Multicriteria integer programming: A (hybrid

Mathematical Programming 21 (198l) 204-223 North-Holland Publishing Company

MULTICRITERIA INTEGER PROGRAMMING: A (HYBRID) DYNAMIC PROGRAMMING RECURSIVE APPROACH Bernardo V I L L A R R E A L Instituto Technologico y de Estudios, Superiores de Monterrey, Monterrey, Nuevo Leon, Mexico

Mark H. K A R W A N Industrial Engineering, State University o[ New York at Buffalo, Amherst, NY, U.S.A.

Received 28 November 1978 Revised manuscript received 29 September •980 Dynamic programming recursive equations are used to develop a procedure to obtain the set of efficient solutions to the multicriteria integer linear programming problem. An alternate method is produced by combining this procedure with branch and bound rules. Computational results are reported. Key words: Multicriteria Optimization, Integer Programming, Dynamic Programming.

I. Introduction This p a p e r presents a new a p p r o a c h to solving multicriteria integer linear programming problems. The a p p r o a c h is an extension of the fundamental dynamic p r o g r a m m i n g recursive equation. Relaxations and fathoming criteria which are fundamental to branch and bound procedures, are suggested to obtain an alternate hybrid approach. The problem of concern is formulated as follows. ( M C I L P ) v-max

{,=~C"xn}={Cx}, N

s.t.

~

anxn 0,

integer.

where C ~ = (ct . . . . . , cpn) t, a n = (atn, ..., aun) t and k = (k~, k2, ..., k~) denote vectors of integers, and v - m a x (vector maximization) is used to differentiate the solution from the c o m m o n single objective maximization. The solution to ( M C I L P ) is a set of integer points, X ° ( b ) , called efficient, such that if x ° E X ° ( b ) , there does not exist any other feasible point, x, with C x >- C x °

with at least one strict inequality. For the case in which kn = 1, for all n, Bitran [3, 4] and Simopoulos [19] have developed procedures that generate the set of 204

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efficient solutions for the problem. Banker et al. [I] have developed a procedure to solve (MCILP) based on an efficient frontier approach. Initial computational results are not favorable in comparison with Bitran [3, 4]. Bowman [5] provides a formulation that ensures the generation of all the efficient integer solutions, however, he does not suggest any computational procedure. The algorithm developed in this paper is an application of the work developed in Villarreal and Karwan [21], in which the serial multistage decision process is extended to a multicriteria framework. It is also a very similar procedure to that developed by Marsten and Morin [12] and Morin and Marsten [14, 15] for the maximization of a single objective problem. The plan of the paper is as follows. Section 2 provides some recursive formulations and a procedure for solving (MCILP). Fathoming criteria which are characteristic of branch and bound schemes are introduced in Section 3 to develop a hybrid procedure, and finally, Section 4 shows some computational results.

2. Development of the algorithm In the development of the algorithm, it will be assumed that all c~i >-0 and a~i --- 0. The resulting problem could be called the multicriteria multidimensional knapsack problem. The dynamic programming recursions can be more difficult to analyze in the case a~j~0. However, this assumption is easily handled in the imbedded state approach developed in Section 2,2 and 2.3. A desirable transformation of the set of constraints, S, is the following: [Y.- UBi,m+I(Y,,).

Notice that any basic feasible dual solution can be used as an upper bound for the particular objective function value, for any of the resource efficient solutions. From this result, one can devise a scheme similar to that suggested in the single objective case by Marsten and Morin [12] for fathoming purposes. A general description of the scheme is the following. Assume that there are Q resource efficient points at stage m, and let Hmx(q) denote the vector of objective function values for the qth point. Step 1': Set I T ( q ) = 1, q = 1. . . . . Q. Solve (D) for C~ and ( b - Y0- Perform sensitivity analysis for each C~, i = 2 .... ,p. Form the upper bound vector UB,~+t(Y0 and test if for any element, LB i E LB

