Resource Allocation with Lumpy Demand: To Speed or Not to Speed?

Transcription

1 Resource Allocaton wth Lumpy Demand: To Speed or Not to Speed? Track: Operatons Plannng, Schedulng and Control Abstract Gven multple products wth unque lumpy demand patterns, ths paper explores the determnaton of both the lot sze for each product and the resource allocaton among the products, gven an nvestment budget for producton rate mprovements. Each product s optmal producton polcy takes on only one of two forms: ether contnuous producton or lot-for-lot producton. A heurstc procedure decomposes the problem nto a mxed nteger program and a nonlnear convex resource allocaton problem. The model can be extended to allow the frm to smultaneously alter both the producton rates and the ncomng demand lot szes through quantty dscounts. by Bntong Chen and Charles L. Munson Washngton State Unversty Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

2 1. Introducton In ths paper we examne the alteraton of the producton rate, a parameter that s often fxed n nventory models. Munson and Rosenblatt (2001) show that the benefts from coordnated lot szng among frms n a supply chan are hghest when the producton rate (fxed n ther model) s ether low (approachng the demand rate) or hgh (approachng nfnty). Ths suggests that t may be frutful to consder the producton rate as a varable n nventory models. Producton rates can be ncreased by addng workers, modernzng equpment, addng machnes, addng shfts, usng overtme, provdng tranng, ncreasng mantenance, etc. (Bretthauer and Côté, 1996). Researchers have studed changng producton rates n varous settngs (see also Chakravarty and Shtub, 1992; and Jones, Moses, and Zydak, 1998). Our research explores the effect of changng producton rates on a generalzed verson of the tradtonal Economc Producton Quantty (EPQ) model that ncorporates lumpy demand patterns. We further consder a frm producng multple products, each wth ts own lumpy demand pattern. Gven an nvestment budget for productvty mprovements, we determne the proper resource allocaton among products that are dentfed as canddates for resource allocaton. Other papers that have examned the allocaton of producton resources among products nclude Btran and ax (1981), DeCrox and Arreola-Rsa (1998), Evans (1967), Glasserman (1996), and Sox and Muckstadt (1996). 2. Optmal Producton Polcy for a Sngle Product wth Lumpy Demand Consder the tradtonal EPQ model, whch ntroduces a fnte producton rate nto the Economc Order Quantty (EOQ) framework. The well-known optmal lot sze and assocated mnmum cost for the EPQ model are respectvely gven by 2DS /[ h(1 D / P)], and 2DSh(1 D / p), where D s the annual demand, S s the setup cost per order, h s the annual holdng cost per unt, and P s the annual producton rate. Typcally the producton rate P s predetermned. owever, f the company could choose the producton rate level, then holdng and setup costs would be mnmzed (equal to 0) when P = D. In other words, the optmal producton polcy would be contnuous producton, where the optmal lot sze equals, only an ntal setup s needed, and nventory never accumulates. Interestngly, unless the frm needs the resources to produce other products, ths observaton suggests that the company has no ncentve (at least from a holdng and setup cost perspectve) to search for more effcent operatng methods to speed up the producton rate. The EPQ and EOQ models assume contnuous unform ncomng demand patterns, whch s equvalent to the company constantly recevng nfntely small customer order szes. Ths assumpton s most approprate for retalers or companes sellng to many customers. owever, many manufacturers sell to one or very few large customers, mplyng that ther customer demand s lumpy,.e. they receve fnte customer order szes ntermttently. Next we nvestgate whether or not the above producton rate observaton for the EPQ model holds n the lumpy demand case. We assume one buyer orderng the same quantty Q (perhaps ts EOQ) at fxed tme ntervals. Note that our model s also applcable ndvdually to multple buyers orderng customer-specfc products accordng to the ncreasngly common ndustry practce of mass customzaton, where each product needs to be produced separately or orders from dfferent customers are never combned. It can be shown that under ths scenaro, the optmal lot sze for the manufacturer Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

