The Value of Demand Postponement under Demand Uncertainty

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1 Recent Researches n Appled Mathematcs, Smulaton and Modellng The Value of emand Postponement under emand Uncertanty Rawee Suwandechocha Abstract Resource or capacty nvestment has a hgh mpact on the frm proftablty. However, ths decson must be made earler when demand s uncertan and t s very dffcult to change later on. Many frms search for other strateges to deal wth uncertanty n order to gan more proft and stay n a busness. Postponement strategy s one of the strateges that many companes use to hedge aganst the uncertanty. A frm can ncrease ts proftablty when t can postpone some decsons or actvtes to the tme later untl the more nformaton s obtaned. In ths paper, we consder a frm producng two substtutable products. The frm needs to make two decsons: the capacty nvestment and producton quantty. The objectve of ths work s to study how the demand postponement affects the frm s capacty nvestment and ts proftablty under demand uncertanty and how degrees of substtuton mpact our fndngs. Based on ths framework, we model the problem as a two-stage stochastc programmng. We characterze the optmal nvestment capacty and producton quantty under dfferent postponement strateges. In addton, the necessary and suffcent condtons for nvestment n flexble capacty are obtaned. eywords Capacty nvestment, emand postponement, emand uncertanty, Stochastc programmng, Substtutable Products. I. ITROUCTIO OSTPOEMET strategy s one of the strateges that P companes can use to hedge aganst uncertanty. Many type of postponement have been studed as stated n [] and [] such as tme postponement, place postponement, and form postponement, etc. There are applcatons of postponement n ndustry and results from [3] ndcated that there s a postve relatonshp between postponement and company performance. The postpone strateges we consder n ths work are the quantty postponement and demand postponement strateges. The producton quantty postponement refers to a frm s ablty to postpone ts producton quantty untl after demand nformaton s obtaned. Consderng the quantty postponement along wth the flexble resources ncreases the Ths work was supported n part by the Thaland Research Fund (TRF under Grant MRG58 and Faculty of Graduate Studes, Mahdol Unvesty. R. S. Author s wth the epartment of Mathematcs, Faculty of Scence, Mahdol Unversty, Bangkok, 4 Thaland (correspondng author to provde phone: ; fax: ; e-mal: scrsw@ mahdol.ac.th. frm proftablty. The frm s flexble resources allow the frm to swtch capacty amount of the multple products and provdes a rsk poolng effect. Many researchers have focus on the flexblty provded by the producton postponement strateges as shown n [], [4], and [5]. The mpact of quantty postponement on the optmal capacty has been studed by [6] for a sngle-product frm. On the other hand, there s a few works studyng on demand postponement. [] formulates and solves the capacty plannng problem for one product. In ths paper, we extend the model to the substtutable products. In addton, we are also nterested n the effect of the demand postponement strategy when the frm has mplemented the producton quantty postponement. Therefore, the objectve of ths paper s to consder the mpact of the demand postponement on the optmal capacty nvestment under dfferent postponement strateges. Specfcally, we consder the no postponement and the producton quantty postponement as base cases. Then, we examne the effect of demand postponement on the optmal capacty nvestment and the frm s proft under both strateges as well as study how the degrees of substtuton between products affect the fndngs. II. OTATIOS, MOELS, A ASSUMPTIOS In ths work, we consder a frm that produces two substtutable products n a monopolstc settng. Ths frm needs to determne capacty, prcng, and producton quantty. Let denote the capacty of the flexble resources that can produce both products, c be a unt nvestment cost, c h be a unt holdng cost, c q be a unt product cost, c be a dscount prce (or rembursement cost for the demand n postponement perod. The producton cost for each tem s ncluded here, d be a demand of product, p be a sellng prce of product and, wthout loss of generalty, let (p p, s be an amount of sellng product, q and q p be producton quanttes of product for regular and postponement perod, respectvely. Tmelne s dvded nto two stages shown n Fg. Stage s a plannng perod and Stage ncludes the regular and postponement perods. ISB:

2 Recent Researches n Appled Mathematcs, Smulaton and Modellng Stage Stage Plannng Perod, q,q emand s realzed Fgure : Tmelne Regular Perod, q,q q p,q p q,q elvery Tme q,q, q p,q p In Fg, the nvestment capacty ( s determned n the plannng perod when demand s hghly uncertan. There are four cases that are consdered, o postponement (, emand postponement (, Producton postponement (P, and producton and demand postponement (P. All the decson varables are determned n the plannng perod under strategy (. For strategy (P, the producton quantty, q, s determned after the demand curves are realzed. When the demand postponement strategy s mplemented, the postponed producton quantty, q p, s determned after the demand curves are realzed and ths products wll be delvered n the postponement perod. We assume that the fracton of unsatsfed demand, -, s retaned n the postponement perod. That s, when the demand exceeds the producton quantty, there s only (-(d -q remanng demand n the postponement perod. Aggregate lnear demand functon for two substtutable products s consdered. It s a functon of the prces of both products: d = ξ αp βp3, =, ξ s an ntercept of product, β s the degree of substtuton ( β α. Let E[.] be an expectaton operator wth respect to random varables, ξ, and Pr(. be a probablty. Let π (. be the proft functon n Stage. III. MATHEMATICAL FORMULATIOS Postponement Perod tmelne Strategy Strategy Strategy P Strategy P From the above, we can dvde the problem nto two stages. Stage corresponds to the tme when the demand s uncertan and Stage s the tme after the demand nformaton s obtaned. The mathematcal model for the problem under demand strategy ( can be formulated as (stage max V E[ π (, q] c ( c c q qp,, = q h = s.t. q ( q, for =, (stage max π ( q, = ps ( p cq sq, p p = = ( s.t. s q for =, (3 s ( ε αp βp3 for =, (4 qp q for =, (5 qp ( ( ε αp βp3 s for =,(6 s, q for =, ( p The objectve of the model n Stage s to maxmze the expected proft functon. The objectve functon n Stage s to maxmze the frm proft by determnng the amount of sell, s, and the postponed producton quantty q p. Constrant ( states that the total producton quantty should not exceed the frm s capacty. Constrants (3 and (4 mply that the amount of sell cannot exceed producton quantty and nonnegatve demand of each product. In addton, (5 and (6 ensures that the postponed producton quantty, q p, does not exceed the producton quantty and t does not exceed retaned demand. Constrants ( and ( are non-negatvty constrants for nvestment capacty, producton quantty, sellng amount, and postponed producton quantty, respectvely. ote that the model for the Problem s smlar except that q p s set to zero. The mathematcal model for the problem under producton quantty and demand postponement strategy (P can be formulated as (stage max V E[ π ( ] c s.t. (8 qq, p (stage max π ( = ( p c q ( p c q q p = = s.t. q q (9 q ( ε αp βp3 for =, ( qp qp ( qp ( ( ε αp βp3 s for =, ( q, q for =, (3 p The objectve of the model n Stage s to maxmze the expected proft functon. In ths case, the frm determnes the producton quantty after the demand s realzed. Therefore, the producton and the holdng costs do not occur n the frst ISB:

3 Recent Researches n Appled Mathematcs, Smulaton and Modellng stage. In Stage, the objectve functon s to maxmze the frm proft by determnng the producton quantty and the postponed producton quantty gven that the nvestment capacty and prces. Constrants (9 and ( state that the total producton quantty and the total postponed producton quantty should not exceed the frm s capacty, respectvely. Constrant ( mples that the producton quantty should not exceed the nonnegatve demand of each product. In addton, ( the postponed producton quantty (q p does not exceed the retaned demand. (8 and (3 are non-negatvty constrants for nvestment capacty, producton quantty, and postponed producton quantty. The model for the producton quantty postponement (P s smlar to Problem (P except that q p s set