H,,x (1) + UBm+,(Y,) -< LB i, with at least one strict inequality is satisfied. If so, eliminate this point by setting IT(1)=0. Setl=l,d=2,/3*=b-Y,,andC*=Ca. Step 2': For q = d . . . . . Q with IT(q) = 1, test if for any LB~ ~ LB Hmx(q) + U B D U A L -< LBi

with at least one strict inequality is satisfied. If so, eliminate the associated qth point by setting IT(q) = 0. U B D U A L denotes the upper bound vector associated with the corresponding qth point computed using the optimal dual solutions for each of the p problems solved for the lth point, and the costs resulting from using the qth solution ((b - Y J ) . Step 3': Choose the next qth point with IT(q) = 1. Set l = d = q and perform sensitivity analysis from /3* to ( b - Y~) and for each Ci from C*. Set /3*= ( b - Yq) and go to Step 2'. If l = Q terminate the bounding procedure and continue with the regular procedure. The regular procedure is meant to be the normal recursive dynamic programming scheme. Other upper bound vectors and modified resource tours which can be used as above can be obtained from the lagrangian and surrogate relaxation of problem (P). These two tours will be called the Lagrangian Resource Tour (LRT) and the Surrogate Resource Tour (SRT) respectively. The lagrangian relaxation of problem (P), for a given resource efficient point, q, is expressed as follows. (PD

max{Cix - A { A x - (b - Yq)}:

k -> x -> 0, integer}

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215

where A denotes the lagrangian multiplier vector. The associated dual problem is given as follows. (D~)

min{ max {(C~ - A A ) x + A(b - Yq)}}. x>-{0} k->x>-{o~, integer

Clearly, if v(.) denotes the solution value for the problem (.), one has that v(D~) -> v(P). Hence, using this relationship one is able to obtain an upper bound vector from the solution of p lagrangian duals. Further, as one can notice from a rearrangement of (Dx), one may use the lagrangian relaxation solutions for any resource efficient solution by substituting for the values of Yq. The re-arrangement is the following. (DD

min{)t (b - Y~) + max (Ci - AA)x}. x>-{O}

k->x~0, integer

Geoffrion [6], Glover [7], and Greenberg and Pierskalla [8] have developed results that imply the following relationships. v(P) >- v(Da) >- v(P), where v(15) represents the solution value of the linear relaxation of problem (P). However, Geoffrion [6] also suggests that v ( P ) = v(Dx) in the case in which the problem (Px) possesses the integrality property defined below. Definition 4. The problem (P~) has the integrality property if v(P~) = v(Px) for all

A, where (P~) is the linear relaxation of (PDIn our case, one can notice that the solution to (PA) can be achieved equivalently even if one relaxes the integrality constraints on x. Thus, (P~) has the integrality property and the solution value obtained for the dual could be achieved solving the linear relaxation of (P). As a consequence, the bounding and fathoming power obtained using an upper bound vector composed of the values for the p lagrangian duals, could also be achieved by solving the p linear relaxations of (P). So, the use of lagrangian duality would be advantageous if its associated computational effort is less than that devoted to obtain v(P). The modified scheme (LRT) is structured in a similar manner to the one above. Let us now develop the concepts needed for a Surrogate Resource Tour (SRT). The surrogate relaxation for problem (P) is expressed as follows. (P~')

max{Cix: v ( A x - b + Yq) -< 0, k >- x -> 0, integer}.

The associated surrogate dual problem is given as (Ds)

min{v(Pe)}. V~0

216


Let a vector Y, be given such that vY~ -O, integer}.

Clearly, v(P ~) > v(PD for the same given v vector since the constraint for (~v) is less restrictive than for (P~). Thus a bound on objective i for resource efficient point q is a valid bound on objective i for point z if vYz v(DA) --> v(Ds) -> v(P). As mentioned earlier, v(P)= v(Da) in our case. A direct consequence of this relationship is that using the value of the surrogate duals for each criterion, Ci (i = 1..... p), to compute the upper bound vector, instead of the lagrangian duals and linear relaxations, will improve the bounding and fathoming power of the testing scheme. This is possible since the surrogate relaxation problem, (P"), does not have the integrality property defined in Karwan and Rardin [10]. A scheme similar to (LRT) is easily structured. The following section presents computational results regarding the use of several of the sets of upper and lower bounds described previously.

