Network-based optimization algorithms for the capacitated multi-item lot sizing problem

Comput. & Indus. Engng Vol. 9, No. 3, pp. 297-305, 1985

0360-8352185 $3.00 + .DO g' 1985 P e r g a m o n Press L t d

Printed in the U . S . A .

NETWORK-BASED OPTIMIZATION ALGORITHMS FOR THE CAPACITATED MULTI-ITEM LOT SIZING PROBLEM JAMES R . EVANS Department of Quantitative Analysis, College of Business Administration, University of Cincinnati, Cincinnati, OH 45221, U.S.A.

(Received in revised form September 1984; received for publication 9 Januao' 1985) Abstract--The capacitated, multi-item lot size problem is modeled as a fixed-charge singlecommodity network flow problem. A branch and bound algorithm is developed which generally finds optimal solutions early in the search process. This property is exploited in developing heuristics that are based on the branch and bound scheme. Computational experience is discussed.

1. INTRODUCTION

In this paper we present a new approach for modeling and solving the capacitated, multi-item lot sizing problem. This problem involves determining lot sizes for multiple products having deterministic demands over a finite number of scheduling periods. Constant unit production and inventory holding costs are incurred in each period, and there is a fixed set-up cost for production in any period. In addition, there are constraints on the aggregate production (and possibly inventory) in each period. The objective is to minimize total cost while satisfying demand within the capacity limitations. This problem typically arises in the context of material requirements planning, and has also been referred to as the multi-item loading problem in the literature. It is a natural extension of the single product model addressed by Manne[10] and Wagner and Whitin[11] some 25 yr ago. The computational difficulty of this problem has been well-noted in the literature. Other than general-purpose mixed-integer programming methods, no optimization techniques other than heuristics have been proposed. These include the work of Eisenhut[3], extensions by Lambrecht and Vanderveken[8] and Dogramaci, Panayiotopoulos and Adam[2]; papers by Dixon and Silver[l] and Karni and Roll[6]. The purpose of this paper is to propose a new modeling approach for this problem which is conceptually simple and unique with respect to prior research. The formulation exploits the underlying multicommodity network flow structure of the problem and draws upon previous results concerning the equivalence between certain multicommodity networks and ordinary (i.e. single commodity) network flow problems. The model lends itself to a class of algorithms which operate explicitly on only the binary variables in the problem. Thus we feel that this approach will open up new avenues of research. Since the multi-item capacitated lot sizing problem is NP hard, little is to be gained from trying to develop a true optimization approach for large problems. We shall present a branch-and-bound algorithm which can be modified to obtain useful heuristics for the problem. Computational results will be discussed. 2. MATHEMATICAL FORMULATION

The capacitated, multi-item lot sizing problem can be formulated as follows. Define X,.t = production of product i in period t, li, = inventory of product i held from period t to t + l, Lt = aggregate production capacity in period t, expressed in units of product, d;, = requirement for product i in period t, 297

298

JAMES R. EVANS

Sit Cgt h. Ht N T 8.

= = = = = = =

set up cost for producing product i in period t, unit production cost of product i in period t, inventory carrying charge per unit of lit, aggregate inventory capacity in period t, n u m b e r of products, n u m b e r of periods, I if product i is produced in period t; 0 otherwise.

We a s s u m e that initial and final inventories are zero. The optimization p r o b l e m is then N

7

P: min ~

~] [ S . 8 . + Cit Xit ~- h . lit]

i=l

t=l

subject to i=1

.....

N

t=l

.....

T

(1)

Xi, <~ L,

t=l

.....

T

(2)

lit <<- H t

t = 1. . . . .

T

(3)

i=1

N

X i t + li,t

i -

lit = dit

N i=l N

E i=1

lit, X i t ~ O; ~it = O, 1

.....

t = 1. . . . .

I

i=1

N

.....

t = 1. . . . .

T.