3 should be an nteger multple n of Q. We assume further that the producton rate s a functon of nvestment n technology. For notonal convenence, we defne the producton rate to be P ( x) = DM ( x), where x s the amount of technology nvestment used to ncrease the producton rate and M(x) s a multpler functon that s assumed to be concave and strctly ncreasng wth respect to x. In addton, we set M(0) = 1,.e., the current producton rate s assumed to equal the customer demand rate. Notce that the above assumptons on M(x) are qute general, and many producton functons commonly used n the mcroeconomcs lterature satsfy the assumptons. For example, the multpler functon may take the form of β M ( x) = 1+ αx, α > 0,0 < β 1. If β=1, M(x) reduces to a lnear functon of x. The precedng functon assumes that an nfnte nvestment leads to an nfntely fast producton rate. On the other hand, f the producton rate s bounded from above, the multpler functon could take the βx form of M ( x) = 1+ α(1 e ), α, β > 0, where the maxmum producton rate s (1 + α)d. Substtutng DM(x) for P and applyng Joglekar's (1988) formula, we obtan the optmal holdng and setup cost functon for the manufacturer when facng lumpy demand of sze Q: c( x, n) = DS /( nq) + [( n 1) ( n 2) / M ( x)] Qh / 2. We are nterested n fndng the optmal nvestment x 0 and manufacturer's lot sze multpler n to mnmze the cost functon c(x,n). Consder the followng two cases dependng upon the value of n: If n 2, the cost functon s nonncreasng n x. Therefore, an optmal x s 0 and we have c ( 0, n) = DS /( nq) + Qh / 2. Clearly, the optmal n when n 2 s and c ( 0, ) = Qh / 2. If n = 1 (lot-for-lot producton), we have c ( x,1) = DS / Q + Qh /[2M ( x)]. 2 2 Denote = Qh / 2, ρ = 2DS /( hq ) = ( EOQ / Q), and f ( x) = ρ + 1/ M ( x). The optmal holdng and setup cost as a functon of x then reduces to c(x) = mn{1, f(x)}. The precedng analyss mples that the optmal producton polcy s ether contnuous producton (n = ) or lotfor-lot producton (n = 1). Furthermore, nvestng to ncrease the producton rate s attractve only n the case of lot-for-lot producton. Let τ = lm1/ M ( x). Snce M(x) s strctly ncreasng, 1/M(x) τ for all x 0. Therefore, a x necessary condton for the product to receve nvestment to ncrease the producton rate and perform lot-for-lot producton s: ρ + τ < 1 or ρ < 1 τ. (1) We call a product that meets ths condton a canddate product. If τ = 0, whch happens when nfnte nvestment leads to nfnte producton rate, then the necessary condton reduces to: ρ < 1. In other words, the manufacture should only consder nvestng n producton rate mprovement f ts EOQ s smaller than the buyer's order sze Q. 3. Optmal Producton Polces for Multple Products 3.1 Resource Allocaton Among Canddate Products We now consder the stuaton where the manufacturer produces multple products. A total nvestment budget of B s avalable to speed up producton. The manufacturer needs to decde the amount of nvestment to allocate to each product n order to reduce total nventory related cost. The lot sze of each product then follows drectly from the resource allocaton as descrbed n Secton 2. Based on the dscusson of Secton 2, resource allocaton should be consdered only among the canddate products. Let I be the set of all canddate products. The mpled optmzaton problem then reduces to the followng smplfed resource allocaton problem (P): Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

4 I Mnmze c ( ) subect to x 0 I, where subscrpts for each product have been added to the notaton of Secton 2. Problem (P) s a nonconvex, nonsmooth, nonlnear knapsack problem and s dffcult to solve n general. 3.2 eurstc Procedure for Resource Allocaton Instead of attemptng to solve (P) optmally by usng an enumeraton-based procedure, we are nterested n desgnng an effcent heurstc procedure by takng advantage of the structure of the problem. Our heurstc s a decomposton procedure that conssts of the followng three steps: 1. Solve a lnear approxmaton of problem (P) to determne a subset of products, J, to receve nvestment allocaton. It turns out that the lnear approxmaton can be formulated as a mxed nteger program (MIP). 2. Solve a convex nonlnear resource allocaton problem to allocate the budget B among products n J. 3. Apply a greedy heurstc to reduce the sze of set J by removng one product from J at a tme and re-solvng the nonlnear resource allocaton problem n Step 2. Repeat ths process untl the total cost no longer mproves Lnear Approxmaton of (P) We need the followng parameters to ntroduce the lnear approxmaton of cost functon c (x ) for each product. x b s the break-even pont of budget allocaton that makes producton rate mprovement worthwhle for product. It s obtaned by solvng the equaton 1 = f (x). m ( b f x ) s the slope of f evaluated at the break-even pont. E s the maxmum possble resource allocaton beyond x b n the lnear approxmaton. It occurs at the pont where the tangent lne from x b reaches the lowest possble cost f ( ) = ( ρ + τ ). Smple calculaton shows that E = ( 1 ρ τ ) / m. As shown by Fgure 1, the lnear approxmaton replaces the nonlnear porton of c (x ) by a tangent lne of f (x) between x b and x b + E, and a flat lne thereafter. See Fgure 1. Defne ξ, J, as a bnary varable equal to 1 f product receves resource allocaton and 0 otherwse. Defne δ, J, as a contnuous varable representng extra resources allocated to product beyond the break-even pont x b. By defnton, the total resource allocaton to product equals δ + x b f ξ > 0, and equals 0 f ξ = 0. Wth c (x) replaced by ts lnear approxmaton for each, the resource allocaton problem (P) can be reformulated as the followng MIP: x x B Mnmze subect to I I + m δ b δ + x ξ B δ E ξ I δ 0, ξ = {0,1} I. Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

5 In our heurstc procedure, only products wth ξ = 1 n the optmal soluton of the MIP wll be consdered for recevng resource allocaton. In addton, the optmal obectve functon value of the MIP provdes a lower bound for problem (P) Convex Nonlnear Resource Allocaton Let J be a subset of I such that J = { I : ξ = 1}. By restrctng resource allocaton only among products n set J we may smplfy problem (P) to a nonlnear resource allocaton problem. Chen and Munson (2001) descrbe the techncal condtons under whch a large varety of multpler functons may be solved. ere we provde results for two specal cases that result n closed-form solutons: lnear and exponental. If M (x) = 1 + α x (lnear form) for all J, the resource allocaton s: α 1 1 x ( J ) = ( B + ), J. (2) J J α α α β x If M ( x) = 1+ α (1 e ) (exponental form) for all J, the resource allocaton s: ( 1 α β β x J ) = ln 2ln 1, 2 + ~ (1 ) α 4 ( ) 4 ~ (3) β ( ) + α λ J λ J where λ ( J ) s the unque soluton of the followng nonlnear equaton wth respect to λ: 1 α β β ln 2ln 1 B α = (4) J (1 ) 4 4 β + α λ λ Notce that the left hand sde of equaton (4) s monotoncally decreasng n λ. Thus many effcent search technques exst to locate λ ( J ). Further note that some solutons to equatons (2) and (3) could theoretcally be negatve. If ths happens, the set J cannot be the optmal set of products to receve resource allocaton n problem (P) and wll be elmnated n the refnement step of our heurstc to be descrbed n the next subsecton Refnements of Resource Allocaton Denote x, I, and C as the optmal resource allocaton to each product n I, the set of products that receve the resource allocaton n the optmal soluton, and the mnmum total cost of (P), respectvely. Denote NC (J) as the correspondng obectve value of the nonlnear resource allocaton problem. By defnton, C = NC ( I ) +. I \ I Snce the maxmum resource allocaton for each product n the MIP s lmted to E, I s n general (but not always) a smaller set than J and s very lkely contaned n J. Indeed, our numercal tests n the next secton support ths observaton. We now present a greedy heurstc procedure attemptng to reduce the sze of set J towards set I. Step (0) Delete all products wth x (J) 0 from set J and re-solve the nonlnear resource allocaton problem. Repeat f necessary. Step () Let k J be the product such that the followng rato s the smallest: {1 f [ x ( J )]} x ( J ), J. Step () Let J_ = J\{k}. If the total cost decreases after product k s removed from set J,.e., NC ( J _) + < NC ( J ) +, I \ J _ I \ J Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