4. Computational results The imbedded state recursive equations described in Section 2.2 were coded in FORTRAN, and various multicriteria integer linear programming problems generated at random were solved on a CDC 6400 computer system. The objective and constraint matrix coefficients are randomly generated within the ranges [-99, 99] and [0, 99] respectively. The constraint matrix has a 90% density, and the right-hand side vector b, corresponds to 0.25, 0.50, and 0.75 times the sum of the coefficients of the associated row matrix. Computational results for the hybrid dynamic programming scheme are also presented for various bicriterion integer linear programming problems. Note that any problems for which k = 1 (i.e. xi = 0 or 1) and p = 2 can be interpreted as capital budgeting problems with dual objectives of risk versus return. Table 1 shows the solution times in CPU seconds and the number of efficient solutions for a sample of 15 bicriterion multidimensional knapsack problems with four constraints, ten variables, and upper bounds on the variables ranging from one to five. Even though the sample size for each value of k is small, there is a slight indication that, for the same given characteristic of b, the average number of efficient points increases as the value of k increases. One would expect this since the number of alternate solutions increases. One can also notice that the solution time tends to increase (average time) as the value of k

B. ViUarreal and M.H. Karwan/ Multicriteria integer programming

217

Table 1 Empirical results for multiobjective integer linear programming problems with varying upper bounds k* Problem

Efficient points

Time (CPU)

Number

Solution time

Mean

Mean

1 2 3

3.00 5.00 6.00

4.66

5.73 6.73 11.12

7.86

4 5 6

4.00 6.00 5.00

5.00

24.12 36.27 10.11

23.50

7 8 9

7.00 6.00 4.00

5.66

13.14 23.89 20.84

19.29

10 11 12

6.00 2.00 6.00

4.66

30.89 6.79 37.24

24.97

13 14 15

6.00 8.00 7.00

7.00

12.28 22.03 21.00

18.43

M = 4 , N = 1 0 , p = 2 , b=0.50. *k - - Upper bound of variables.

increases because of the larger number of alternate feasible solutions that must be tested for feasibility and dominance. One should expect, however, that this time will increase at a decreasing rate as the upper bound constraint on the variables becomes redundant to the knapsack type constraints. Table 2 contains more detailed information about the solution of a sample of 15 problems with five constraints, eight variables, three criteria, upper bound k equal to two, and b value equal to 0.75 times the sum of the associated row coefficients. These 15 problems are divided into sub-samples of three problems, each with different density on the number of negative coefficients in the objective functions or criteria. Notice that as the density of negative cost coefficients increases, the dominance concept becomes more important, and there seems to be a tendency for an increase in the number of efficient solutions. This may be a result of having alternate feasible solutions containing variables

218

z~z~a

.=.

0

.,~D

.g E m "N

e,.

~'~ ~D

~ZlZ

i

B. Villarreal and M,H. Karwan[Multicriteria integerprogramming

219

that would decrease the function values as they increase in value (they have negative cost coefficients). This will f a v o r the f o r m a t i o n of more n o n - r e s o u r c e efficient points making dominance testing more effective. H o w e v e r , the resulting set of resource-efficient solutions will contain larger sets of points such that if x and x* are included, then

•

C"x,~ ~ C"x*

n=l

n=l

n=l

n=l

and

As this subset increases, the resulting set of efficient solutions will obviously increase. One final c o m m e n t is that since dominance testing tends to b e c o m e more important, the sets of resource efficient points will tend to decrease, as will the solution times (average), A general implication that one m a y expect from the last discussion would be that dominance and the n u m b e r of efficient points are more relevant for problems that contain densities of negative criteria and constraint coefficients within the range of 4 0 % - 60%. Table 3 illustrates the usefulness of employing the hybrid dynamic procedure. In using this scheme, the heuristic developed by Loulou and Michaelides [I1] was applied to obtain sets of lower bounds to the original problems. All the problems are multidimensional k n a p s a c k problems. This p r o v e d to be very useful since m a n y of the bounds generated turned out to be efficient solutions. For this reason, it was decided that extra computational effort for improving this set was not necessary. The sets of upper bounds for the residual problems (RP) were determined by setting each of the remaining variables to their upper bound. One can notice that the reduction in the solution times to the problems (obtained f r o m Villarreal and K a r w a n [21]) is significant. In particular, for those with b-value equal to 0.75 times the sum of the associated row coefficients. Notice

Table 3 Empirical results (CPU seconds) of varying the number of lower bounds b

No bounds

Two bounds Three bounds

Four bounds

0.75*

107.75 (24.33)

7.10 (4.52)

5.99 (2.70)

6.48 (3.23)

0.50**

8.152 (0.67)

2.68 (1.56)

--

--

M = N = 10, p = 2, k = 1, 0% neg. coeff. * Mean of l0 problems (standard deviation). ** Mean of 5 problems (standard deviation).


