average cost estimation of lot-sizing heuristics

International Journal ofproduction Economics, 26 ( 1992) 341-345 Elsevier

341

Average cost estimation of lot-sizing heuristics

Knut Richter

Technical University Chemnitz, Sektion Wirtschafiswissenschaften, O-9010 Chemnitz, Germany

Abstract

The average cost for the dynamics lot-sizing model is defined as sum of all costs associated with feasible solutions, divided by the number of these solutions. This benchmark is used to estimate the MCA heuristic (Groff) and the Silver- Meal-heuristic. The first heuristic is found to produce solutions with cost not higher than average in all inventory periods and some ordering periods while the second one is shown to provide solutions with cost higher than average in the worst case.

1. Introduction

In the dynamic lot-sizing model the demand for all periods and the set-up and holding costs are assumed to be known. It has to be decided when orders should be placed and for how many periods the demand will be covered with the lot ordered (camp. Wagner, Whitin [ 1 ] ).

Increasing the number of periods covered by the lots we decrease the set-up cost and increase the holding one and reversely, and so the problem for an optimal ordering policy arises. The optimal solution of the model provided by the method of dynamic programming has not been accepted by practice and many people have tried to develop heuristic procedures to solve the problem. These heuristics have several properties in favour against the dynamic programming approach: They are simple and can easily be understood and they reflect the way people usually decide problems.

Heuristic procedures have - of course - un- pleasant properties. They usually do not find an optimal solution. If this fact is accepted, it can- not be accepted not to know how good or bad such a solution provided by a heuristic is.

Generally heuristic procedures are estimated using the minimum cost as benchmark (camp. Miiller-Merbach [ 2 ] ). Since minimum cost it- self is not known, only their lower bounds can be

applied and thus the estimation is only approxi- mate. Iff_ denotes the cost for the heuristic solution, f * the minimum cost and b a lower bound for f *, then fH/b is usually applied as criterion for the quality of a heuristic, although&,/f * is the correct ratio.

If looking for another benchmark to be used as quality criterion, the average cost, or “policy average cost”, introduced in Richter and Voriis [ 3 1, may qualify heuristic procedures. Then heuristics securing fH/fA < 1, where fA is the average cost, can be regarded as sufficiently good. In this paper Groff’s Marginal Cost Approach [ 41 is shown to provide sufficiently good results for inventory and some other periods, i.e. fH/fA 6 1 holds, for example, if the time horizon ends in a period with no set-up. Axdter [ 5,6] noticed some time ago, that the Silver-Meal-heuristic [ 71 may perform so bad that fH/f * - + co in the worst case. We prove a similar result. For every time horizon not less than three there is a data set implying _&/ fA > 1. The investigation of heuristics in this paper does not imply new practical consequences for the inventory management. The policy average cost approach may, however, help to qualify heuristics from a uniform point of view.

09255273/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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2. The policy average cost problem

Let d,, d,, . . . . d, be the demand values for the periods i= 1, 2, . . . . T, s be the set-up cost and h the holding cost per unit and period. The set of data {d,, d2, . . . . d, s, h} determines an instance of the dynamic lot-sizing problem. A heuristic procedure applied to this problem has to provide solutions for all possible instances. It is, there- fore, of interest to know, how such procedures perform generally and for extremely chosen instances.

A policy, or feasible solution, places the orders

x, =OV C;!!;d,

to such periods that every period’s demand will be covered. There are m= 2T-’ distinct feasible solutions if d,> 0 is measured for all t. Every feasible solution has some cost value. For example, costs ST and s+h(d,+zd,+...+ (T- l)d,) oc- cur if the demand is ordered in each period or if the demand for all periods is ordered at the beginning of the first period, respectively.

The policy average cost criterion is defined as the sum of cost values associated with feasible solutions divided by their number. Let the different feasible solutions be labelled by i= 1,2, . . . . m and let the cost corresponding to i be denoted by C(i). Then the total average costf, is

m

Sometimes the symbolf, ( T) will be used to stress the planning horizon in the problem. For fA we have proved in [ 3 ] the following Theorem 1:

Corollary. fA ( T) =fA ( T- 1) +s/2 + dT( l- 2’-T)hforT>,2andf,(l)=s.

Example. Let d, = 3, d,=2, dg= 1, s= 5, h=2. Then

X4(1)=5,

f,(2)=5+2.5+2*2* ( 1-21-2)=9.5

f,(3)=9.5+2.5+2*1*(1-2’-3)=13.5

Fig. I. The policy average costs_&(t)

Graphically the policy average cost for this example is displayed in Fig. 1.

3. Estimating heuristics

3.1. The marginal cost approach (Groff[])

In this heuristic the demand d, of period t is ordered at the beginning of the first period, if

2s/h> (t- l)td, (1)

holds. If this inequality is not fulfilled, then period t becomes a new set-up or ordering period. More formally, the procedure does the following: (1) k:=l (set-upperiod).

(2) xk : = dk (xk = ordered CpaUtity in period k) . (3) Ifk=TStop. (4) t:=k+l. (5) If2s/h> (t-k) (t-k+ l)d, then (2)xk:=xk+dtandx,:=0,elsek:=t,goto

(2). Let now Cost(t) denote the cumulative cost for periods 1, 2, . . . . t if Groff’s heuristic is applied to an instance of the lot-sizing problem. For the example from Section 2 we then have

k= 1, t=2: 2:=5>4=2d* and

t=3: 2;=5<6=6d,, thus

x,=5,x,=O,Cost(2)=9

k=3,x,=l,Cost(3)=9+5=14

It can be seen that Cost(2) <fA(2) holds, i.e. x,=0 implies that the cumulative cost is not higher than average. This property will be proved now using several steps.

Lemma 1. Let relation ( 1) be fulfilled for t = 2, 3, . . . Then the Marginal Cost Approach provides a solution x with x,=0, t=2, 3, . . . such that Cost(t) <h(t).

Proof. If t = 2, then

fA(2) =3;+hd,zs+hd, =Cost(2) L L

If t = 3 then it follows from

f,(3)=2s+y+3?

=S+hd’+3hd,+2S 2 4 2

( 1) that s> 3hd, and

>s+hd +3%+jhd, 2 4 2

=s+hd +ghd, 2 4

>s+hd2+2hd,

=Cost(3)

Let now t 2 4 be arbitrary and let the statement be true for all periods less than t. Then

;+hd’(l-2l-‘).(t-l)hd, (3)

will implyf,(t) aCost( If (3)is not fulfilled, then

2s<4hd,(t-I-1+2’-‘)<4hd,(t-1)

<t(t-l)hd,

holds for all t2 4, which contradicts relation (1). 0

Remark. Groffs’ method adds the demand values d, to x1 as long as relation ( 1) is fulfilled. If ( 1) is not fulfilled, the corresponding period k will be used as an ordering period, i.e. x,> dk holds.

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Now, two different situations are possible. (i) Relation (2 ) holds for t = k+ 1 and the de-

mand of another period will be added to xk. (ii) Relation (2) does not hold and t=k+l

becomes also an ordering period. According to these different cases distinct asser- tions can be proved.

Lemma 2. Let Cost(k-l)<f,(k-I), 2s/h <(k-l)kd,and

2;> (t-k)(t-k+l)d,

for t=k+ 1, k+2, . . . where ka.3. Then the Marginal Cost Approach provides a

solutionxwithx,=O, t=k+l, k+2, . . . such that Cost(t)<f,(t).

Proof. It is sufficient to show that

Cost(t,k)=Cost(t)-Cost(k-1)

&(&k)=&(t)--fA(k-1)

where

Cost(t,k)=s+hC:,,+,d,(r-k)

and

f,(t,k)=(t-k+l);+&d,.(l-21-‘)

(i) t=k+ 1. Then the inequalities

2;~ (k-l)kd, and2:>2d,

hold and

2d,< (k- l)kd& (2’-4)dk

for ka 3 is fulfilled. It follows from this relation that

d,2-k<dk( 1-2l-!=)

This inequality implies

fA(t,k)=s+hdk(l-2’-“)+hd,(l-2’-‘)

>s+hd,=Cost(t,k)

(ii) t>k+2. In this case it is sufficient to prove the inequal-

itys/2+hd,(l-2’-‘)a(t-k)hd,;or

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+zd,(l-k- 1+21-l) (4)

Let t=k+2. Then s>3hd,, 3>2(1+2’-‘) for t35, and s/2>hd,( 1+2’-‘), i.e. relation (4) holds.

The case t b k+ 3 can be handled similarly, thus Lemma 2 is proved. 0

Lemma 3. Let k be an ordering period if the marginal Cost Approach is applied and let 2s/ h d 2d,, , be fulfilled. Then

xk+,=dk+, andCost(k+l)&(k+l)

Proof. If k= 1 then

Cost(k+1)=2sd3s/2+hdZ/2=f,(2)

Let nowk>l andCost(k-l)gf,(k-1). Then Cost(k+l)=Cost(k-1)+22sand

&(k+l)=fd(k-l)+hd,(l-2’-“)

+hd,+,(l-2-“)+s

The lemma is proved if

s<hd,J1-2’-k)+hdk+,(l-2--“) (5)

is fulfilled. It follows from the fact that k is ordering period that

2iaU(+l)dkforsome2<v<k

Let (5) be not true. Then

,,2s( l-Z’-“) (v’_V) +hd,+Jl-2-?

and

l_2(1-2’-i-) (v’-v) (6)

One can easily see that the multiplier of hd,, , in (6) is greater than 1 and thus s> hd,, 1, which contradicts the conditions of the lemma. 0

Theorem 2. Let the Marginal Cost Approach be applied to an instance of the dynamic lot-sizing problem. Then it provides a solution satisfying

Cost(t) <14(t)

for all periods t with x,= 0 or with s d hd,.

Proof. The application of Lemmata l-3 yields the result immediately. 0

Interpretation. The MCA heuristic is not worse than average if the planning horizon ends with an inventory period or with an obvious order period of type s d hd,.

Let to the example a fourth period with d., = 3 be added. Then s=5<2d,=6 and Cast(4)= 14+5= 19<Jd (4)~ 13.5 +2.5 +2*3*7/8= 21.25.

3.2 The Silver-Meal heuristic (71

In this method the demand d, of period t is ordered at the beginning of the first period, if

PCost(t-l,l)>(t-l)Cost(t,l) (7)

holds, where Cost(t,k) has been defined in the proof of Lemma 2 and Cost (0) = 0 is assumed. If this inequality is not fulfilled, then the period t becomes a new set-up period. In general, if k is the last set-up period, the test

(t-k+l)Cost(t-l,k)>(t-k)Cost(t,k) (8)

will decide, if d, is to be ordered in period k or not. Similar to Axsater [ 5,6], in this section it will be shown that for any time horizon T there exists a sequence of demands d,, d2, . . . . d,. such that the cost of the Silver-Meal heuristic is worse than the average. We restrict ourself to the cost factors s = h = 1. Let the sequence of demands

d,>O,d,=d,O<d<l,d,=d/(t-1)

be given.

(9)

Theorem 3. Let s= h = 1. Then the Silver-Meal heuristic generates for the sequence of demands (9) a solution with x1 = C,‘= ,d, and cost function

Cost(t,l)=l+(t-1)d (10)

Proof. It follows from (7 ) that such a solution will be generated if and only if Cost(t-l,l)>(t-l)2d,holdsforallt.Let t=2. Then Cost(l,l)=l>O and Cost(2,1)=1+d. Generally,

Cost(t-l,l)=l+(t-2)d

and

>(t-1)2d/(t--l)=(t--l)%,

Cost(t,l)=l+(t-2)d+(t-l)d/(t-1)

=l+(t-1)d. 0

Theorem 4. Let the conditions of Theorem 3 be fulfilled. Then for every Ta3 there is some O<d< 1 such that

Cost(t,l)>f,(t)fort=3,4 ,..., T

Proof. Let t=3. Then Cost(t)>f,(t) if and only if 1+2d>2+d/2+3d/8ord>8/9. Let Cost ( t - I,1 ) >fA ( t ) be fulfilled for d > d’ and t < T. Then Cost ( t, 1) >fA ( t ) holds if and only if

d>i+d( l-21--1)

2 t-l or

2d(t-l-1+2’-‘)>t-1 or (11)

d> t-1

2(t-2+21-f)

Since the right hand side of inequality ( 11) is less than one, there is some dad’ satisfying ( 11) which secures the inequality of this theorem for period t and, consequently also for all t=3, 4, . . . . T. 0

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Interpretation. This extreme behaviour of the method brought Silver and Miltenburg [ 81 to improve the heuristic.

Acknowledgement

The author thanks two referees for their help- ful comments on the paper.

References

Wagner, H.M. and Whitin, T.M., 1958. Dynamic version of the economic lot size model. Manage. Sci., 5( I ): 89- 96. Miiller-Merbach, H., 198 1. Heuristics and their design: A survey. European J. Oper. Res., 8(8): l-23. Richter, K. and Voros, J., 1990. The average dynamic lot size model. In: Proc. Lagerhaltungssysteme und Logistik, Math. Cesellschaft der DDR, Leipzig, pp. 267-273. Groff, G.K., 1979. A lot sizing rule for time phased com- ponent demand, Prod. Invent. Manage., 20(4): 66-74. Axslter, S., 1982. Worst case performance for lot sizing heuristics. European J. Oper. Res., 9(4): 339-343. Axslter, S., 1985. Performance bounds for lot sizing heuristics. Manage. Sci. 3 I(5 ): 634-640. Silver, E.A. and Meal, H.C., 1973. A heuristic for select- ing lot size requirements for the case of deterministic time varying demand rate and discrete opportunities for re- plenishment. Prod. Invent. Manage., 14(2): 64-74. Silver, E.A. and Miltenburg, J., 1989. Two modification of the Silver-Meal lot sizing heuristic. Infor, 22( 1 ): 56- 69.

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