78 10.8 10-14 10X20 Equal to 37.704 12.457/58.262 1031.6 n.a.

15 15X30 Equal to No - _ 1074.0 solution

16-25 5X15 Greater 0.617 0.346/0.943 19.22 n.a. than

26-29 10X30 Greater 26.408 0.512/81.677 497.4 n.a. than

II gives basically the same results for each of the nine transportation problems. The previous times in this case are from Gray's "decomposition approach" on a Burroughs 5500. Problems 15 and 34 were not successfully solved because of storage limitations.

Discussion of Computational Results

Two primary results may be noted from the computational experience. First, the algorithm discussed here solves 36 of the problems tested and solves problems from each type tested. In those problems that were solved, the time comparisons appear to be favorable. One favorable comparison not shown for problems 1-9 is that an average of 32.6 nodes was needed in

TABLE II FIXED CHARGE TRANSPORTATION TEST PROBLEMS

Summary of Results

Problem Rows X Degenerate? Solution Vertices Prev. sol. number columns egenera e. time (secs) scanned time

30 4X6 No 1.318 3 26.3 30a 4X6 Yes 1.388 3 n.a. 31 5X7 No 2.2286 5 3262.8 31a 5X7 Yes 4.709 6 n.a. 32 3X4 Yes 0.540 10 7.7 32a 3X4 No 0.433 12 n.a. 33 3X3 Yes 0.517 21 n.a. 33a 3X3 No 0.333 15 n.a. 34 4X6 Yes No - 32.6

solution



1190 Technical Notes

Steinberg's search procedure, as compared to 20.8 vertices in our procedure. Secondly, as Murty first surmised, the relative sizes of the fixed-charge and continuous components appear to have more effect on the efficiency of the algorithm than does the size (mXn).

If we look at Tables II and III, we can see how the relative sizes of the fixed and continuous costs affect the solution procedure. Problem 31 (5X7) is solved very quickly, while a smaller Problem 35 (4X6) is not solved at all. The difference is in the relative sizes of costs. Our procedure is more efficient for problems where the fixed costs do not dominate.

A second comparison from Table II is the effect of degeneracy. On the basis of these four problems, the effect does not seem to be very great. However, the sample is not large enough to indicate any definite results.

Three indications for future work in this area are (i) to develop a com- panion algorithm that works from the fixed-charge side for problems with relatively large fixed charges; (ii) to specialize parts of this algorithm to the

TABLE III RANGE OF PARAMETER VALUES FOR TRANSPORTATION PROBLEMS

Problem cii range di1 range Supply/demand Highest degree range of degeneracy

30/30A 2/114 0/43 8/55 1 31/31A 2/114 0/43 8/55 1 32/32A 0.64/7.60 5/14 5/50 1 34/34A 0.64/1.08 11/20 10/45 1 35 0.64/2.83 10/20 5/45 2

transportation problem, i.e., finding adjacent vertices, better lower bounds, etc., and (iii) to compare the use of Gray's bounding procedure, Pg, with the one discussed here to determine the efficiency of each for various types of problems.

ACKNOWLEDGMENTS

The author wishes to acknowledge the invaluable aid given by D. S. Rubin in the initial preparation of this paper, Many thanks are also due to Roland Minch for his careful reading and helpful suggestions.

REFERENCES

1. M. L. BALINSKI, "Fixed Cost Transportation Problem," Naval Res. Log. Quart. 7, 41-54 (1961).

2. N. V. CHERNIKOVA, "Algorithm for Findiiig a General Formula for the Non- negative Solutions of a System of Linear Inequalities," U.S.S.R. Comp. Math. and Math. Phys. 5, 228-233 (1965).




3. PAUL GRAY, "Mixed Integer Programming Algorithms for Site Selection and Other Fixed Charge Problems Having Capacity Constraints," SED-Special Report, Stanford Research Institute (1967).

4. W. M. HIRSCH AND G. B. DANTZIG, "The Fixed Charge Problem," Naval Res. Log. Quart. 15, 413-424 (1968).

5. P. G. McKEOWN, "An Extreme Point Ranking Algorithm for Solving the Linear Fixed Charge Problem," Ph.D. Dissertation, University of North Carolina (1973).

6. KATTA G. MURTY, "Solving the Fixed Charge Problem by Ranking the Extreme Points," Opns. Res. 16, 268-279 (1968).

7. D. S. RUBIN, "Vertex Ranking and Cardinality Constrained Linear Programs," Opns. Res. 23, 555-565 (1975).

8. D. I. STEINBERG, "The Fixed Charge Problem," Department of Engineering and Applied Science, Washington University, St. Louis, Mo., Report No. COO-1493-15 (1968).

9. D. I. STEINBERG, "The Fixed Charge Problem." Naval Res. Log. Quart. 217-235 (1970).



A Vertex Ranking Procedure for Solving the Linear Fixed-Charge Problem

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