Dynamic Price Quotation in a Responsive Supply Chain for OneofaK ind Production


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1 Dynaic Price Quotation in a Resonsive Suly Chain for OneofaK ind Production Jian (Ray) Zhang Deartent of Mechanical and Manufacturing Engineering, University of Calgary, 5 University Drive NW, Calgary, Alberta TNN, Canada Barrie R. Nault Haskayne School of Business, Scurfield Hall, University of Calgary, 5 University Dr. NW, Calgary, Alberta TNN, Canada Wensheng Yang School of Econoics and Manageent, Nanjing University of Science and Technology, Nanjing, Jiangsu, P R China Yiliu Tu Deartent of Mechanical and Manufacturing Engineering, University of Calgary, 5 University Drive NW, Calgary, Alberta TNN, Canada A bstract This aer studies the setting in which a oneofakind roduction (OKP) fir o ers two tyes of orders (duedate guaranteed and duedate unguaranteed) at di erent rices to the sequentially arriving custoers, who are also OKP roduction firs. The rices for two tyes of orders are quoted to each custoer on its arrival. We study two robles in this setting. First, we odel a dynaic ricing strategy (DPS) and coare our DPS with a constant ricing strategy (CPS). Through a nuerical test, we show that both the fir and its custoers are better o when our DPS is eloyed, so that the DPS iroves overall erforance of the suly chain. Through an industry case, a custo window roduction fir, we show how to aly the roosed DPS when roducts are colex. We also develo a ethod to adatively estiate the fir s available caacity, the nuber of future arrivals and the distributions of the custoers willingness to ay and iatience factor. The siulation result shows that, when ultile distribution araeters are unknown, the roosed araeter estiating ethod results in estiates close to the true values. K eywords: Dynaic ricing, aketoorder, araetric learning, caacity anageent, resonsive suly chain, oneofakind roduction Corresonding author. Tel.:+(); fax:+()886. Eail addresses: (Jian (Ray) Zhang), (Barrie R. Nault), wensheng (Wensheng Yang), (Yiliu Tu) Prerint subitted to Elsevier Seteber,
2 . Introduction Constrained by global coetition in rices and costs, the large firs have liited incentives to rovide highvariety custoization (Cavusoglu et al. 7), which becoes the core coetence of local sall or ediu sized enterrises (SMEs). Oneofakind roduction (OKP) is a strategy that allows local firs to rovide highly custoized roducts at an effective roduction rate. OKP features a once successful aroach for roduct develoent and roduction according to secific custoer requireents, and accordingly no rototye or secien is ade (Tu et al. ). The fir usually does not kee inventory of soe inuts (e.g., arts) and only akes orders on deand. As a consequence, this requires that the entire suly chain rovides fast and reliable delivery. We study the OKP suly chain, in which the custoers of an OKP fir are also OKP firs. An OKP suly chain is characterized by SMEs, batches of orders that are sall or just one ite, aketoorder roduction, downstrea firs that cannot guarantee future orders to ustrea firs, and detailed requireents that change fro order to order. Because of these features, it is difficult for firs within the sae OKP suly chain to for a reliable cooerative relationshi through contract or integration based on revenue and inforation sharing (iannoccaro and Pontrandolfo, Disney et al. 8, Kelle et al. 9). In this work, we exaine suly chain coordination without revenue and inforation sharing. Because it is usually the case that custoers in a suly chain are heterogenous in leadtie requireents, different tyes of custoers should be treated with different riority to realize coordination. However, as an OKP fir usually receives discrete orders which arrive sequentially, then the roble is to allocate caacity to custoers fro different riority classes. That is, whether to allocate caacity to the current custoer or save it for future arrivals that ight be fro higher riority classes and hence generate higher rofits (Keskinocak and Tayur ). In the literature, a wellacceted ethod to solve this roble is rice differentiation between riority classes. The research in ricing under sequentially arriving custoers starts fro the ricing roble in a riority queue. That is, the custoers ay different rices for different ositions within the queue, but the fir does not guarantee the due dates e.g., (Kleinrock 967, Adiri and Yechiali 97, Mendelson and Whang 99, etc). A detailed review of the research on riority ricing can be found in Bitran and Caldentey (). We focus on the case where soe custoers have strict requireents on leadtie guarantee while others do not. In this case the fir offers two tyes of orders: duedateguaranteed (order) and duedateunguaranteed (Uorder), and only the orders are roised duedate delivery. We coare two ricing strategies: a dynaic ricing strategy (DPS) and a constant ricing strategy (CPS). orders and Uorders are riced differently, but in DPS the fir dynaically changes the rice for each tye of order to axiize its rofit. That is, the rice quotes to a custoer are also deterined by its arrival tie. We analyze how the suly chain benefits when each fir rices each order accounting for the order s arrival tie together with the fir s roduction caacity and schedule. To atch the DPS for largerscale robles, we also introduce a eriodic ricing strategy (PPS), which is a coroise between the features of DPS and CPS. Our results suggest a roising aroach to integrate (or coordinate) a twoechelon suly chain with no doinant echelon. A nuber of articles on dynaic ricing can be found in the literature (Yano and ilbert
3 5), but ost research focus on ricing in aketostock (MTS) roduction, e.g., Transchel and Minner (9), Ray et al. (5), etc. In contrast, we use the Bellan equation (Bellan 957) to coute the rice quote for OKP aketoorder firs. The earliest work we have found on ricing using Bellan equations is Kinacaid and Darling (96), and ore recent studies used siilar ethods to odel suly chain dynaics, e.g., Stadje (99), allego and Ryzin (99) and Zhao and Lian (),etc. Using nuerical analysis, we coare rice quotes under the two different ricing strategies. The results show that in largescale robles the CPS rices can be a good aroxiation for DPS rices. We exlain the alication of the roosed DPS through an industrial case fro a custo window roducing fir. In the case study, we roose a ethod to evaluate the fir s available caacity and future order arrivals, which we require to coute rices. We chose a custo window anufacturer for case study because it is a tyical OKP fir whose roducts have hierarchical structures and are rocessed through ultile roduction lines. The DPS studied in this work can also be alied in other OKP firs where leadtie is a factor of quality but the leadtie is constrained by the fir s caacity, e.g. olding coanies, hightech coonent coanies, or service roviders such as transortation coanies. We also develo a set of ractical araeter estiation ethods for our roosed ricing strategy. In the literature, ost authors assue that the distribution of the custoers iatience factors, which can be defined as how uch it costs a custoer for each tie unit that the lead tie of its order is increased, is exogenous and known by the fir. Based on this distribution of iatience factors, otial rices are couted following the disciline of third degree rice discriination (Perloff 9). However, in ractice, this iatience factor is usually hard to easure or obtain. In