to zero. IV. RESULTS The optmal soluton to Problems,, and P can be characterzed n the smlar procedures. Hence, n what follows, we focus on Problem P. To analyze the problem, we solve the problem backward. Frst, for a gven set of p and p, we derve the optmal soluton for the problem n Stage and show that the expected proft functon n Stage s strctly jontly concave n q, and q. Consequently, the optmal soluton s unque and the arush-uhn-tucker (T frstorder condtons are necessary and suffcent for optmalty to the problem. Ths leads to the followng results. Theorem The producton quanttes q and q are the unque optmal soluton to Problem P f and only f there exsts v that satsfes the followng condtons: [ c cp ( pc]pr( 5 [ c cp ( p c]pr( 6 9 [ p cp ( ( p c]pr( [ pcp c]pr( [ p cp c]pr( 8 = c v v = 5 = { αp βp < ξ < αpβp, ξ < αp βp} 6 = { ξ < αpβp, αp βp < ξ < αp βp}, = { ξ > αpβp}, 8 = { ξ < αp βp, ξ > αp βp}, 9 = { αp βp < ξ < αpβp, ( α β( p p < ξ ξ < ( α β ( p p} = { αp βp < ξ < αp βp, ξ ξ > ( α β( p p} = { ξ > αp βp, ξ > αp βp, ξ ξ < ( α β( p p} = αp βp < ξ < αp βp, ξ ξ > ( α β( p p Proof: See Appendx. Theorem The producton quanttes q and q are the unque optmal soluton to Problem f and only f there exsts v and v that satsfes the followng condtons: ppr( ξ > q αp βp = c ch cq v ppr( ξ > q αp βp = c ch cq v q v =, for =, Theorem 3 The producton quanttes q and q are the unque optmal soluton to Problem f and only f there exsts v and v that satsfes the followng condtons: ( p cpr( ξ > q αp β p ( p ( cpr( = c ch cq v ( p cpr( ξ > q αp β p ( p ( cpr( = c ch cq v q v =, for =, = { q αp βp < ξ < q αpβp}, = { q αp βp < ξ < q αp βp}. Theorem 4 The producton quanttes q and q are the unque optmal soluton to Problem P f and only f there exsts v that satsfes the followng condtons: ppr( ξ > αp β p, ξ > αp β p ppr( ξ > αp β p ppr( = c v v = = { αp β p < ξ < αp β p, ξ ξ > ( α β( p p } The optmal capacty nvestment under each postponement strategy follows a threshold polcy. Corollary The optmal nvestment polcy under postponement strategy s one of the followng cost threshold polces: If c ch cq < c, then >. Otherwse, =. * = q q the threshold values s gven as ISB:

4 Recent Researches n Appled Mathematcs, Smulaton and Modellng c = max{ p Pr( ξ > αp β p, p Pr( ξ > αp βp } c ch cq c If <, then >. Otherwse, =. * = q q the threshold values s gven as c = max{ ( p cpr( ξ > αp βp,( p cpr( ξ > αp βp } 3 c c If <, then >. Otherwse, =. the threshold values s gven as c = p Pr( ξ > αp βp p Pr( < ξ < αp βp, ξ > αp βp 4 c c If <, then >. Otherwse, =. the threshold values s gven as c = ( pc cppr( ξ > αp βp ( p c cppr( < ξ < αp β p, ξ > αp βp Proof: It follows drectly from Theorems -4. The above corollary suggests that f the total unt cost c ch cp s less than the cost threshold, the frm always nvests n the capacty under Strateges and. On the other hand, f the total unt cost s hgher than the cost threshold, t s better for the frm not to nvest. Smlarly, under the Strateges P and P, f the unt nvestment cost s less than the cost threshold value ndcated n Corollary, then the frm should nvest n the capacty. Corollary c c and c c Proof: It follows drectly from Proposton. Corollary ndcates that there are some regons that the frm should nvest n ts capacty to gan more proft when the demand postponement s consdered. ext, we perform a numercal study to nvestgate the mpact of the optmal nvestment capacty changes n demand substtuton parameter β. Results from the numercal studes suggest that the optmal * nvestment capacty s ncreasng n β for all strateges (,, P, P as shown n Fg.. The results show that the frm wll nvest more when the products are more substtutable. Ths s because the frm can take advantage from the substtuton. However, the rate of change of the optmal nvestment capacty s less when producton postponement s mplemented. Ths s because of the flexble resources whch s also used to hedge aganst demand varablty. It s not surprsng that when the products are ndependent or not substtutable the frm nvests more under strategy P. Ths s because the frm has ablty to deal wth the demand varablty by usng the producton quantty postponement from the flexble resource. The results also show that the demand postponement strategy leads to nvest less