(4)

(5)

We note that this formulation is more general than those previously a d d r e s s e d in the literature due to the inclusion of the aggregate inventory capacity restriction (3). Such constraints can easily be a c c o m m o d a t e d by the approach that we shall describe. In fact, lower bounds on X , and I , can easily be incorporated. These might represent m i n i m u m required production or safety stock levels in each period. Constraint (2) a s s u m e s that all products use the same a m o u n t of capacity. If this a s s u m p t i o n cannot be made, then by a transformation of variables, one can reformulate the p r o b l e m to be equivalent to P, provided that H, is sufficiently large; that is, there are no aggregate inventory restrictions. This is done as follows. S u p p o s e that (2) is written as ,%'

(2)

kiX. <~ L, i=1

where ki is a " c a p a c i t y absorption coefficient" that reflects the amount of capacity used in producing one unit of product i. Then k i X , represents a m e a s u r e of equivalent units of capacity. Define yi, = k i X , . Then (l) becomes y./k

+ li,,

J

--

lit

=

di,

or

y , + k,L~

I - kiI,, = k, di,

Ill

Network-based optimization algorithms

299

By defining J i t = kill, and eit = k i d i t and appropriate substitution, one obtains the form of problem P in which the variables are now expressed in equivalent units of capacity. Constraint (3) h o w e v e r , becomes N

Jit/k~ <~ Ht

(3)

i=1

If Ht is sufficiently large, then this can be replaced by ~ = ~ J . <~ H,. In the models discussed in prior literature, aggregate inventory restrictions were ignored, so our model is sufficiently general. 3. NETWORK FLOW PROPERTIES OF THE MULTI-ITEM LOT SIZING PROBLEM The capacitated multi-item lot sizing problem P can be regarded as a classic multicommodity network flow problem with fixed charges. Define z~ to to be the v e c t o r (Sll, ~ 2 . . . . . ~lr, 821 . . . . . 8N~). Then, if A is fixed, the problem [called p(A)] becomes an ordinary linear multicommodity network flow problem. Problem P(A), however, has special structure that lends itself to efficient solution as a single commodity network problem. This result follows from[4] regarding transformations of multicommodity networks. Specifically we may state the following: Proposition 1: Problem p(A) is equivalent to N

T

p ' (A): ~] • /=t

N

T

Sit ~it "~- min ~

~

t=l

/=1

[Cs, X , + hs, Is,]

t=l

subject to: Xit -- lit + l i , t - 1 = dit

N E S i t -- OLt =

-Lt

i=1

.....

N

t=

1,...,T

t=l

.....

T

(7)

t=l

.....

T

(8)

i=1

.....

N

(6)

i=l N

c~ +

f3~_l

-

13t =

Lt

-

Ht

+

Ht-1

-

~. di~ i=1

Xit ~ L t ~it

t=

1. . . .

,T

(9) (10)

13o = 13r = 0

Ho=Hr=O lit, Xit ~ 0

i = 1. . . . .

N

(11)

The variables O/. t and [3, are the slack variables for constraints (2) and (3) respectively. The p r o o f follows from T h e o r e m 1 of [4]. Since e v e r y variable in constraints (6), (7), and (8) appears twice with opposite signs, problem P ' ( ~ ) is an ordinary network flow problem. An example for N = 3. T = 4 is shown in Fig. 1. The r.h.s, of (6)-(8) determine the supply/demand at the nodes, with a positive (negative) value corresponding to a supply (demand). Constraints (9) serve as simple upper bounds on the production arcs and are actually redundant. T h e r e f o r e , for any vector A, optimal values of production and inventory can be found by solving a simple network flow problem. If any Xi, = 0 in the optimal solution and ~i, = I, then 8;, is set to zero since the fixed cost would not be incurred. Therefore, the solution o f the original problem P can be effectively reduced to developing an enumerative algorithm involving only the variables ~i,.

300

JAMES R. EVANS dll

111 7~ L I - HI - Zd

I21 I31

x22

"~,~ L 2 - H 2 + H 1-~0i2

x32

II2 /32

132

t) d23

13 x23

('3

x33

L3 -H3+H2-Zdi 3 i

,

If3 123

B3

d14 x14

I33

~4

x24 ~ x34

Q4

L4 ÷H3_Zdi4

Fig. 1. Network flow representation of P'(A).