220

that the increase in the size of the set of lower bounds (from two to four) is not of much help for fathoming purposes (with various exceptions in individual problems). This increase in the number of members of the set of lower bounds is accomplished by considering more )t values with the characteristics given in Section 3.1. The increase in the solution times for the cases in which the set of lower bounds becomes larger is due to the increase in the computational times for the extra members, and their lack of effectiveness for fathoming purposes. One main characteristic that was noticed when solving multicriteria multidimensional knapsacks is that elimination by dominance was not as helpful as we would have wished. Elimination by infeasibility and the use of bounds was more relevant. Table 4 illustrates a comparison of the solution times achieved to solve the same ten problems of Table 3 with b = 0.75 using two slightly different type of hybrid dynamic programming recursions, and a branch and bound type algorithm. The first column corresponds to the solution times achieved using the dynamic programming recursion in which sets of lower bounds are obtained using the heuristic of Loulou and Michaelides [11], and an upper bound vector is computed by setting the remaining variables to their corresponding upper bounds after stage 3. Columns 2 to 'all' are associated with the solution times used by a dynamic programming recursion that at each stage (after stage 3) uses an upper bound vector similar to the one just described for fathoming purposes, and at stage 6, the upper bound vector is formed by using the values of the dual solution for the linear relaxation of the residual problem (i.e., resource tour). Finally, the last column corresponds to the solution time obtained by using this same last recursion without the dominance test. This last approach is called 'branch and bound' because its similarity with the standard branch and bound procedures. In this procedure the branching and variable selection strategies are defined by the following stage and the values of the variable associated with that stage. This last approach was the most successful in solving the sample of problems selected. More detailed information about the behavior (per stage) of the sets of resource efficient, dominated and fathomed points in a sample of four

Table 4 Comparison of solution times (CPU seconds)* of hybrid dynamic programming recursions and branch and bound approach Simple bounding

7.10 (4.52)

Modified LP bounds 2

3

4

5

All

6.89 (4.45)

6.75 (4.48)

6.64 (4.44)

6.66 (4.46)

6.96 (4.61)

M = N = 1 0 , p = 2 , k = l , 0 % n e g , coeff. * Mean for ten problems (standard deviation).

Branch and bound 1.19 (0.18)

221

.=_ E 9

E e~

.,-

I

r~

7:

0 0 e-

0 0

r~

.=. 0

J

0

E e~

9

r~

._=

E

0

0

"te~

o

[..,~

~ . . 6 Z


222

problems, from Table 4 using the four different variations of algorithms developed in the paper is given in Table 5. One can observe the substantial decrease in the number of resource efficient points to be stored and analyzed in the later stages, that result from the application of a bounding and fathoming scheme. Table 6 illustrates a rough comparison of the solution times of samples of two problems with different b values solved by the dynamic programming recursion, the hybrid dynamic recursion, the branch and bound scheme. These are compared to the mean solution time for problems of the same characteristics solved by Bitran [4]. Comparison of solution times with those of Bitran's must be considered in view of different computer systems (Bitran employs a Burroughs B6700), and different problems generated with the same characteristics. Based on this small sample, one may say that the solution times obtained for solving problems with b = 0.25, solved by the hybrid procedure, are comparable to those obtained using Bitran's procedure. However, for values of b = 0.50 and b = 0.75, the hybrid procedure yields greater solution times than both, the branch and bound scheme and Bitran's procedure. In these cases, the solution times achieved by the later procedures seem comparable. All the problems are multidimensional knapsack problems. After analyzing these computational results, it seems very reasonable to devote more effort to the development of better bounding and fathoming schemes. In particular, the use of the lagrangian and surrogate resource tours in being currently analyzed and compared computationally. Notice that these two schemes can very well be applied for single-objective problems, and used in a similar fashion as the one developed by Marsten and Morin [12].

Table 6 Comparison of solution times of various methodologies Hybrid procedure b value

Problem number

No bounds

Bitran's scheme*

Two bounds

Branch and bound

Range

Mean

0.75

I 2

63.32 75.32

5.55 9.18

0.82 0.87

0.630.91

0.77

0.50

3 4

8.75 •2.80

2.64 7.•2

0.59 0.81

1.521.98

1.65

0.25

5 6

0.30 0.33

0.51 0.49

---

1.111.98

1.40

M = 4 , N=lO, p=3, k = l . * Sample of five problems.


223

References [1] R.L. Banker, M. Guinard and S.K. Gupta, "An algorithm for generating efficient solutions to the multiple objective integer programming problem", Working Paper, The Wharton School, University of Pennsylvania (1978). [2] R.E. Bellman, Dynamic programming (Princeton University Press, Princeton, N J, 1957). [3] G.R. Bitran, "Linear multiple objective programs with zero-one variables", Mathematical Programming 13 (1977) 121-139. [4] G.R. Bitran, "Theory and algorithms for linear multiple objective programs with zero-one variables", Technical Report No. 150, Operations Research Center, Massachusetts Institute of Technology (1978). [5] V.J. Bowman, "On the relationship of the Tchebycheff norm and the efficient frontier of multiple criteria objectives", in H. Thiriez and S. Zionts, eds., Multiple criteria decision making (Jouy-en-Josas, France, 1975) (Springer-Verlag, Berlin, 1976) pp. 76-85. [6] A.M. Geoffrion, "Proper efficiency and the theory of vector maximization", Journal of Mathematical Analysis and Applications 22 (1968) 618-630. [7] A.M. Geoffrion, "Lagrangian relaxation and its uses in integer programming", Mathematical Programming Study 2 (1974) 82-114. [8] F. Glover, "Surrogate constraint duality in mathematical programming", Operations Research 23 (1975) 434-451. [9] H.J. Greenberg and W.P. Pierskalla, "Surrogate mathematical programming", Operations Research 18 (1970) 1138-1162. [10] M.H. Karwan and R.L. Rardin, "Some relationships between lagrangian and surrogate duality in integer linear programming", Mathematical Programming 17 (1979) 320-334. [11] R. Loulou and E. Michaelides, "New greedy--like heuristics for the multidimensional 0-1 knapsack problem", Worker Paper, McGil! University (1977). [12] R.E. Marsten and T.L. Morin, "A hybrid approach to discrete mathematical programming", Mathematical programming 14 (1978) 21-40. [13] T.L. Morin and A.M.O. Esobgue, "The imbedded state space approach to reducing dimensionality in dynamic programs of higher dimensions", Journal of Mathematical Analysis and Applications 48 (1974) 801-810. [14] T.L. Morin and R.E. Marsten, "Branch and bound strategies for dynamic programming", Operations Research 24 (1976) 611-627. [15] T.L. Morin and R.E. Marsten, "An algorithm for nonlinear knapsack problems", Management Science 22 (1976) 1147-1158. [16] G.L. Nemhauser, Introduction to dynamic programming (Wiley, New York, 1966). [17] G.L. Nemhauser and R.S. Garfinkel, Integer programming (Wiley, New York, 1972). [18] C.C. Petersen, "Computational experience with variants of the Balas algorithm applied to the selection of R & D projects", Management Science 13 (1967) 736-750. [19] A.K. Simopoulos, "Multicriteria integer zero-one programming: A tree-search type algorithm", Masters Thesis, Naval Postgraduate School (Monterey, CA, 1977). [20] Y. Toyoda, "A simplified algorithm for obtaining approximate solutions to zero-one programming problems", Management Science 21 (1975) 1417-1427. [21] B. Villarreal and M.H. Karwan, "Dynamic programming approaches for multicriterion integer programming", Working Paper 78-3, State University of New York at Buffalo (1978). [22] B. Villarreal, Personal notes for PhD dissertation, State University of New York at Buffalo (1978). [23] P.L. Yu and M. Zeleny, "The set of all nondominated solutions in linear cases and a multicriteria simplex method", Journal of Mathematical Analysis and Applications 49 (1975) 430--468.