n the frm s capacty and t leads to lower nvestment when addton producton quantty s mplemented along wth. Moreover, our fndng ndcates that the frm nvests less when the demand postponement s addtonally mplemented. ue to the ablty of the frm to postpone part of the demand to tme later, t may cause the frm nvests less Proft Fgure 3: The frm's proft versus β/ν. Smlar results are obtaned for the frm s proft. Fg 3 shows that the proft s ncreasng n substtuton parameter for all strateges. The demand postponement generates the hgher proft for the frm. P P Optmal nvestment capacty Fgure : Optmal nvestment capacty * versus β/ν. P P V. COCLUSIO In ths paper, we study the mpact of the demand postponement on the optmal capacty under demand uncertanty for a two-product frm n a monopolstc settng. The frm makes decsons nvestment capacty and product quantty to maxmze the proft functon. We formulate mathematcal models for dfferent postponement strateges. The necessary and suffcent condton for the optmal capacty nvestment s obtaned for each strategy. Our fndngs show that mplementng the demand postponement leads to nvest less n optmal capacty and ncrease n the frm s proftablty. Our results come wth lmtatons. We consder the lnear demand functon whch depends on the prce of the products. ISB:

5 Recent Researches n Appled Mathematcs, Smulaton and Modellng It would be nterestng to consder other realstc demand functon. We also assume a frm that s n a monopolstc settng. Consderng the competton would be another nterestng extenson to our models. APPEIX Consder problem, we frst solve the Stage Problem. The demand space can be decomposed nto dfferent dsjont sets; see Fg. 4. αp β p α p β p α p β p α p 4 5 β p α pβ p Fgure 4: The demand space for Problem P We can obtan the closed form expressons for the producton quanttes and postponed producton quanttes for each regon. It s easy to verfy that the optmal soluton to the Stage Problem s unque and the optmal Stage proft can be obtaned drectly. Before we solve the Stage problem, let ε be a realzaton of ξ and f (, be a jont probablty densty functon of ξ and ξ, respectvely. To solve the Stage Problem, the Stage proft s consdered. We derve E[ Π( ] = [ c cp ( p c]pr( 5 [ c c ( p c]pr( p 6 9 p p cp c cp c]pr( [ p c ( ( p c]pr( [ ]Pr( [ p 8 α p β p E[ Π( ] = αp β p αp β p αp β p [( p c ( ( p c] p f( αp βp, ε dε ( p p f( αp βp, ε dε [ p ( c cp] f( ε, ( α β( p p dε <. αpβ p ( ( p c f ( ε, ( α β( p p ε dε αpβ p It can be shown that V = E[ π ( ] c s strctly concave n. Therefore, the optmal soluton to the Stage Problem s unque, and snce the constrants are lnear, the T frst-order optmalty condtons, gven n Theorem, are necessary and suffcent for optmalty. Smlar method can be used to obtan the results n Theorems -4. ACOWLEGMET I would lke to thank Thaland Research Fund (TRF for the fnancal support and r. ancht Malavongs for all hs comments and support for ths research. REFERECES [] J. M. Swamnathan and H. L. Lee, esgn for Postponement, Handbook of OR/MS on Supply Chan Management, Ed., Graves, S. and de ok, 3. [] B. Yang,.. Burns, and C. J. Backhouse, Management of Uncertanty through postponement, Int. J. Prod. Res. Vol 4, o. 6, 4, pp [3] B. Yang,.. Burns, and C. J. Backhouse, The Applcaton of Postponement n Industry, IEEE Trans. Eng. Manage., Vol. 5, o., May 5, pp [4] C. H. Fne and R. M. Freund, Optmal Investment n Product-Flexble Manufacturng Capacty, Manage. Sc., Vol. 36, 99, pp [5] J. A. Van Meghem, Investment strateges for flexble resources, Manage. Sc. Vol. 44, 998, pp. -8. [6] J. A. Van Meghem and M. ata, Prce versus Producton Postponement: Capacty and Compeptton, Mange. Sc., Vol 45, 999, pp [] A. V. Iyer, V. eshpande, and Z. Wu, A Postponement Model for emand Management, Manage. Sc., Vol. 49, o. 8, August 3, pp Rawee Suwandechocha receved a B.S. n mathematcs from Unversty of Rochester, Y, USA, n 999, and M.S. and Ph.. n Operatons Research from epartment of Industral and System Engneerng, Vrgna Polytechn c Insttute and State Unversty, Blacksburg, VA n 6. She s a lecturer n the epartment of Mathematcs, Faculty of Scence, Mahdol Unversty, Bangkok, Thaland. Her research nterests nclude appled operatons research, supply chan and logstcs, nventory, and capacty plannng. ISB: