THE MULTIPLE SET-UP ORDERING COSTS IN INVENTORY PROBLEM

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1 J. Operatios Research Soc. of Japa Vol. 19. No. 1. March The Operatios Research Society of Japa OPTIMAL BATCH (8,S) POLICIES FOR THE MULTIPLE SET-UP ORDERING COSTS IN INVENTORY PROBLEM JUNICHI NAKAGAMI Ch iba Uivl' r s ity (Received April ; Rised Se,Ptemher ) ABSTRACT Cosider the dyamic ivetory problem whe the orderig cost fuctioyl is liear with multiple set-up costs. I geeral, a optimal ivetory policy is sesitive to the form of the orderig cost, so that the purpose of this ote is to defie a ew policy, ie, a batch (s,s) policy ad to show the sufficiet coditios uder which this policy is optimal. 1. INTRODUCTION Cosider the sigle item, periodic :~eview, stochastic ad dyamic ivetory model whe the orderig cost fuctio is liear with multiple set-up costs rather tha oe with a sigle set-up cost. This type of cost is either covex or cocave, but has a :practical meaig whe the ordered quatity i each period is delivered by trasportatio's vehicle which has certai limited capacity. I geeral, a optimal ivetory policy is sesitive to the form of the orderig cost, so that util ow some types of ivetory policies have examied ad studied for several authors. Scarf [4J proved that a (s,s) policy is optimal for a liear cost with a sigle set-up,ad this case was ivestigated i detail by Iglehart [lj, Veiott [5J et al. Recetly Porteus [3J proved that a geeralized (s,s) policy is optimal for a cocavely icreasig cost. The purpose of this ote is to discuss the Lippma [2J's model i which the orderig cost has multiple set-ups, ad show the optimality of "batch (s,s) polices". 91

2 92 Juichi Naka,qami K+cM c(z) ~ ~ I I K o M 2M 3M Fig.l Graph of c(z) 2.FORMULATION OF THE MODEL I this sectio we discuss the model which explicitly allows for the ucertaity i demads ad make assumptios to keep the otatio simple. Less restrictive assumptios uder which the results of this paper still valid are give i Sectio 4. Let c(z) deote the orderig cost fuctio with multiple set-up as follows: (1) c(z) = K{~} + cz for z > 0, where c ~ 0, K, M > ad {z} is the miimum iteger ot smaller tha z. We illustrate this by Fig. 1. 'l-.1he we iterpret M as the capacity of a trasportatio vehicle, K as the cost of its use ad c as the uit cost of the treated item, the c(z) is more reasoable for if vehicles of the trasportatio are trucks the orderig cost is a fuctio oly of the umber of trucks required to satisfy the order ad ot of the fractio of truck space used. (if excess space Gaot be used.) It is specifically assumed that orders are delivered immediately, shortages are backlogged ad the objective is to miimize the total expected cost attributed to orderig, holdig ad pealty for shortages over periods. The quatities demaded i each period are idepedet, idetically distributed, oegative radom variables with COmmo p.d.f. ~('). Costs to be icurred periods i the future are dis couted by the factor a, where 0 < a ~ 1. Let holdig ad shortages costs charged o edig ivetory i each period be deoted by 1('), the the oeperiod expected holdig ad shortage cost for the level y of ivetory after Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

3 Optimal Batch Policies i Ivetory Problem 93 orderig is deoted by We assume g(y) exists for each y. As usual, let f (x) be the miimum expeeted cost over periods as a fuctio of the :"evel x of ivetory before orderig. We have, f (x) if [ c(y-x) + g(y) + a!~ f_l(y-~).(~)d~ J y>x ( 1,2,..., fo(x) - 0 ). Let (4) The G (x) f (x) = if [ G (y) + K{Y-x} J.. cx M y~x ( = 1, 2,... ; fo(x) " 0 ). Let Y (x) deote the optimal ivetory policy i the period, ie, the optimal level of ivetory after orderig i the first of periods whe the level of ivetory before orderig is x. Y (x)-x f (x) = G (y (x)) + K{ M } -cx 3. OPTIMALITY OF BATCH (s,s) POLICIES The we have for every x. I this sectio we shall give a defiitio of batch (s,s) policy ad some sufficiet coditios uder which this policy is optimal i the fiite horizo problem. Defiitio 1. A batch (s,s) policy is a. ivetory policy defied by parameters s, S with s < S ad M (>0), su(~h that Y(x) x for x > s Y(x) mi s-x ) S, x + M{M} for x < s. We illustrate this by Fig. 2. A bateh (s,s) policy has a followig S-s ecoomic iterpretatio. I case 2, where {M} ~ 2, M is smaller tha S-s, ie, the maager has small-sized trucks for trasportatio use as compared with a satisfig level regio (s,sj. the he orders a miimum amout of the item with full-loaded trucks so as to rai:3e the ivetory level upto the Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

4 94 Juichi Nakagami Case 1: {8-s} ~l 1 8 s --- x s-2m 8-2M s-m 8-M s Case 2: {S-s} > 2 M Y(x) x s-3m s-2m s-m s 8 Fig. 2 Batch (s,8) policy rigio (s,8j if the iitial level is less tha s; batch policy. I case 1, 8-s where {~} = 1, the maager has large-sized trucks, the he caot order the item with full-loaded trucks so as to raise the ivetory level ito the regio (s,8j. 80 that he raise the ivetory level ot to exceed 8 with trucks which are ot always full-loaded if the iitial level is less tha s; batch policy + (s,8) policy. to the well-kow (s,8) policy. If M is sufficietly large this policy is idetical Theorem 1. If G (x) is covex ad bouded below, the a batch (s,8) policy is optimal i period. Proof. From otatioal coveiece, we abbreviate the subscript. By our assumptio o G(x), there exist the smallest real umbers sad S with s < 8 (which may be :00.), such that Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

5 Optimal Batch Policies i Ivetory Problem 95 (6) G(S) ~ G(x) for all x, [G(x) - G(x+m)J < K for all x ~ s. S-s Case 1, {M} = 1 : (i) We have, for ay x ad y with s ~ x < y G(y) + K{y~x} > G(y) + K > G(x) [by (7)J. Thus it follows that (8) Y(x) = x o [s,oo) (ii) We have, for ay x ad y with S-H < x < s, x < y, Hece we get G(y) + K{y~x} ~ G(y) + K ~ G(S) + K [by (6)J. y ( x ) = S = mi [ S, x+m im s-x} J o [S-M,s). For ay x ad y with S-2M < x < S-M, x < Y ~ x+m, we have G(y) + K ~ G(x+M) + K (eluality holds iff y = x+m). Thus it is easily show by iductio that for ay x with S-(d+l)M ~ x < S-dM, d = 1, 2,..., (10) mi [ G(y) + K{y~x} J ~ G(x+dM) + dk. x+dm~y~x (iii) Therefore we have for ay x with S-(d+l)M ~ x < S-dM, d=l,2,..., mi [ G(y) + K{Y~x} J = mi [G(y) + K{y-~-dM} + dk J y~x y>x+dm [by (lo)j G(S) + (d+l)k if S-M < x+dm < s = [by (9)J G(x+dM) + dk if s ~ x+dm < S [by (8)J. Hece, if S-M < x+dm < s the {s~x} ( ) i s-x} d+l, so that Y x = S ~ x + M M ' ad if s < x+dm < S the {s~x} = d. so that Y(x) = x + M{s~x} < S. The, Y(x) = mi [ S. x + M{s~x} J o (_00, s). Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

6 96 Jrmichi Nakagami Case 2, {S-s} > 2 M = Similarly to Case 1, we ca prove that Y(x) = x o [s,oo), y(x) = x + M{s~x} < So (_oo,s), ard Y(x) = mi [S, x + M{s~x} ] o (_oo,s), ad hece a batch (s,s) policy is optimal i period, which completes the proof of the theorem. It is clear that ar. ivetory policy which is optimal to use i the first of perjods is also optimal to use whe there are periods left i a m periods problem. We will therefore oly be iterested i derivig coditios which isure that G (.) is covex for every ad hece that a batch (s,s) policy is optimal for the first of periods. Defiitio ;:0. A desjty ~(') is called M-idifferet, if it satisfies for ~ f,; < M. If we divide the demaded quatities by M, the M-idifferet desities give o iformatio ab01;t vrhich quatities left are likely to occur, that is, such desities are idifferet (igorat) of the remaiig quatities. M For example, let, for > 2 ' v > 0, the a M-idifferet desity ~(f,;) v Of M ~ M 1-2 ~ ~ < + 2 otherwise is give by, where v(o) is a geeralized probability desity defied o [~,oo) ad!dv() 1 =M"' Theorem 2. If g(o) is covex ad bouded below ad ~(o) is M-idifferet, the a batch (s,8) policy is optimal for ay fiite horizo problem. Proof. Here we will show by iductio that G (.) is covex for all. For = 1, Gl(x) = g(x) + cx is covex. Assume that Gk(o) is covex. The by Theorem 1 there exist two levels sk' Sk with sk < Sk such that (11) + dk - cx 1, 2,..., Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

7 Optimal Batch Policies i Ivetorory Problem 97 Hece we have (12) if x < mi (Sk,sk+M) - s*, ad fk(x) - fk(x-dm) is odecreasig i x. Now we exami G k + 1 ( ) defied by (4). We have from cotiuity ad piecewise covexity of f k ( ) give by (11) For x < s* we have (13) = Ei:O f~-f~(x-i;:-im) <j>(i;:+im) dl;: Ei:O f~-f~(x-i;:) <j>(i;:+im) 0_1;: [by (12)J = - f'(x-i;:) dl;: 1 fm- M 0 k [by the 1<. )'s M-idifferece] = 1- r*g' (I::) dl;: - c = costat (= -C) M sk x-s* Ad for x > s* let d* = {-M---}, we have [by (11) J. f x o - s * ( [f~(x-i;:) - f~ X-I;:-Cl*M)J<j>(I;:)dl;: + f~ f~(x-i;:-d*m)<j>(i;:)dl;: [by (12)J. The the secod term is -C, it is therefore sufficiet to show that the itegrat of the first term is o-egative ad o-decreasig i x. x -s* x -s* 1 xl < x 2 ' let di {-M---}, d:2 = {-t-} respectively, the di ~ d:2. For [by (12)J f' (x -1;:-d*M) k 1 2 < f k '(x 2-1;:) - f'(x -1;:-d*M) k 2 2 [by (12)J. Thus the proof of Theorem 2 is complete. Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

8 98 Juichi Nakagam 4. CONCLUDING REMARK I this ote we have show that a batch (s,s) policy is optimal i a stadard ivetory model whe the orderig cost is liear with multiple setups. The most crucial assumptio is that the demad pdf is to be M-idifferet. Our results will also valid for the ostatioary case; c (z) = c z ~ (.) is M-idifferet, the we have If G (.) is covex ad bouded below ad f (x) = if [ c (y-x) + g(y) + a J~ f l(y-~)~ (~)d~ J - y;,x where fo(x) = lower-bouded covex fuctio, for = 1,2,... Ufortuately covexity of f (x)'s will ot be obtaied for geeral demad pdf ad hece ay batch (s,s) policy may ot be optimal. However, i may cases the demad pdf is ot determied precisely, ad a M-idifferet pdf gives a good approximatio to the demad desity by exploitig a least-square method. For istace, whe the actual demad pdf is ~(.), the M-idifferet pdf ~(.) is give by i which v*() is the miimizig v() of the itegral J (~(~) Although it is a rough approximatio it is useful to put v*(') ~(~))2 d~ ~(. )/M Ackowledgemet The author expresses his thaks to Professor M. Sakaguchi of Osaka Uiversity for his helpfull discussios ad suggestios. Refereces [IJ Iglehart, D.L., "Optimality of (s,s) Policies i the Ifiite Horizo Ivetory Problem," Maagemet Sciece, 9 (1963), [2J Lippma, S.A., "Optimal Ivetory Policy with Multiple Set-Up Costs," Maagemet Sciece, 16 (1969), [3J Porteus, E.L., "O the Optimality of Geeralized (s,s) Policies," Maagemet Sciece, 17 (1971), [4J Scarf, H., "The Optimality of (S,s) Policies i a Dyamic Ivetory Problem," i MathematicaZ Methcx1s i the SociaZ Scieces, K.J.Arrow, S.Karli, ad P.Suppes (eds.), Staford Uiversity Press, Staford,1960. [5J Veiott, A.F.,Jr., "O the Optimality of (s,s) Ivetory Policies: New Coditios ad a New Proof," J.SIAM AppZ. Math., 14 (1966), Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.