the suly chain anageent literature the distribution of rando variables are usually obtained fro a learning rocess. For exale, Chen and Plabeck (8) develo a learning ethod to obtain the robability of a custoer choosing a substitute, and Tolin (9) uses a Bayesian learning rocess to dynaically udate the sulier s yield distribution. In our odel, we develo a axiulikelihoodestiation (MLE) based learning ethod to estiate the distribution of the custoer s willingness to ay (WTP) as well as the distribution of the iatience factor. There are also several articles that focus on estiating the distribution of the custoer s WTP. Bisho and Heberlein (979), Haneann (98) and Caeron (988) roosed ethods to estiate the ean of the custoer s WTP when the distribution tye is known. Kriströ (99) studied the case where the distribution tye is not known and roosed a nonaraetric estiator of the distribution of the custoer s WTP, which requires larger sale sizes. Due to sale size liitations in OKP suly chains, we extend the revious literature and study the case where the distribution tye is known, but the distribution araeters, such as the ean and variance of a noral distribution, ust be estiated. This is realistic when the fir has soe rough inforation on its custoers WTP distribution. As for the custoer s iatience factor, we have not found any work studying the estiation of its distribution. Our article is organized as follows. In Section, we describe the roble and define the notation and assutions. In Section a dynaic ricing ethod is resented with a olynoial algorith to find the otial solution. In Section, a CPS is resented. In Section 5, we coare the fir and its custoers welfare under the two different ricing strategies. In Section 6, we roose a ethod to estiate araeters required to coute the dynaic rices. In Section 6., we resent an industrial case to show how to evaluate
4 the fir s available caacity and future custoer arrivals, while in Section 6., a learning rocess is designed to estiate the distribution of the a custoer s WTP and iatience factor. In Section 6., we discuss the anageent insight of this research by referring a ractical case study. Section 8 contains final rearks and a suary of future work.. Notation and Assutions We study an OKP fir which accets two tyes of orders, orders and Uorders. These orders are for the next roduction eriod as shown in Figure. Our roble is to decide the otial rice quote for each tye of order. Every newly received order is guaranteed delivery by the end of the next eriod. To guarantee the roised due date of the  orders, the fir disatches a higher riority to the orders in roduction. The quantity of unallocated caacity is used to guarantee the delivery of orders. We use the ter available caacity to reresent the quantity of unallocated caacity. The fir does not accet orders when it has no available caacity. In addition, the fir stos acceting orders at the beginning of the next eriod, which we nae as deadline. After the deadline, the roduction schedule in that eriod is frozen. The fir does not allow new orders to be inserted into a frozen schedule because at the beginning of each roduction eriod, the fir needs to reallocate the available resources, e.g., the nuber of workers at each achine, internal and external logistics, etc. Inserting an order usually incurs extra cost. The sueriority of freezing roduction schedules has been roven by Sridharan et al. (987). [Figure about here.] Notation. We denote the two different rice quotes the fir offers for order and Uorder by, U :, U R +, resectively. The available caacity, which reresents the nuber of caacity units (i.e., anday, anhour, etc), is denoted by : Z +. The nuber of future arrivals before the deadline is denoted by n : n Z +, which is adatively estiated, noting that each arrival does not necessarily result in an acceted order. (, n) reresents the case in which the fir has available caacity and exects n future arrivals. We use r : r R to reresent a custoer s WTP for a order. A custoer s WTP is deterined by two factors, i.e., its valuation on the fir s roduct and the substitutes fro the fir s coetitors. Suosing that a custoer values the fir s roduct at V : V R and the rofit it can obtain by choosing the best substitute is S : S R +, then we define a custoer s WTP as r = V S. Because V and S are both rando variables, then r is a rando variable. We use v : v R + to reresent the custoer s iatience factor. We define the iatience factor as the cost incurred to the custoer when there is no duedate guarantee. Without loss of generality, we use f(r, v) to denote the joint robability density function (JPDF). We also use the notation f(r, v; θ) to reresent the JPDF of r and v when the for of the distribution functions deends on the vector θ. The adative control rocess is described as in Figure. As shown in Figure, the dynaic ricing odule coutes the rices and U, and then the fir quotes the rices to arriving custoers. The sale collecting odule gathers custoers choices and their arrival rate, and the roduction onitoring odule onitors the workload of the fir s each roduction line in real tie. The sale collecting odule and the roduction onitoring
5 odule eriodically ass the inforation to the araeter estiating odule. Based on the current fir s roduction status, custoer arrival rate and custoer choices, the araeter learning odule adjusts the estiation of, n, θ r and θ v, and then asses the new estiators back to the dynaic ricing odule. [Figure about here.] Assutions. We ake the following assutions to for our odel: Assution. The fir akes a takeitorleaveit offer. We assue that the fir akes rice quote to each custoer. If the custoer accets the rice quote, then it laces the order; otherwise the custoer leaves and does not coe back. A siilar assution can be found in revious research, e.g., allego and Ryzin (99), where they assued that custoers do not act strategically by adjusting their buying behavior in resonse to the fir s ricing strategy. This is also coon in ractice. As an OKP fir usually kees little or no arts inventory and only akes orders on deand, but when the fir orders arts, it requires fast delivery. Therefore, if the custoer does not accet the current offer, then a substitution has to be found iediately and hence long ter strategic behavior does not haen. Assution. The rocessing tie of an order is constant. Here we assue that the workload of a single order is constant and equal to unity, which we take as a caacity unit. This assution is consistent with the characteristic of OKP that the batch size of an order is sall, or even just a single unit. For the case where the orders are heterogeneous in rocessing tie, we can aroxiate the otial rices through our odel, and the ethod is shown in Section 6.. We abstract fro the issues of differing setu costs between orders. Assution. The variable cost of roduction is zero. As entioned in earlier, we treat the labor cost as a fixed cost, that is, an added order does not incur additional labor cost. We do not consider aterial cost as the ricing strategy is our focus recognizing that the roble can be easily generalized by subtracting a constant unit roduction cost fro the unit rice. A siilar assution can be found in the literature on roduction lanning and scheduling when ricing is considered, for exale, Chen and Hall () and Deng and Yano (6). There is little loss of generality as with a constant rocessing tie, we can take the variable cost of each order to be fixed, and reinterret rices as net of costs.
6 . Dynaic ricing strategy (DPS) Our setting can be regarded as an extension of the literature in dynaic ricing, which studies cases in which a fir has a stock of goods to disose of within a secified tie, otential custoers arrive sequentially and stochastically, and the robability distribution of rices they are willing to ay is known. The fir sets rices so as to axiize exected cash receits during the sale, recognizing that unsold ites are worthless to the fir (Bitran and Caldentey ). Our setting is siilar because if the available caacity is not fully allocated after the beginning of a roduction eriod, then it is worthless. We use the Bellan equation to coute the otial rice quotes when the fir offers two tyes of orders. Suose that a custoer arrives and receives rice quotations for order and U for Uorder. If the custoer chooses a order, then its net gain, denoted by ξ, is ξ = r. Otherwise, if the custoer chooses a Uorder, then its net gain, denoted by ξ U, is ξ U = r U v. The concet of net gain associates with the ricesetting fir. If a custoer has ositive net gain by urchasing fro the target fir, it eans the net rofit to be obtained fro the target fir is higher than the one to be obtained fro the best offer of other firs (outside otions). We define the ter absolute gain as the difference between the custoer s valuation of the roduct and the roduct s rice. Then the net gain equals to the difference of the absolute gains the custoer akes by choosing the ricesetting fir s roduct rather than its best outside otion. The net gain easures the difference between the net rofits that can be obtained fro a suly chain with the target fir and a suly chain without the target fir. The custoer chooses the otion that creates the higher net gain and only urchases when its net gain is nonnegative. Otherwise it leaves without urchasing and its net gain (fro the fir) is zero. First, we coute the robabilities of the custoer choosing each tye of order. The custoer chooses the order only if ξ and ξ ξ U. Thus, given, U and the distribution of r and v, the robability of the custoer choosing the order, denoted by P (, U ), can be obtained as P (, U ) = Prob(ξ and ξ ξ U ) = Prob(r and r r v U ) = + + U f(r, v)dvdr () The custoer chooses the Uorder under two circustances, i.e., ξ and ξ < ξ U, or ξ < and ξ U. Thus, given, U and the distribution of r and v, the robability of the custoer choosing the Uorder, denoted by P U (, U ), can be obtained as P U (, U ) = Prob(ξ and ξ < ξ U ) + Prob(ξ < and ξ U ) = Prob(r and r <r v U ) = +Prob(r < and r v U ) + U f(r, v)dvdr + r U U f(r, v)dvdr () 5
7 The Bellan equation used to coute the otial rice quotes is based on the robabilities obtained fro () and (). Suose that when a custoer arrives, the fir faces an (, n) case. For clarity of exression, we let n include the current custoer. We exaine the otial rice quote under two different situations: when caacity is greater or equal to the nuber of future arrivals, and when caacity is less, i.e., n and <n. When n, the fir estiates that the nuber of future arrivals will not exceed the current available caacity. Then the (, n) case can be treated as a straightforward uncaacitated roble. Because there is no caacity constraint, a single arrival has no iact on the rice quotation to the other arrivals. Thus, all the arrivals are quoted the sae rice. The fir otiizes the rice quotation by axiizing the exected rofit contributed by each custoer. Thus the otial rice quotation for roble (, n), denoted by { n, U n}, can be obtained by [ ] { n, U n} = arg ax {, U } P (, U )+ U P U (, U ), () which can be solved through the firstorder conditions. Because there are n future arrivals, the axiu exected total rofit, denoted by π n, is π n = n [ np ( n, U n)+ U np U ( n, U n) ]. () When <n, the (, n) case is a constrained roble. Because the rice quoted to a later custoer deends on the behavior of earlier custoers, we use a Bellan equation to solve the otial rice quote for the current custoer. The iact of the current custoer is suarized as follows: If the current custoer chooses the order, a unit of available caacity is allocated. The fir earns and then faces an (,n ) roble; If the current custoer chooses the Uorder, it does not affect available caacity reserved for orders. Thus, the fir earns U and then faces an (, n ) roble; If the current custoer does not choose either tye of order, then the fir earns no rofit and faces an (, n ) roble. Considering the three ossibilities, the otial rice quote for the current custoer, { n, U n}, can be obtained by solving the following Bellan equation: [ πn = ax P (, U )[ + π {, U } n ]+P U (, U )[ U + π n ] +[ P (, U ) P U (, U )]π n ], (5) where π n can be solved recursively. 6
8 When the fir has no available caacity, the custoers can only urchase Uorders. Thus π n for any n can be obtained as π n = ax U n U Prob(ξ U ) = ax n U Prob(r v U ) U = ax n U U + r U U f(r, v)dvdr. The exale in Figure illustrates the rocess of solving for exected rofit when available caacity is and the nuber of future arrivals is 5, π5, through recursion. To solve each πj i where i and j n, πj i and π i j are first solved sequentially. Note that the values of π and π are known when accessed the second tie. Thus, a tabu list can be eloyed to substantially reduce the coutational tie. We reset an n array as a tabu list to store every solved value of πj. i At the beginning of each recursion for couting πj, i we check the tabu list first and see if it is already solved. If πj i is solved already, then the current recursion stos and directly returns the solution stored in the tabu list. In Figure, each node reresents a recursion, and there are recursions in the entire coutation rocess. [Figure about here.] Let T (, n) be the nuber of recursions required in couting π n n> when the tabu list is incororated. Then by analyzing the branched figure as in Figure, it is not difficult to obtain: T (, n) =(n )+ O(n). (6) Fro (6), we can conclude that when the tabu list is eloyed, π n n> can be couted within olynoial tie, which eans that solving n and U n is not an NPhard roble.. Constant ricing strategy (CPS) In this section we resent a CPS in which the fir does not adjust the rice quoted to the custoers, and then we coare the rices obtained by the CPS and our DPS. Suose with a CPS, the fir quotes rices and U to every arriving custoer. The robability of a custoer choosing the order is P (, U ), and so the nuber of custoers that choose the order follows a binoial distribution with n trials, each of which yields success with robability P (, U ). As the nuber of orders is constrained by, then the exected nuber of orders, denoted by N (, U ), is obtained as N (, U )= [ B ( ; n, P (, U ) )] + ib ( i; n, P (, U ) ), (7) i= where b(k; n, Pr) and B(k; n, Pr) are resectively the f (robability ass function) and the cdf of k when k is a rando integer following a binoial distribution with n trials and success robability Pr. 7
9 Siilar to the orders, the nuber of custoers that choose the Uorder follows a binoial distribution with n trials, each of which yields success with robability P U (, U ). Note that besides the custoers that initially refer Uorders, soe custoers choose U orders because the fir has no available caacity. Thus, the exected nuber of Uorders, denoted by N U (, U ), is obtained as N U (, U ) = np U (, U )+ r U n i=+ [i ]b ( i; n, P (, U ) ) κ(, U ), (8) where κ(, U )= [ + f(r, v)dvdr ] /P U (, U ). κ coutes the robability that a custoer s net gain fro choosing a Uorder is ositive, given that it initially refers a order. Based on (7) and (8), we can solve the otial and U by ax N (, U )+ U N U (, U ). (9), U We resent a nuerical analysis to coare the rices obtained fro the DPS and CPS. In the nuerical analysis, we set r and v to be distributed following three different distributions as shown in Figure. In the case where r and v are indeendent, as in Figure (a), the df of each rando variable is not affected by the value of the other rando variable. In Figure (b) and Figure (c), v and r are correlated. We use the distribution in Figure (b) as an exale where r and v are ositively correlated because the exected value of v is increasing in the value of r, and use the distribution in Figure (c) as an exale where r and v are negatively correlated because the exected value of v is decreasing in the value of r. In the indeendent case, we set r and v to be both distributed fro U(, α ), while in the ositively correlated and negatively correlated cases, we set the r and v to be jointly distributed fro the unifor distribution within the area as shown in Figure (b) and Figure (c). We set α =, α =.8 and α =.8 for the nuerical test. In the three cases, we set /n =. We use nuerical ethods to solve and U fro (9). In order to avoid the colexity incurred fro couting the df and cdf of a binoial distribution, we aroxiate the binoial distribution with noral distribution when n is large (n ). [Figure about here.] The rices for order and Uorder under different roble scales in the three cases are dislayed in Figure 8. We observe that when the roble scale is sall ( and n are sall), the difference between two ricing strategies is substantial. The results also show that with increased roble scale, the stochastic roble aroaches to the deterinistic roble. That is, for both DPS and CPS, the rice quotes converges to the otial solution of deterinistic roble. ax P (, U )+ U P U (, U ), Subject to: np (, U )., U [Figure 5 about here.] 8
10 In the nuerical analysis, we can also observe that when the roble scale is larger than a certain oint, the difference between the rices obtained in DPS and CPS is very sall. Under the constraint of constant ricing, couting the CPS rices can be uch ore efficient. Thus, based on the results of nuerical analysis, we can conclude that CPS rices can be a good aroxiation for DPS rices when the roble scale is large. Although in large scale robles the CPS rice is a good aroxiation for DPS rices, the welfare of the fir and of soe custoers ight be significantly different because in CPS the fir does not change rice as tie asses. Thus, the otial rice quote at any tie ay deviate fro the original setting. In ractice, a ore coon ricing strategy, a eriodic ricing strategy (PPS), can be found where the fir changes the rice setting eriodically when the roble scale is large and it is not feasible to dynaically coute a rice quote for every custoer arrival. In our version of PPS, we assue that the fir adjusts rice quote each tie the new custoer arrivals reach a fixed nuber, which we nae as the ricing interval. In the next section, we also investigate erforance of the suly chain under different settings of ricing interval in PPS. 5. Welfare analysis In this section, we analyze the fir s rofit and the custoers welfare when different ricing strategies are eloyed. 5.. The fir s rofit in DPS and CPS The fir s exected rofit can be obtained through (), (5) and (9). In rincile, the DPS should always yield weakly greater rofits for the fir because the dynaic rice could be set to a constant. The ga between the fir s rofits obtained fro our DPS and CPS in the three distribution cases are shown in Figure 8. In Figure 8, the curves show the ercentage by which the exected rofit obtained fro our DPS is higher than the one obtained fro the CPS. We observe that when the exected value of v increases, the sueriority of DPS is ore substantial. [Figure 6 about here.] 5.. The custoer s net welfare obtained fro the ricesetting fir in DPS and CPS We define the custoers net welfare obtained fro the ricesetting fir as the difference of welfare obtained fro a suly chain with the ricesetting fir and the one fro a suly chain without the ricesetting fir. The custoers net welfare in case (, n) is denoted by W n. In an extree case, if all no custoers buy fro the target fir, the custoers net welfare (fro the ricesetting fir) is zero. First we coute any arriving custoer s exected net gain which can be divided into three arts.. If r and v U, then the custoer chooses the order. The exected net gain, denoted by ξ (, U ), can be couted as ξ (, U )= U [r ]f(r, v)dr.
11 . If r and v < U, then the custoer chooses the Uorder. The exected net gain, denoted by ξ (, U ), can be couted as ξ (, U )= + U [r v U ]f(r, v)dvdr.. If U r < and r v U, then the custoer chooses the Uorder. The exected net gain, denoted by ξ (, U ), can be couted as ξ (, U )= r U U [r v U ]f(r, v)dvdr. When the custoer s r and v are not within the ranges secified above, it does not urchase and its net gain is zero. Based on the exected net gain of each arriving custoer, the custoers net welfare can be obtained as W n = ξ ( n, U n)+ξ ( n, U n)+ξ ( n, U n) +P ( n, U n)w n + [ P ( n, U n)]w n. () In (), the first line coutes the welfare of the current custoer when rice quotes are n and U n, resectively, while the second line coutes the total welfare of later arriving custoers. When n, all custoers are quoted the constant rice regardless of the ricing strategy. Thus, the custoers net welfare can be obtained as W n = n[ξ (, U )+ξ (, U )+ξ (, U )], () where and U are couted by (). When there is no available caacity, =, the custoers can only choose Uorders, and then the custoers net welfare is W n = n + r U U [r v U ]f(r, v)dvdr. () In order to obtain the custoers net welfare in DPS and CPS, we substitute n and U n in (), () or () with the rices couted under DPS and CPS for case (, n). Note that in CPS, n and U n are constant during the recursion. Figure 8 shows the ercentage by which the custoers net welfare obtained by DPS is higher than the one obtained by CPS. We observe that in both the indeendent and correlated cases, the custoers net welfare obtained by DPS is higher. We can also observe that when the ean of v increases, the custoer benefit ore fro DPS. This is because the CPS increases the chance that the rice is underestiated or overestiated when the future sulydeand ratio (/n) changes. If the future /n decreases, as entioned in Section, should increase while U should decrease, and hence in CPS orders will be underriced and Uorders will be overriced. When this haens, custoers that arrive early and urchase orders, are better off. However, desite the custoers that are better
12 off, the net gain of the custoers that urchase Uorders are reduced, and also because the orders are underriced, it increases the risk that the fir does not have enough available caacity for the late arriving custoers that would urchase orders. On the other hand, if the future /n decreases, orders will be overriced and Uorders will be underriced when CPS is eloyed. In this case, custoers that urchase Uorders will be better off, but the net gain of the custoers that urchase orders will be reduced. [Figure 7 about here.] Consider an extree case in which there are unit available caacity and arriving custoers whose r and v are both distributed fro U(, ) as shown in Table. We coute the rice quotes for each custoer when DPS and CPS are eloyed. We can observe that although there is a reduction for the welfare of the custoer who arrives earlier, the increase of the welfare of the custoer who arrives later leads to the increase of the total custoer welfare. As the fir s exected rofit also increases in DPS, we conclude that the welfare for the entire suly chain is increased. [Table about here.] Because in DPS, both the fir s and the custoer s net welfare are increased, the net welfare of the global suly chain is increased, and that is the value of DPS over CPS. In the following section, we roose a learning ethod to estiate the required araeters. These araeters are usually hard to be easured or observed in ractice. 5.. The custoers absolute welfare obtained fro the ricesetting fir in DPS and CPS We define the custoers absolute welfare as the welfare obtained by the custoers suosing all the custoers choose their outside otions after rejecting the ricesetting fir. Thus, when couting the custoers absolute welfare, a custoer who rejects the ricesetting fir ay obtain ositive rofit. We use a nuerical test to show the custoers absolute welfare in both DPS and CPS. In the nuerical test, we suose that by choosing their best outside otion, a custoer s absolute gain, S, is distributed fro U(, 5). We set a custoer s valuation of the ricesetting fir s roduct, R, to be distributed fro U(5, ). Suosing that a custoer s iatience factor, v, is distributed fro U(, ), then based on the definition of a custoer s WTP, r = R S, in Section, we have f(r, v) as in Figure 8. If a custoer chooses order, then its absolute gain is R ; if it chooses Uorder, then his absolute gain is R U v; if it rejects the ricesetting fir, then its absolute gain is S. It is straightforward that without the ricesetting fir, the absolute welfare of n custoers would be ne(s), where E(S) is the exected value of S. [Figure 8 about here.] We kee n/ = and coare the custoers absolute welfare for robles at different scales. Due to the colexity of couting the custoers absolute welfare, we evaluate the custoers absolute welfare based on siulation. That is, we siulate the case where a nuber of custoers arrives sequentially. In DPS, each custoer are quoted rices based on
13 current and n, while in CPS, each custoer are quoted the sae rices couted for the first custoer. By suation, we can obtain the custoers total absolute gain. We reeat the siulation exerient for ties, and then the ean of the total absolute gain in each reetition aroxiately equals to the custoers absolute welfare. The ga between custoers absolute welfare using DPS and CPS under different roble scales is shown in Figure 9. The dashed line in Figure 9 is the ower trendline generated by MS Excel. It can be observed that in DPS the custoers absolute welfare is higher the one in CPS, which is consistent with the result of coarison of custoers net welfare between in DPS and in CPS. [Figure 9 about here.] 5.. The relationshi between ricing interval and custoer welfare In large scale robles a coroise ricing strategy, PPS, is often eloyed. We exaine the relationshi between the rice setting interval and the custoer s welfare is studied through a siulation exerient. In the siulation exerient, we set a custoer s WTP and iatience factor to be both identically and indeendently distributed fro U(, ). We set = 5 and n = 5 to siulate a large scale roble. We show in Figure the ercentage by which the custoers net welfare obtained by PPS is higher than the one obtained by CPS under different settings of the ricing interval, denoted by l. Fro Figure, we can see that the PPS increases the custoer s welfare, and the sueriority of l decreases when l is increased. We observe in Figure that when l is less than, the change of the sueriority of PPS is not very significant. This observation indicates that in ractice, the fir can increase the ricing interval within certain range without significantly affecting the custoer welfare. [Figure about here.] 6. Paraetric estiation for dynaic ricing As stated in the descrition of the control syste in Section, the araeter learning odule needs to adatively change the estiates of, n and other araeters related to the distribution of r and v. 6.. Estiate available caacity () and future arrivals (n) In this work, we consider a case where the overall roduction syste within the fir is coosed by any subsystes, and the overall deand has various requireents for caacity of each subsyste. We use an industrial case fro a custo window anufacturer, which has been referred to in a revious study (Hong et al. ), as an exale to deonstrate the rocess of identifying the estiators of and n. [Figure about here.]
14 As introduced by Hong et al., the design of a custo window can be described by an ANDOR tree as shown in Figure. For the urose of clarity, We use N# to nae a node in the tree. When a node of the tree, say node N, is chosen, the custoer then need to choose its child node(s). Let the set of N s direct child nodes be DC (N), and then the conditional robability of choosing a child node e : e DC (N) is denoted by PROB e. If DC (N) is doinated by an AND relationshi, then PROB e = e DC (N). If DC (N) is doinated by an OR relationshi, then e DC (N) PROB e = We suose that any node in the ANDOR tree, N, ust be rocessed by a roduction line, which is denoted by L N. Note that the roduction line here can be a virtual roduction line. For exale, N and N are just nodes reresenting feature otions, no real roduction line is assigned to these nodes. The window roduction is accolished through five real roduction lines, of which three roduce fraes (i.e., wood, etal and vinyl), one cuts glass, and one is for final assebling. In Figure, each node which requires a real roduction line is arked with *. Here we define two tyes of available caacity relating to a node (N), i.e., the available caacity of the roduction line, denoted by CL N, and the overall available caacity of node, denoted by C N. Suose that the axiu workload that can be rocessed on L N in a roduction eriod is MAXW N, and the current total workload of the jobs for orders to be rocessed on L N is CURW N. Then if N is a node with *, CL N can be obtained as CL N = MAXW N CURW N MAXW N If N is a node with no *, since no real roduction line is required, then CL N =. C N is utually deterined by CL N and the available caacity of N s direct child nodes. C N can be obtained as C N = in{cl N, PROB e C e } if DC (N) is doinated by OR relationshi, () e DC (N) or C N = in{cl N, in C e} if DC (N) is doinated by AND relationshi. e DC (N) In (), we use PROB e as the weight of C e when couting C N because when the custoers are ore rone to choose child node e, then e is ore critical when couting the overall caacity. Because each custoer ay choose different custoization for their roducts, the axiu nuber of windows the fir can roduce varies fro eriod to eriod. Thus, we denote the average of the axiu nuber of windows roduced in each eriod by AMAX. Then the available caacity,, can be obtained as = C N AMAX The future arrivals n is the future arrivals before the deadline. In ost theoretical work, the assution is ade that the custoers arrive according to a Poisson rocess, and so n can be estiated as λt, where λ is the arrival rate and t is the tie left to the deadline. However, in any cases, the arrival rate is not just a function of tie. For exale, when
15 the fir s target arket is liited, then the nuber of future arrivals can also be related to the ast arrivals. A variety of ethods to redict the exected nuber of arrivals can be found in Arstrong and reen (5). Even though we assued the rocessing tie of an order is constant (Assution ), in ractice a custoer s order ay vary in its caacity requireent (or rocessing tie). In such cases, we can use our ethod to coute the rice for an order with average caacity requireent. Suosing that after acceting the order, the available caacity becoes, then the rices for order and Uorder can be aroxiately obtained at n( ) and U n( ), resectively. 6.. Learning the distribution of r and v Because it is hard to observe the custoer s WTP and iatience factor in ractice, in this section we focus on develoing a learning rocess to estiate the distributions of r and v. The for of a distribution ight be deterined by any features, such as the distribution tye (i.e., exonential, noral, unifor, etc.). In ractice, the fir ight have ore inforation about soe articular features but less inforation about others. We study the case in which the fir knows the distribution tye of the custoer s WTP and iatience factor, but is uncertain about values of the distribution araeters. Without loss of generality, we suose that the fir knows that r and v are indeendent, and the distribution tyes of r and v are noral, but the eans (µ r and µ v ) and the variances (σr and σv) are unknown. Thus, the goal of the learning rocess is to estiate µ r, µ v, σ r and σ v so that the distribution functions required for couting the otial rice can be fored. We let f r (r; µ r, σ r ) and F r (r; µ r, σ r ) resectively be the df and cdf of r, and let f v (v; µ v, σ v ) and F v (v; µ v, σ v ) resectively be the df and cdf of v. Because extra araeters (µ r, σ r, µ v and σ v ) are needed as inuts in the cdf and df, we odify the arguent in the notation for the robabilities that a custoer urchases a order or a Uorder in () and (). Suosing that the fir has recorded K custoers urchases and custoer k : k {,..., K} is quoted rices k and U k for each tye of orders, then the robabilities that a custoer urchases θ a order or a Uorder can be reresented as P (, U θ ) and P U (, U ) resectively where θ =(µ r, σ r,µ v, σ v ). We develo a axiulikelihoodestiation (MLE) based araetric estiation ethod to find the estiates of µ r, µ v, σ r and σ v. The likelihood, denoted by L, is constructed as where L = K k= [ [ P θ (, U θ ) P U (, U ) ] yk [ θ P (, U ) ] yk [ θ P U (, U ) ] ] yk U, () y k = y k = y U k = { if custoer k leaves without urchase; otherwise. { if custoer k chooses order; otherwise. { if custoer k chooses Uorder; otherwise.
16 To silify the for of the likelihood, we take natural logs both sides of (). After sile algebra, we can obtain ln L = K k= [ yk ln [ θ P ( k, U θ k ) P U ( k, U k ) ] ] +yk θ ln P (, U )+yk U θ ln P ( k, U k ). (5) Because of the colexity of solving the firstorder condition of (5), we use trustregion ethod (Conn et al. ) to search the otial setting of araeters. We resent a neural network (NN) based regression ethod as a benchark for the MLE based araeter estiation ethod. In NN, we train the network to regress the robability of a custoer choosing each tye of orders with resect to the rice quote for each tye of orders. That is, k and U k are the inuts, and P ω (, U ) and PU ω (, U ) are the oututs, where ω is the vector of weights which will be deterined through LevenbergMarquardt backroagation training. In the training rocess, y and y U are the target values for P ω and PU ω, resectively. We set hidden layer which includes neurons for the NN. In the siulation, r and v are each norally distributed such that r N(, ) and v N(, ). The rices quoted to each custoer are randoly generated such that U(, 5) and U U(, ). iven a cobination of r, v, and U, we obtain each custoer s choice indicators, y, y and y U. We test the two regression ethods under different volues of historical sale records. Then we coare the oututs of MLE based ethod and NN based ethod with the true result couted by () and () in Section. In the coarison, we randoly choose and U fro U(, 5) and U(, ), resectively, and then show the coefficient of variation of the root ean square error, CV RMSE, couted as E [ (y ŷ) ] CV RMSE = ŷ, where y is the outut of the regressed robability functions, ŷ is the outut of the true robability functions (() and ()), and ŷ is the ean of ŷ. The coarison of MLE based regression and NN based regression is shown in Table. In Table, each value of CV RM SE is the average of rounds of siulation. We observe that the MLE based regression ethod can achieve an outut uch closer to the true value couted by () and (), which suorts the sueriority of the MLE based regression. However, this MLE based regression ethod is only alicable when the distribution tyes of r and v are known. In the case where r and v are distributed with unknown distribution, the NN based regression is the only otion because r and v are not required to for P ω (, U ) and P ω U (, U ). [Table about here.] Figure grahically shows the oututs of the true robability functions and the regressed robability functions using different regression ethods when the sale size is 5. We exect that the oututs of NN based regression aroxiates better when the nuber of neurons and sale sizes are increased, whereas it is usually not realistic to obtain a large enough sale. However, because the NN based regression is an substitute of the MLE 5
17 based regression when the fir cannot estiate the distribution tye of r and v, then it is a good ractice for the fir to siultaneously run the two regressions so that the fir can cature the failure of estiating the distribution tye of r and v. [Figure about here.] 7. A ractical case study of a local window and door coany As a ractical case study we investigate a local door and window coany and analyze its sales data to study the otential to increase the coany s exected rofit and the custoers welfare. Hereafter, we use the ter rincial coany to refer the local door and window being studied. As an OKP coany, the rincial coany rovides custoized doors and windows to the local custoers that include local builders and renovators. Custoer orders can be divided into two categories: NewConstruction (NC) and Sales&Installation (SI). In the NC category, the custoers order the custoized doors and windows for building new hoes or other kinds of new estate roerties, while in the SI category, the custoers order relaceents of broken, inefficient, or outofstyle windows or doors. In the coany s business, the NC orders and SI orders take u the roortions of 55% and %5 (by dollar sales), resectively. The two categories of orders have different features. In the NC category, the custoers usually order the roducts early before the due dates direct fro the coany. For exale, soe builders usually order the doors or windows with secified requireents several onths before the due dates and request ontie delivery services. Because of the longter cooeration with the builders, the rincial coany schedule the NC orders with riority. In the SI category the windows and doors are usually ordered for iediate use. Thus, the custoers are sensitive to the leadtie. In the rincial coany the roduction is lanned in weeks. At any tie, the roduction schedule for the current business week is deterined and frozen. Thus, new orders can only be scheduled after the current business week, which eans that at earliest, the rincial coany can only guarantee the delivery before the end of the next week. In the SI category, the rincial coany treats the orders which require delivery before the end of the next week as orders, and treats the other orders as Uorders. For the Uorders, the coany only roises a leadtie of  weeks. The rincial coany roduces roducts of ultile roduct failies, and the roducts in different roduct failies are rocessed in different roduction lines. Thus the caacity of the roduction line deterines the roduction caacity for a roduct faily. The rincial coany rices each roduct faily searately. The custoers ake Uorders by default, and for each roduct faily, there is a fixed rice for every detail settings of roduct features. If a custoer requires a order, then an extra aount is charged on to of the Uorder rices. The nuber of orders is constrained by the coany s available caacity. The available caacity is couted by subtracting the workload of rescheduled jobs, which include NC orders and the Uorders which have been laced for weeks, fro the caacity of the 6
18 roduction line. Here, we only show the coany s sales data of one roduct faily, vinylfraed windows. The sho floor layout of the designated roduction line, V, is dislayed as in Figure. [Figure about here.] Figure shows the week sales data of V orders in Aril 9. Note that the sales data is scaled in this article because of the nondisclosure agreeent. At the beginning of the week, the roduction anager counicates with the sales deartent the available caacity to guarantee the delivery of orders received in that week. The sales deartent stos acceting orders when the available caacity is coletely allocated. The nubers of orders and Uorders as well as the rescheduled orders are dislayed in the for of accuulated barchart. The dash line in Figure shows the V roduction line s caacity. The rincile coany eloys a sile differential ricing strategy for orders and U orders, where the custoers are charged 5 ore than the Uorder rice for a order. [Figure about here.] Fro Figure, we can see that aong the SI orders received within the four weeks, the roortion of orders takes u 5%, 6%, % and 6%, resectively. It is obvious that in the first week, the available caacity is not fully utilized while in the third and the forth weeks, the roortions of orders are bounded by the coany s available caacity. Fro Section 5, we know that in this case the coany and the custoers welfare are not axiized due to iroer ricing. Thus, a otential exists where both the coany and the custoers welfare can be increased by eloying DPS or PPS. In order to estiate the distributions of the custoers WTPs and iatience factors, the coany can first dynaically change the extra charge for orders by adding or subtracting a constant aount. The coany can also add a eriodically varying rootion discount to Uorders. When the coany has enough sales data under varying rices, the custoers WTPs and iatience factors can be estiated using MLE, and then the ore efficient extra charge for orders and rootion discount for Uorders can be couted through the ethod introduced in this work. 8. Conclusion In this research we odeled a DPS when the fir s caacity is liited and a duedate guarantee ay be required or favored by soe custoers. We forulated a dynaic Bellan odel to coute the otial rice quotes. We analyzed the coutational colexity of the roosed dynaic ethod, roving that the colexity of the dynaic ethod to coute the otial rice quotes is olynoial. Usually it is believed that if the fir dynaically changes rices to axiize its rofit, the arket deand is exloited and so the custoers welfare is lowered. However, by coaring DPS with the constant ricing strategy (CPS), we show that when the fir dynaically change the rices for each tye of orders, both the fir and the custoers are better off. This finding strongly suorts the sueriority of DPS. In the literature, we have not found any research studying the how the DPS increases benefits to the entire suly chain. 7
19 For ractical alications, we also roosed ethods to estiate the araeters, which are coleentary to the roosed ricing strategy. We introduced the ethods to evaluate the fir s current available caacity, future arrivals, and the distribution of the custoer s WTP and iatience factor. In the estiation of the distribution of WTP and iatience factor, the roosed estiation ethod can be easily extended so that the distribution with ore unknown araeters can be estiated. We also resented a case study where DPS and CPS could be eloyed. As future work, the roosed ethod can be studied within a suly chain which includes one sulier that sulies arts for ultile anufacturers. In this setting, the sulier is OKP anufacturer and faces a ricing roble with resect to the delivery date, while the downstrea anufacturers choose the best otion and then face a job sequencing roble. Acknowledgeent We thank the Natural Sciences and Engineering Research Council of Canada (NSERC) under the discovery grant rogra, titled Adative Planning and Control of Suly Chain in Oneofakind Production, and Natural Science Foundation (NSF) of China under its grant 7876, titled Otiization and Synchronization of Leadtie Sensitive oneofkind Production Suly Chain for suort. We also thank the Robson Professorshi Endowent, and the Inforatics Research Center in the Haskayne School of Business at the University of Calgary for suort. References References Adiri, I., Yechiali, U., 97. Otial riorityurchasing and ricing decisions in nononooly and onooly queues. Oerations Research (5), Arstrong, J. S., reen, K. C., 5. Deand forecasting: Evidencebased ethods. Forthcoing chater, Monash University, Deartent of Econoetrics and Business Statistics. Bellan, R., 957. Dynaic Prograing. Princeton University Press, Princeton, NJ. Bisho, R. C., Heberlein, T. A., 979. Measuring Values of Extraarket oods: Are Indirect Measures Biased? Aerican Journal of Agricultural Econoics 6 (5), Bitran,., Caldentey, R.,. An overview of ricing odels for revenue anageent. Manufacturing & Service Oerations Manageent 5 (), 9. Caeron, T. A., 988. A new aradig for valuing nonarket goods using referendu data: Maxiu likelihood estiation by censored logistic regression. Journal of Environental Econoics and Manageent 5 (), Cavusoglu, H., Cavusoglu, H., Raghunathan, S., 7. Selecting a custoization strategy under coetition: Mass custoization, targeted ass custoization, and roduct roliferation. Engineering Manageent, IEEE Transactions on 5 (), 8. Chen, L., Plabeck, E. L., 8. Dynaic inventory anageent with learning about the deand distribution and substitution robability. Manufacturing Service Oerations Manageent (), Chen, Z.L., Hall, N..,. The coordination of ricing and scheduling decisions. Manufacturing Service Oerations Manageent (), Conn, A., ould, N., Toint, P.,. Trustregion ethods. MPSSIAM series on otiization. Society for Industrial and Alied Matheatics. Deng, S., Yano, C. A., 6. Joint roduction and ricing decisions with setu costs and caacity constraints. Manageent Science 5 (5),
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LAB VIII. LOW FREQUENCY CHARACTERISTICS OF JUNCTION FIELD EFFECT TRANSISTORS
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