Optimal firm's policy under lead time- and price-dependent demand: interest of customers rejection policy

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1 Optiml firm's policy under led time- nd price-dependent demnd: interest of customers rejection policy Abduh Syid Albn Université Grenoble Alpes, G-SCOP, F Grenoble, Frnce Ynnick Frein Université Grenoble Alpes, G-SCOP, F Grenoble, Frnce Rmzi Hmmmi ESC Rennes School of Business, Rennes, Frnce Abstrct Considering led-time- nd price-sensitive demnd, we investigte whether client rejection policy, modeled s M/M//K system, cn be more profitble thn n ll-client cceptnce policy, modeled s M/M/ system. We provide nlyticl insights for the cses with nd without holding nd penlty costs by compring M/M// to M/M/ models. Keywords: Led-time quottion, Pricing, M/M//K. INTRODUCTION The delivery led-time, which represents the elpsed time between the plcement of the order by the customer nd the receipt of this order, hs become fctor of competitiveness for compnies nd n importnt purchse criterion for mny customers. Gery nd Zonnenberg (000) reported tht top performers mong 0 orgniztions conducted inititives not only to reduce costs nd mintin relibility, but lso to improve delivery speed nd flexibility. Bker et l. (00) found tht less thn 0% of end consumers nd less thn 30% of corporte customers bse their purchsing decisions on price only; for substntil mjority of purchsers both price nd delivery led time re crucil fctors tht determine their purchse decisions. Thus, in order to increse their profit, compnies must not focus only on price but lso need to quote the right delivery led time to their customers. A short quoted led time cn led to higher demnd but cn lso result in lte delivery, which ffects the firm s reputtion for on-time delivery nd deters future customers (Slotnick, 04). In ddition, compnies risk to lose mrkets if they re not cpble of respecting the quoted delivery led-times (Kpuscinski nd Tyur, 007). A long quoted led time cn reduce the risk of lte delivery but leds to lower demnd. This rises the following relevnt question: Wht is the best led time tht must be quoted by compny when customers re not only sensitive to price but lso to led time? Some uthors tried to nswer this question while considering n M/M/ system (Plk et l., 998; Pekgün et l., 008; So nd Song, 998). One of the chrcteristics of the M/M/ is ccepting ll customers, which cn led to congestions in the queue.

2 In order to cope with this sitution, firms cn rect by quoting longer led times in order to mintin the desired service level. However, this leds to lower demnd nd revenue. Strting from this observtion, we investigte in this pper whether customer rejection policy cn be more beneficil for the firm thn n ll-customers cceptnce policy. Indeed, our ide is bsed on the fct tht rejecting some customers might help to quote shorter led time for the ccepted customers, which might led to higher demnd nd profit. We model this rejection policy bsed on n M/M//K system. We nlyticlly determine the optiml firm s policy (optiml price nd quoted led time) in cse of M/M// system. Then, we compre the optiml firm s profit under M/M// with the optiml profit obtined by M/M/. Two situtions re considered: system without holding nd penlty costs nd system where these costs re included. The rest of this pper is orgnized s follows. In the next section, we develop the formultion of the M/M//K system with price-nd led-time-sensitive demnd. Lter, we nlyticlly solve the M/M//K system for K= without holding nd penlty cost nd compre the results to those obtined with M/M/. We then focus on the cse with holding nd penlty cost for the rest of the pper. We finlly conclude nd give future work directions. PROPOSED MODEL (THE M/M//K) As in Plk et l. (998), we consider mke-to-order firm where the cpcity is ssumed to be constnt while price, quoted led-time, nd, consequently, demnd re decision vribles. Customers re served in first-come, first-served bsis (FCFS). The rrivl process is ssumed to be Poisson process. The processing times of customers in the system re ssumed to be exponentilly distributed. Contrry to the ssumptions of M/M/ model where ll customers re ccepted (s in Plk et l., 998 nd Pekgun et l., 008), we reject customers when we hve lredy K customers in the system, therefore we model the system by n M/M//K. Similrly to Liu et l. (007); Plk et l. (998); nd Pekgün et l., (008), the demnd is ssumed to be liner decresing function in price nd quoted led-time. p p l b p b,, l () where: = price of the good/service set by the firm, = quoted led-time, p,l = expected demnd for the good/service with price p nd quoted led-time l, = mrket potentil, = price sensitivity of demnd, = led-time sensitivity of demnd, l b b Since the demnd is downwrd sloping in both price nd quoted led-time, b nd b re restricted to be non-negtive According to Plk et l. (998), this liner demnd function is trctble nd hs severl desirble properties. For instnce, with such liner demnd, the price elsticity is incresing in both price nd quoted led-time. Customers would be more sensitive to long led-times when they re pying more for the goods or service. Similrly, customers would be more sensitive to high prices when they lso hve longer witing times.

3 In order to prevent the firms from quoting unrelisticlly short led-times, we ssume tht the firm mintins certin minimum service level. The service level (s) is defined s the probbility tht the ctul led-time (W) stisfies the quoted led-time (l) (P (W l) s). Since we ssume n M/M/l/K queueing system with men service rte, μ, men rrivl rte (or demnd), λ, nd throughput rte (effective demnd),, then the expected number of customers in the system is given by Ls (see eq. ()), nd the ctul led-time (time in the system) is exponentilly distributed with men (see eq. (3)). The probbility tht the firm is ble to meet the quoted led-time (P (W l)) nd the probbility tht job is lte (P (W > l)) re given in eq. (4). These equtions re bsed on Gross et l. (008). L s W L s P W l K K L s K with P K nd with () i PK K k l e k 0 i0 i! l Pk P K K K 3 if nd PW l or P K if K i K k l e k 0 i0 i! l The objective of the firm is to mximize its revenue, which includes the following three prts: () Expected revenue (net of direct costs) is represented by p m, where m is the unit direct vrible cost. () Totl Congestion cost is expressed s the men number of jobs in the system multiplied by the unit holding cost (Ls F). This cost typiclly represents the in-process inventory holding cost. (3) Totl Lteness penlty cost is expressed s (penlty per job per unit lteness) (number of overdue clients) (expected lteness given tht job is lte). The number of overdue clients is equl to: (throughput rte) (probbility tht job is lte). The penlty cost per job per unit lteness (denoted by c) reflects the direct compenstion pid to customers for not meeting the quoted led-time. Mthemticlly, this totl Lteness penlty cost is given by c PW lw. Thus, the firm s optimiztion problem cn be formulted s follows: (P0) l, p, Subject to ml F c PW l P k P p s W (5) b p b l (6) i K k l l Pk e k i i 0 0! PK s (7) (8) K PK if nd P K K if K (9) P K (0) K (3) (4)

4 , p, l, 0 () where, Decision Vribles = Price of the good/service set by the firm, l = Quoted led-time, b = Men rrivl rte (demnd), b p m s P K K Prmeters = Mrket potentil = Price sensitivity of demnd, = Led-time sensitivity of demnd, = Men service rte (Production cpcity), = Unit direct vrible cost, = Service level defined by compny, = Probbility of rejected customer, = System cpcity. In this formultion, constrint (6) imposes tht the men demnd (λ) does not exceed the demnd obtined with price (p) nd quoted led-time (l). Constrint (7) expresses the service level constrint. Constrint (9) clcultes the probbility of rejecting customers. Constrint (0) is the number of customers tht re served nd exit the system. Constrints () re the non-negtivity constrints. Solving nlyticlly this generl cse seems to be difficult. So, in the following section, we only consider the cse of K =. We will consider two situtions: the cse without penlty nd holding cost; nd the cse where these costs re included. In both cses, we will compre the obtined optiml solution with the optiml solution of the M/M/ pproch nd derive insights. M/M// WITHOUT PENALTY COST AND HOLDING COST We consider here the cse without holding nd penlty costs. Thus, the objective function P W l e, hence consists only in the mximiztion of the expected revenue. The Probbility l implying tht the service level constrint cn be written s formultion of the problem becomes: e l s. Consequently, the (P), l, p, Subject to p m () b p b l (3) e l s (4) (5) P (6) P (7), p, l, 0 (8) 4

5 Eq. (4) cn be rewritten s l ln s. Then, by integrting the equlity constrint (eq. (5-7)) into the objective function, we cn simplify the formultion s: (P ),l, p Subject to p m (9) b p bl (0) l ln s () (), p, l 0 Using the new formultion (P ) we will now trnsform the problem into single vrible optimiztion problem. Becuse of the spce limittion, ll the proofs re reported in Albn et l. (06). Firstly, the demnd constrint (eq. (0)) is binding t optimlity, thus: b p bl p bl b (Albn et l., 06). Secondly, the service level constrint (eq. ()) is l s ln s lso binding t optimlity (Albn et l., 06). Hence, ln. We denote l z l z by z, nd get:. Substitute into the expression of p, we obtin: p b z b. Substituting p into the objective function, we get new formultion of the problem with single vrible (λ) s: b f z mb (P ) 0 b b Proposition. There exists λ 0 such s 0 (06)). b z f iff m b (3) (proof see Albn et l. In order to find the optiml solution, we use the first derivtive conditions. Indeed, df d d b z mb d b b b z mb b 0 b b 0 (4) The numertor of eq. (4) must be equl to zero. It cn be proven tht the discriminnt (Δ) of this numertor under condition of proposition is greter or equl to 0. Hence eq. (4) hs two rel roots: b z m nd b z mb (5) b Given tht λ is negtive, there is only one fesible sttionry point λ. Under proposition, λ is positive. It cn lso be shown tht the objective function is concve function in λ,l,p 0 (Albn et l., 06). Hence, λ is lso the optimum point. This leds to the results presented in proposition. 5

6 Proposition. For problem P, the optimum demnd is b z mb with z ln s, the optimum led-time l ln s, the optimum price p b l b p m., nd the optimum profit = COMPARISON BETWEEN M/M// AND M/M/ WITHOUT HOLDING AND PENALTY COST In this section, we compre our model (M/M//) with the existing M/M/ tken from Pekgün et l. (008) s they don t consider holding nd penlty costs. We use bse cse with prmeters: b = 4; μ = 0; s = 0.95; m = 5. We vry the mrket potentil () nd the led-time sensitivity (b). For ech pir of vlue (, b), we clculte the reltive gin obtined by using M/M// insted of M/M/. This reltive gin is clculted s follows: M/M// Profit Profit M/M/ Profit M/M/ 00% (6) A positive vlue mens tht the pproch with rejections (M/M// model) is better thn the pproch without rejections (M/M/ model). Tble - Comprison for different vlues of nd b b M/M/ vs M/M// % 7.94% 8.9% 3.4% 0.66% % 5.7% 6.%.57% -0.99% %.6% 3.96% -0.30% -.65% % 9.96%.79% -.8% -4.33% 6 6.3% 7.3% -0.40% -4.07% -6.03% 5.57% 4.66% -.59% -5.99% -7.74% %.0% -4.80% -7.9% -9.48% % -0.65% -7.03% -9.87% -.4%.% -3.3% -9.9% -.86% -3.04% 8.68% -6.0% -.57% -3.88% -4.87% 0 5.6% -8.74% -3.90% -5.94% -6.74% 9.85% -.49% -6.8% -8.06% -8.67% % -4.30% -8.7% -0.3% -0.65% % -7.7% -.% -.48% -.7% % -0.3% -3.83% -4.83% -4.86% 5 -.9% -3.0% -6.56% -7.30% -7.4% We cn see in tble tht the rejection policy cn be better thn the ll-customers cceptnce policy, in prticulr when the mrket potentil is smll nd the led-time sensitivity is high. 6

7 Thus, the client rejection policy cn be better in some cses even when the penlty nd holding costs re removed. This motivtes the purpose of next section where we will study the cse with penlty nd holding costs M/M// WITH PENALTY COST AND HOLDING COST With the ddition of penlty nd holding costs, the objective function includes the expected revenue, the totl congestion costs, nd the totl lteness penlty costs. The formultion of this objective function hs been presented erlier. The service level constrint is similr to the previous cse. Thus, the formultion of the problem is: (P), p, l, Subject to F c (7) l p m e bp bl (8) e l P s (9) (30) (3) P (3),, l, p 0 (33) Integrting the equlity constrints (eq. (30 3)) to the objective function nd rewriting l e s l ln s, we get the following formultion of the problem: s (P ),, l p Subject to l p m F ce (34) b p bl (35) l ln s (36), p, l 0 (37) It cn be shown tht the demnd constrint (eq. (35)) is binding t optimlity (Albn et l., 06). Thus, the formultion becomes: (P ), l l b l b m F ce (38) l ln s (39) l, p 0 (40) Subject to In ddition, there is lso fesibility condition of this problem s explined in proposition 3. 7

8 Proposition 3. The problem P is fesible (λ,l 0 nd profit 0) iff l b l mb Fb cb e 0 (proof see Albn et l. (06)). bl b m nd Unlike the first cse where the penlty nd holding costs re not considered, the service level constrint is not necessrily binding in this cse, which complictes the solving pproch. Indeed, for lrge vlues of c, the ctul service level hs to be very high (close to ) to void high penlty cost. This indictes tht the ctul service level cn be greter thn the imposed service level (s). We now present the min steps to get the optiml solution given in proposition 4. The detiled proof is given in (Albn et l., 06). To solve the problem, we pply the Lgrngin multiplier method. The sttionry points of problem (P ) must stisfy: where, L 0, L 0 l, L 0 b l b m F ce L, l,, 0 nd l ln s l l ln s (4) As we hve lredy explined, we hve two situtions: () the service level constrint (39) is non-binding: s sc, () the service level constrint (39) is binding: s sc, where the criticl vlue for the service level (sc) equls to b bc. With this criticl service level Mx s, (Albn et l., 06). (sc) we cn prove tht the ctul service level is equl to s c l e Mx s, s c l l ln Mxs s c, ln x where, x Mx s, b c b (4) To find the optiml demnd, we derive eq. (4) by λ. L 0 0 l b l mb Fb b ce b (43) The numertor of eq. (43) should be equl to zero. It is proven tht, with the fesibility condition, the discriminnt (Δ) of eq. (43) is positive (Albn et l., 06). Hence eq. (43) hs two roots which re: bl bm Fb b ce l nd 8

9 bl bm Fb bce l (44) λ is negtive. Under fesibility condition of proposition 3, λ is positive. It is proven tht the objective is concve function for λ,l,p 0 (Albn et l., 06). Thus, the led-time (l ) nd demnd (λ) provide the optiml solution. As summry, the optimum point of this problem is given in the proposition 4. Proposition 4. For problem P:. The optimum led-time l ln x with x Mx s, b c b.. The optimum demnd cn be found by using eqution l bl bm Fb bce. b l, the optimum price l 3. The optimum profit m F ce p. Bsed on the result found in this section, we will investigte whether our M/M// model with penlty nd holding costs cn be better thn the M/M/ model of Plk et l. (998). COMPARISON BETWEEN M/M// AND M/M/ WITH HOLDING AND PENALTY COST In this section, we compre our model (M/M//) with the existing M/M/ tken from Plk et l. (998). We vry the mrket potentil () nd the led-time sensitivity (b) (see Tble ). For ech pir of vlue (, b), we clculte the reltive gin obtined by using M/M// insted of M/M/. This reltive vlue follows eqution (6). We use the sme bse cse s in the previous comprison with dditions of F = nd c = 0. As expected, we hve more cses where M/M// is better thn M/M/ in this sitution with penlty nd holding costs compred to the sitution without holding nd penlty costs. In the M/M/, becuse ll clients re ccepted, the led-time cn be very long which cn cuse high congestion costs. As b increses, the demnd becomes more sensitive to led-time, which fvors the rejection policy modeled by the M/M//. CONCLUSION We hve provided the generl formultion of M/M//K with penlty nd holding costs where demnd is sensitive to both price nd led-time. We solved nlyticlly the cse with K= in two situtions: with nd without penlty nd holding costs. We conducted numericl experiments bsed on the nlyticl solutions nd showed tht the rejection policy (M/M//) cn be better thn the ll-customers cceptnce policy (M/M/). This pper cn be extended in mny wys such s considering the cse of K> or modeling system of type M/D/. We re currently working on these issues. p b 9

10 Bibliogrphy Tble - Comprison for different vlues of nd b b M/M/ vs M/M// % 6.95% 5.50% 9.58% 6.% % 4.3% 3.33% 7.74% 4.49% %.53%.7% 5.90%.86% 7 4.6% 8.84% 9.0% 4.05%.% % 6.7% 6.86%.9% -0.43% % 3.5% 4.69% 0.33% -.09% % 0.85%.5% -.54% -3.76% % 8.0% 0.34% -3.43% -5.45% 3.77% 5.56% -.85% -5.34% -7.6% 0.4%.9% -4.05% -7.6% -8.89% % 0.5% -6.7% -9.% -0.65% 9 3.8% -.4% -8.5% -.9% -.43% % -5.0% -0.80% -3.9% -4.5% 7 6.4% -7.8% -3.% -5.4% -6.0% 6 3.0% -0.56% -5.47% -7.33% -8.0% % -3.35% -7.88% -9.49% -9.97% Albn, A.S., Frein, Y., Hmmmi, R., 06. Study of Client Reject Policies under Led-Time nd Price Dependent Demnd (Working Pper No. hl ). Avilble t: Bker, W., Mrn, M., Zwd, C. 00. Price smrter on the Net. Hrvrd Business Review 79(): 7. Gery, S., Zonnenberg, J.P., 000. Wht It Mens To Be Best In Clss. Supply Chin Mngement Review 4(3 (July/Aug)): Gross, D., Shortle, J. F., Thompson, J. M., Hrris, C Fundmentls of queueing theory (Fourth Edition). John Wiley & Sons, New Jersey. Kpuscinski, R., Tyur, S., 007. Relible Due-Dte Setting in Cpcitted MTO System with Two Customer Clsses. Opertions Reserch 55(): Liu, L., Prlr, M., Zhu, S.X., 007. Pricing nd Led Time Decisions in Decentrlized Supply Chins. Mngement Science 53(5): Plk, K., Erlebcher, S., Kropp, D.H., 998. Led-time setting, cpcity utiliztion, nd pricing decisions under led-time dependent demnd. IIE Trnsctions 30(): Pekgün, P., Griffin, P.M.P., Keskinock, P., 008. Coordintion of mrketing nd production for price nd ledtime decisions. IIE Trnsctions 40(): 30. Slotnick, S.A., 04. Led-time quottion when customers re sensitive to reputtion. Interntionl Journl of Production Reserch 5(3): So, K.C., Song, J.-S., 998. Price, delivery time gurntees nd cpcity selection. Europen Journl of Opertionl Reserch, ():