6 we set J = J_ and return to Step (). Otherwse, we stop the refnement and the current set of products J wll receve the resource allocaton x (J). Notce that n Step () the numerator represents the cost ncrease f product does not receve resource allocaton x (J). Furthermore, the numerator wll be negatve when x (J) < x b, mplyng that product dd not receve enough allocaton to make producton rate mprovement worthwhle at all. Step () elmnates the product n ths case. 4. Numercal Examples To test the performance of the heurstc, we conducted four numercal studes of 50 random samples each. Both lnear (Equaton (2)) and exponental (Equatons (3) and (4)) nvestment multpler functons were used for both a 10-product and a 15-product scenaro. Optmal solutons were obtaned by performng full enumeraton over all possble combnatons of products n set J contanng the products recevng allocaton. We chose our study szes because full enumeraton CPU tmes were about 30 seconds for the 15-product case, and would approxmately double for each product added after that. The random data sets were generated n Excel, and the solutons were performed n GAMS/OSL2. We used the followng parameters: = Trangular(50, , 25000), ρ = b Trangular(.01,.99,.75), B = Unform(1/10, 1) ( E + ) [10-product case], and B = Unform(1/15, 0.8) + b x Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL. x ( E ) [15-product case]. For the lnear multpler functons, α = Trangular(.01,.00001,.001). For the exponental multpler functons, α = Dscrete Unform(1, 99), and β = Trangular(.00001,.001,.0001). In support of Step 3 of the heurstc (the refnement step), the MIP soluton usually contaned more products n the allocaton set than the optmal soluton dd. Apparently ths occurs because the tangent lne reaches the lower bound on cost more quckly than the actual cost functon c (x ) does. Furthermore, n most cases (but not all), the products recevng allocaton n the optmal soluton were ncluded n the MIP soluton's allocaton set. Thus, the greedy heurstc step of elmnatng the product wth the least gan per dollar spent usually works well and elmnates the approprate products. Step 3 has shown to be an mportant addton to the heurstc. The heurstc tself performed qute well. Out of all 200 samples, the maxmum cost devaton of the heurstc was 2.89%, and the average was %. Furthermore, the heurstc provded the optmal soluton n 175 samples (87.5% of the tme). For comparson purposes, we also calculated the costs of the MIP allocaton (the actual values of δ from the MIP), elmnatng steps 2 and 3 from the heurstc. The resultng allocatons were generally qute poor, wth an average devaton from optmal of %. Ths suggests that whle the MIP set (the ξ values) provdes an excellent startng pont, the actual MIP allocatons are not good, necesstatng the soluton of the convex nonlnear resource allocaton problem (Step 2 of the heurstc). 5. Extensons 5.1 Selectng the Best Total Resource Allocaton Level B Followng numerous other resource allocaton models n the lterature, we have assumed that the budget level B s predetermned and wll be fully allocated. owever, one common drawback of these models s that they usually do not ustfy the resource allocaton,.e., they do not determne whether the beneft resultng from the resource allocaton outweghs the nvestment tself. Our model can be extended to ncorporate ths ssue n one of the two ways.

7 Frst, the user can solve the model for teratve levels of budget B up to a certan captal budget lmt. For each budget level, the net present value (NPV) of annual savngs (net of B) are calculated over the approprate tme horzon. The budget level yeldng the hghest NPV s then chosen. Second, one may alternatvely nclude the resource allocaton tself as a separate annualzed cost n the obectve functon. Chen and Munson (2001) explan how the heurstc procedure proposed n ths paper can be easly modfed to ncorporate ths extenson. The downsde of makng ths change, however, s that any closed-form solutons for x are lost, even f M s a lnear functon. 5.2 Smultaneously Alterng P and Q va Quantty Dscounts For a number of years, researchers have examned the benefts, n the form of lower holdng and setup costs for manufacturers, of usng quantty dscounts to entce customers to change the sze of ther orders. Ths technque effectvely coordnates the supply chan by fndng lot szes that mnmze total system holdng and setup costs. In the precedng analyss, we assumed that ncomng order szes Q were fxed. Chen and Munson (2001) show how the model and heurstc n ths paper can be extended to ncorporate the possblty of changng Q along wth the producton rate P. 6. Concluson In ths paper we have examned the EPQ model wth lumpy demand to determne under what condtons that producton rates should be ncreased. We have also presented an effectve heurstc for allocatng lmted resources among products that are canddates for productvty mprovement. The answer to the queston, To speed or not to speed, depends largely on the relatve holdng and setup costs of the manufacturer and buyer. In general, large ncomng order szes suggest that the manufacturer should use lot-for-lot producton and speed up as much as possble. On the other hand, small ncomng order szes suggest that the manufacturer should use contnuous producton and not speed up at all,.e. only produce at a rate (nearly) equal to the demand rate. Moreover, to the extent that the sze of ncomng orders can be altered, costs wll be further reduced as follows. Large orders (when productvty s ncreased) should be made even larger, and small orders (when productvty s not ncreased) should be made even smaller. Chen and Munson (2001) shows how the model n ths paper can accommodate the smultaneous determnaton of both producton rate and ncomng order sze. In a sense, both producton polces mpled by the model n ths paper have a JIT flavor. A frm usng contnuous producton produces no faster than necessary. And a frm usng lot-forlot producton (hopefully) apples a very hgh producton rate to the process, mplyng no buldup of nventory. Producton s delayed untl the last possble moment. Fnally, the manufacturer's answer to a customer that s orderng JIT (small order quanttes), s to slow down ts producton rate (usng the contnuous producton polcy). Ths polcy s perhaps counter-ntutve to the dea that a manufacturer should utlze extremely hgh producton rates wth JIT customers to be able to respond rapdly to customer orders. References Btran, G.R. and A.C. ax. Dsaggregaton and Resource Allocaton Usng Convex Knapsack Problems wth Bounded Varables. Management Scence. Vol. 27, No. 4 (1981). pp Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

8 Bretthauer, K.M. and M.J. Côté. Nonlnear Programmng for Multperod Capacty Plannng n a Manufacturng System. European Journal of Operatonal Research. Vol. 96 (1996). pp Chakravarty, A.K. and A. Shtub. The Effect of Learnng on the Operaton of Mxed-Model Assembly Lnes. Producton and Operatons Management. Vol. 1, No. 2 (1992). pp Chen, B. and C.L. Munson. Resource Allocaton n a Lumpy Demand Envronment. Workng Paper. Washngton State Unversty, DeCrox, G.A. and A. Arreola-Rsa. Optmal Producton and Inventory Polcy for Multple Products under Resource Constrants. Management Scence. Vol. 44, No. 7 (1998). pp Evans, R. Inventory Control of a Multproduct System wth a Lmted Producton Resource. Naval Research Logstcs Quarterly. Vol. 14 (1967). pp Glasserman, P. Allocatng Producton Capacty Among Multple Products. Operatons Research. Vol. 44 (1996). pp Joglekar, P.N. Comments on A Quantty Dscount Prcng Model to Increase Vendor Profts. Management Scence. Vol. 34 (1988). pp Jones, P.C., L.N. Moses, and J.L. Zydak. Inventory Investment, Product Cycles, and the Imperfectly Compettve Frm. Internatonal Journal of Producton Economcs. Vol. 54, No. 3 (1998). pp Munson, C.L. and M.J. Rosenblatt. Coordnatng a Three-Level Supply Chan wth Quantty Dscounts. IIE Transactons. Forthcomng, Sox, C.R. and J.A. Muckstadt. Mult-Item, Mult-Perod Producton Plannng wth Uncertan Demand. IIE Transactons. Vol. 28 (1996). pp Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.

9 Cost c ( x ) Slope = m (ρ +τ ) E b x x Fgure 1. Cost Functon c x ) and Its Lnear Approxmaton ( Proceedngs of the Twelfth Annual Conference of the Producton and Operatons Management Socety, POM-2001, March 30-Aprl 2, 2001, Orlando, FL.