4. B R A N C H - A N D - B O U N D ALGORITHM

In this section we shall discuss a branch-and-bound algorithm based on solving P(A). Throughout this discussion, A0 represents an ordered list of the variables 8i,. At any stage of the algorithm, the variables 8,-t are either fixed at 0 or "free", that is 8,, = 1 only ifXi, > 0 in a particular solution. Figure 2 illustrates the logic of the algorithm. It employs a straightforward LIFO strategy based upon the ordered list ~o, that is, the last variable fixed at zero is made free when backtracking. Initially all 8i, are free and P(A) is solved. The resulting solution Zo yields an upper bound UB on the optimal solution. At any node in the tree, the next member of Ao is set to zero and the linear programming relaxation of P(A) is solved, yielding an objective value ZLp. If Zt.p < UB, then the unallocated portion of the fixed costs is computed and added to ZLt, yielding the true objective value Z for P(A). If the LP is infeasible or if ZLe ~ UB we backtrack. If 2 < UB a new incumbent is found; otherwise the search continues. We next discuss some implementation details. In preliminary experiments, four methods were selected for ordering the list Ao. These are: (1) sort ~it by largest &, first; (2) sort g,-, by smallest di,/Si, first; (3) sort 8;, by smallest di, first; and (4) soft 8it by largest d , first. Each of these methods has some intuitive rationale. Rule (1) attempts to reduce fixed costs by the largest amount during the initial phases of branching. Rule (2) is based on the reasoning that it makes little sense to produce a small amount for a relatively high fixed cost. Rule (3) is similar, but ignores the fixed cost. It assumes that small amounts

301

Network-based optimization algorithms

Sort the vector

/~.

Inilialize

UB.Z°

of AO tO zero

~

r !

,k Branch to next node

J

Yes Yes

No

LP 1

Solve relaxation

~c kfrQckto 1 k ~edecessor node

I

Yes

No

Yes

T Fig. 2. B r a n c h - a n d - b o u n d algorithm.

of production can be absorbed in other periods. Finally, rule (4) attempts to produce infeasibilities early in the branching process since large quantities of production are unlikely to be absorbed in other periods. This effectively reduces the number of binary variables that need be considered very early. For 10 test problems ranging in size from 3 to 5 periods and 3 to 5 products, rule (4) performed better in overall computation time but had one distinct disadvantage: the optimum solution was not obtained until the end of the search. In contrast, optimum solutions using rule (3) were consistently found very early in the search; thus if premature termination of the algorithm were necessary, an optimal or near optimal incumbent would be available. This rule was chosen for further experiments which we will discuss later. Rule (1) was consistently the worst while rule (2) showed little difference from (3). A second consideration is the choice of a lower bounding scheme. Many branchand-branch algorithms use a linear programming relaxation for lower bounds. Traditional LP approaches to this problem (such as Manne's[10]) are not computationally feasible since they contain large numbers of columns. For this problem, however the transformed network flow model provides a convenient method for lower bounding via the linear programming relaxation. Consider the following general formulation of the fixed-change network flow problem: min ~ E CijXij ~- E E fij~ij i

j

i

j

Z x , j - E xj, = b, J

J

X,j <~ U,~ au X,j >~ O 5ij -~" O, 1.

302

JAMES R. EVANS

In this model, (i,j) and bi is sink, and bi = on ~q. Define and Uq. Then (I) min

fir represents the fixed charge on arc (i, j); Uq the upper bound on arc the net supply at node i. That is, bi > 0 if i is a source, bi < 0 if i is a 0 if i is a transshipment node. Now, let us relax the integrality restriction Yq = U~j ~q. Yi2 is thus a continuous variable with values between 0 the linear programming relaxation can be expressed as

i

j

i

Ex

j

j-

J

J

X~j -

Y~s <~ 0

Yq ~ Uq

Xi~, Yij >! 0 N o w consider the following related problem: (II) min

E E (Cij + fij/Uij)Xij i

j

Ex,j j

- Exj

= hi

j

O <~ Xis <~ Uq. Proposition 1. Problems (I) and (II) are equivalent. Proof." Let (x, y) be any optimal solution to (I). Suppose Xq < Yij. We note that (fij/ Uij) S i j "~ (fij/Uij) Yij. Thus setting Yq = Xq will result in a better feasible solution. We conclude that the constraint Xij - Yq <~ 0 must be tight at optimality, and the result follows. This is a more general statement of a similar result used by Kennington[7] for fixed-charge transporation problems. The third consideration that we wish to discuss is the solution of the network flow subproblems. The two most common algorithms for minimum cost network flow problems are the out-of-kilter algorithm and primal methods based on the simplex algorithm[5]. The out-of-kilter algorithm was chosen since it is widely available and easy to implement. Second, and more important, it can be initiated with any solution satisfying flow conservation at the nodes of the network. In this respect, flows can be saved from one subproblem to the next during the course of the branch and bound algorithm. It is easily adaptable to updating optimality/feasibility when costs or capacities are changed. Such is the case when solving the L P relaxation or fixing some to zero. If a primal code were used, dual simplex iterations would be it necessary, requiring substantial modifications to existing codes. With the out-of-kilter, only a few iterations are usually required to update the solution to the last network problem that was solved. 5. COMPUTATIONAL EXPERIENCE AND HEURISTIC MODIFICATIONS

We first wish to emphasize that these computational results are preliminary and are intended to illustrate the flexibility of the modeling approach, and not meant as a comparative computational study with other processes. We envision the basic model to be used on a interactive basis, and perhaps even tied into "expert systems" of artificial intelligence. This is plausible since the only required decisions are whether or not to produce a product in each time period. Production planners need not explicitly worry about production quantities and can include other relevant, non-quantifiable variables into the decision-making process.

Network-based optimization algorithms

303

Table 1 presents computational results for a set of randomly-generated test problems using the mind;, sort rule (3). The algorithm was coded in FORTRAN and run on an Amdahl 470/V6. In addition to these, 2 problems each of 4 periods, 8 products; 6 periods, 6 products; and 6 periods, 8 products were run, none of which terminated within 300 sec. Despite these seemingly grim results, the time to optimum clearly shows that optimum solutions are found very early in the procedure. Also, analysis of problem solution statistics showed that a large number of infeasible network problems were being solved. In an effort to reduce computational times, two heuristic modifications to the algorithm were added. Consider a particular solution to P(A) somewhere in the tree. Suppose that ~,s is set to zero in branching from that node. A sufficient condition for the resulting solution to be feasible is that there is enough residual capacity in prior periods to absorb the demand in period s; that is d,s

(L,

-

t
i

where Xi, is the optimal production of product i in period t for P(~). This condition is not necessary since it might be possible for a reallocation of production of other products to free up enough capacity to be feasible. The inclusion of this simple condition enhances the performance of the algorithm. Computational results are given in Table 2. The average total time was reduced by over 70% from that in Table I and the optimum was found in nearly all cases. However, this modified algorithm still could not solve larger problems (T × N > 32) within 300 sec. A second modification was added, drawing from the fact that optimum solutions are generally found early in the tree. A limit of 5 × T × N on the number of feasible solutions generated by the search was made. This value was chosen since statistics showed that most optimal solutions of the first 16 test problems were Table I. Computational results with branch and bound algorithm Problem

Time to Optimum(sec.)

Periods

Products

I

4

3

.06

Total Time (sec.) .64

2

4

3

.31

1.65

3

4

3

.45

3.44

4

4

3

.08

1.44

5

4

4

.48

1.10

6

4

4

.18

2.06

7

4

4

1.08

5.05

8

4

4

8.25

45.17

9

4

5

.20

1.19

10

4

5

.83

13.21

11

4

5

1.16

137.18

12

4

5

11.62

164.71"

13

4

6

.75

14

4

6

18.39

275.17"

15

4

6

.63

284.90*

16

4

6

52.17

>300

* execution terminated when 2000 feasible solutions have been generated; "optimum" is best incumbent

304

JAMES R, E;'ANS

Table 2. Computational results with infeasibility condition check

Problem

Time to Best (see.)

Total Time (sec.)

Optimum found(*) or deviation f_romoptimalitZ

Periods

Products

I

4

3

2

4

3

.08

.77

3

4

3

.33

2.04

*

4

4

3

.07

.60

*

.07

.26

*

.2%

5

4

4

.21

.46

*

6

4

4

.17

.84

*

7

4

4

.28

.57

*

8

4

4

3.80

22.41

*

9

4

5

.22

.94

*

I0

4

5

.44

2.60

*

II

4

5

.55

9.74

.2%

12

4

5

4.80

13

4

6

,52

14

4

6

17.48

15

4

6

.56

13.82

*

16

4

6

.50

28.74

*

98.93**

*

7,43

*

185.50"*

.06%

** execution terminated when 2000 feasible solutions have been generated.

Table 3. Computationalexperience for larger problems: total time (percent deviation from best solution) Periods

Straight P r o d u c t s Branch & bound

Infeasibility check only_

4

8

> 300 (0)

27.31

4

8

> 300 (.26)

6

6

> 300 (.49)

> 300

6

6

> 300 (1.15)

Infeasibility check + node l i m i t

(1.75)

18.33

(1.75)

(0)

15.60

(0)

(0)

33.62

(0)

> 300

(0)

29.92

(3.03)

143.56

6

8

> 300 (0)

> 300

(9.26)

6

8

> 300 (2.24)

> 300

(0)

111.79 62.92

(10.48) (.45)

found well within this limit. Table 3 shows the computation times and quality of solution for these larger problems. In all but one case, this heuristic either found the best solution or was within 3% of the best found by the other approaches in significantly less time. A larger node limit will obviously increase computational time but also improve the likelihood of terminating with the optimal solution. 6. S U M M A R Y

We have developed a new method of modeling the capacitated multi-item lot sizing model that lends itself to a variety of heuristic search techniques and opens up new avenues of research for this problem. Heuristics discussed in this paper were based on a branch-and-bound algorithm, although the model can be used in conjunction with local search and other heuristic strategies. No effort was made to develop tighter bounds

Network-based optimization algorithms

305

than the linear programming relaxation; the inclusion of improved bounds will improve the overall performance of the algorithm and its heuristic modifications. REFERENCES 1. P. S. Dixon and A. S. Silver, A heuristic solution procedure for the multi-item, single level, limited capacity, lot sizing problem. J. Ops Magmt 2, 23-39 (1981). 2. A. Dogramaci, J. C. Panayiotopolous and N. R. Adam, The dynamic lot-sizing problem for multiple items under limited capacity. AIIE Trans. 13(4), 295-303 (1981). 3. P. S. Eisenhut, A dynamic lot-sizing algorithm with capacity constraints. AHE Trans. 7(2), 170-176 (1975). 4. J. R, Evans, A single-commodity transformation for certain multicommodity networks. Ops Res. 26, 673-680 (1978). 5. P. A. Jensen and J. W. Barnes, Network Flow Programming. Wiley, New York (1980). 6. R. Karni and Y. Roll, A heuristic algorithm for the multi-item lot sizing problem with capacity constraints. lEE Trans. 8, 241-247 (1976). 7. J. Kennington, The fixed-charge transportation problem: a computational study with a branch-and bound code. AIIE Trans. 11(4), 319-326 (1976). 8. P. R. Kleindorfer and E. F. P. Newson, A lower bounding structure for lot-size scheduling problems. Ops Res. 23(2), 299-311 (1975). 9. M. R. Lambrecht and H. Vanderveken, Heuristic procedures for the single operation, multi-item loading problem. AIlE Trans. 11(4), 319-326 (1979). 10. A. S. Manne, Programming of economic lot sizes. Mgmt Sci. 4(2), 115-135 (1958). 11. H. M. Wagner and T. M. Whitin, Dynamic version of the economic lot size model. Mgmt Sci. 5(1), 8996 (1958).