Integration of capacity, pricing, and lead-time decisionsin a decentralized supply chain

Stuart X. Zhu n

Department of Operations, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands

a r t i c l e i n f o

Article history:Received 14 November 2013Accepted 25 February 2015Available online 5 March 2015

Keywords:CapacityPricingLead timeStackelberg gameIncentives and contracting

a b s t r a c t

We consider a decentralized supply chain consisting of a supplier and a retailer facing price- and lead-time-sensitive demand. The decision process is modelled by a Stackelberg game where the supplier, as aleader, determines the capacity and the wholesale price, and the retailer, as a follower, determines thesale price and lead time. The equilibrium strategy of these two players is obtained. By comparing withthe performance of the corresponding decentralized chain without capacity decision as a benchmark, wecharacterize the impact of capacity decision on the players prot, i.e., the supplier's prot may be alwayssignicantly increased while the retailer's prot is only increased when the capacity is underestimatedin the benchmark model. Further, we demonstrate that the integration of capacity decision can alsosignicantly reduce the prot loss caused by double marginalization. Finally, we nd that the revenue-sharing and two-part tariff contracts cannot coordinate the decentralized channel. Instead, we propose afranchise contract with a contingent rebate that can achieve channel coordination and a winwinoutcome.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

In various industries, rms frequently face capacity, pricing andlead-time decisions. The integration of these three critical deci-sions between manufacturing and marketing/sales is widely citedas a weapon to gain a competitive advantage in the market (e.g.,Shapiro, 1977; Braham, 1987; Crittenden, 1992; O'Leary-Kelly andFlores, 2002). However, most 21st-century supply chains aredecentralized. For example, in the healthcare industry, PhilipsHealthcare Netherlands, a world-known medical-equipment man-ufacturer, has several manufacturing plants in Europe and NorthAmerica. To fulll the demand in Asia, the rm has a major dealerin Hong Kong, LABandTECH (http://www.labntech.com/). Whencustomers want to purchase the products, the dealer usuallydetermines both the sale price and the promised delivery time(lead time) to customers. Because of diversied and uncertaincustomer requirements, the manufacturer only starts the produc-tion/assembly after customers nalize their orders. Therefore, todeliver the orders to customers on time, the capacity decision isvery imperative. Another example is Internet retailing. Whencustomers visit Amazon.com or BestBuy.com, they can observethe on-line information of the price and the delivery time

determined by the e-commerce rms. After customers conrmtheir orders, the e-retailers may ask UPS or the postal service tohandle the delivery. Those logistics providers have to guarantee asufcient capacity to meet the requirement, specially in the peakdemand season. From the above examples, we observe that tomaintain customer satisfaction and achieve competitive advan-tages, it is essential to integrate the capacity, pricing, and lead timedecisions by aligning the interests of different companies, which isa fundamental step to achieve a triple-A supply chain (Lee, 2004).However, as Narayanan and Raman (2004) point out, mostcompanies do not worry about their partners' strategy whenbuilding supply chains to deliver goods and services to customers,which results in poor supply chain performance.

From the perspective of supply chain integration, the purposeof this paper is to investigate the impact of integration of capacity,pricing, and lead time decisions in a decentralized supply chain(DSC) on the marketing strategies and prots of the rms and howto coordinate such a DSC in order to achieve a winwin outcomefor all the rms. As a long term decision, suppliers usually makethe capacity decision based on historical demand information butignore the impact of the capacity decision on their supply chainpartner, which may cause a negative effect on her own. Therefore,the supplier's capacity decision should be aligned with herpartner's pricing and time decisions. Moreover, it is necessary tomotivate the supplier to make the rst-best capacity decision inorder to yield high prots for all the participants in the chain.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijpe

Int. J. Production Economics

http://dx.doi.org/10.1016/j.ijpe.2015.02.0260925-5273/& 2015 Elsevier B.V. All rights reserved.

n Tel.: 31 503638617; fax: 31 503632032.E-mail address: x.zhu@rug.nl

Int. J. Production Economics 164 (2015) 1423

However, in most of the previous literature, the capacity decisionis made independently of the pricing and lead time decisions. Tothe best of my knowledge, the coordination of capacity, pricing,and lead time decisions in the DSC has not been well studied yet.Hence, this paper addresses the following questions. What are theoptimal capacity, pricing, and lead time decisions in the DSC? Canthe incorporation of capacity decision signicantly increase boththe supplier's and the retailer's prots? Can it also reduce theprot loss caused by double marginalization? How can an incen-tive mechanism be designed in order to align the interests of thetwo individual rms and achieve channel coordination? Webelieve that the answers to the above questions can providehelpful guidelines and decision-making tools for managers.

We consider the DSC consisting of a supplier (or a manufac-turer) and a retailer (or a distributor). We model the dynamicdecision process between the supplier and the retailer by using aStackelberg game. The supplier, as a leader, determines thecapacity and wholesale decisions. Given the supplier's decisions,the retailer, as a follower, decides the sale price and promiseddelivery time (PDT). First, we develop the procedure to obtain theoptimal decisions of capacity, pricing, and PDT for the DSC. Second,by using the model without capacity decision as a benchmark, wend that when the capacity decision is underestimated in thebenchmark model, the incorporation of capacity decision can yielda shorter PDT, a higher retail price, and a larger market share. As aresult, the prots of both rms increase greatly. However, whenthe capacity decision is overestimated, only the supplier's prot isincreased while the retailer's prot is decreased. Third, we ndthat the integration of capacity, pricing and time decisions maysignicantly reduce the negative impact of double marginalizationon the DSC. Finally, we demonstrate that the revenue-sharing andtwo-part tariff contracts cannot coordinate the DSC and propose afranchise contract with contingent rebate that can achieve channelcoordination and a winwin outcome.

The remainder of the paper is organized as follows. In Section2, we review the related literature. Section 3 formulates thedecentralized decision models for the retailer and the supplier.In Section 4, we derive the equilibrium. Section 5 investigates theimpact of the integration of capacity decision on the rms' protsand double marginalization. Next, we examine how to achievechannel coordination and a winwin outcome in Section 6. Section7 summarizes the paper and points out some potential directionsfor further research.

2. Literature review

There are two streams of research related to our work: leadtime quotation and capacity management, as well as coordinationof the DSC. There exists an extensive study on the rst stream. Byassuming an exogenous capacity, Palaka et al. (1998) and So andSong (1998) study the optimal selection of price and delivery time(which is uniformly applied to all customers' orders) so as tomaximize the prot of an M/M/1-type Make-to-Order system. So(2000) extends the work to multiple rms and derives theequilibrium solution. Ray and Jewkes (2004) study the optimaldelivery time strategy for a single rm with capacity investmentand a service-level constraint by explicitly modelling the relation-ship between price and delivery times. Boyaci and Ray (2006)analyze the impact of capacity cost on the lead time quotation fortwo substitutable products. Shang and Liu (2011) consider a Nashgame where multiple rms make their lead time and capacitydecisions. In these papers, a static PDT is used as a decisionvariable that affects demands directly like the price. This is similarto our work. However, these papers consider either a singledecision maker or multiple decision makers at the same stage of

a supply chain while we consider two independent decisionmakers at two stages in a DSC and provide a more in-depth studyof the system performance under different decision scenarios. Thedynamic PDT quotation and/or pricing problem has been studiedby Duenyas and Hopp (1995), Plambeck (2004), Celik andMalglaras (2008), and Feng et al. (2011).

Wang and Gerchak (2003) study capacity games in push andpull systems in which both the assembler and the componentsuppliers have to make capacity decisions before the uncertaindemand is realized. As for the integration of pricing, lead time andcapacity decisions, Boyaci and Ray (2003) examine the threedecisions of two substitutable products for a centralized rm withprice and time-sensitive demands. Ray et al. (2005) explore theoptima pricing and stocking decisions in a serial two-echelondecentralized supply chain with uncertain price-sensitive demandand random delivery time. Liu et al. (2007) consider pricing andleadtime decisions and show that the inefciency in the DSC isstrongly affected by market and operational factors. Those papersconsider either pricing and capacity decisions or pricing andleadtime decisions while our paper studies the integration ofpricing, lead time, and capacity decisions. Recently, Pekgn et al.(2008) study the capacity, pricing, and leadtime decisions for twodecentralized departments in a rm and treat the marketing as therevenue center and the production as a cost center. The authorssuggest a revenue-sharing contract with transfer price to coordi-nate the pricing and leadtime decisions. Different from this paper,we consider two individual players in a decentralized channel andeach of them aims to maximize their own prot rates.

For the second stream, there is a signicant body of literatureon DSCs for channel coordination. Various incentive schemes havebeen proposed to achieve coordination, including two-part tariffby Zusman and Etgar (1981) and Moorthy (1987), buy-backcontracts by Pasternack (1985), franchise arrangement by Lal(1990), quantity exibility contracts by Tsay (1999), sales-rebatecontracts by Taylor (2002), and revenue-sharing contract by Wanget al. (2004) and Cachon and Lariviere (2005). These papers focuson the coordination of pricing and replenishment quantity deci-sions in a DSC. Pekgn et al. (2008) consider the coordination ofpricing and time decisions between manufacturing and marketingdepartments in a rm. Leng and Parlar (2009) show that a prot-sharing contract can coordinate the decision of lead time betweena manufacturer and a retailer. Hu et al. (2011) analyze a coordina-tion scheme with an internal price for the manufacturing and thesales departments. Xiao and Qi (2012) study a two-stage chainconsisting of one supplier and one manufacturer. The manufac-turer determines the sale price and quoted lead-time, and main-tains a delivery time standard. Under their setting, the authors ndthat all-unit quantity discount mechanism can coordinate thedecentralized channel for most cases. Different from these papers,we address the coordination of pricing, lead time, and capacitydecisions. Interested readers can refer to a recent comprehensivereview by Cachon (2003).

3. Decision models

We consider the DSC consisting of one supplier and oneretailer. Upon receiving an order from the retailer, the suppliercompletes the nished product and delivers it to the retailer.Potential customers are sensitive to both price and PDT, whichrequires the retailer to offer an appropriate retail price and PDT. Asindependent decision makers, the supplier and the retailer maketheir own decisions in order to maximize their individual protrates. The supplier controls her production facility. Naturally, thesupplier should determine the production capacity (). Then, thesupplier also chooses the wholesale price (ps). In response to the

S.X. Zhu / Int. J. Production Economics 164 (2015) 1423 15

supplier's price and capacity decisions, the retailer determines thebest retail price (pr) and PDT () to be quoted to customers so thatthe retailer's own prot rate is maximized. In making thesedecisions, the retailer will also be fully responsible for any latedelivery penalties. Since the retail price together with the PDT willaffect the level of demands and thus the supplier's operation costand prot, the supplier must consider the reaction of the retailerwhen making her decisions. Thus, the two members of the supplychain interact in a Stackelberg game with the supplier as theleader and the retailer as the follower. We assume that all theinformation is the common knowledge for both players.

Suppose that the utility of individual customers follows auniform distribution (Mendelson and Whang, 1990; Stidham,1992). A customer will purchase only if his or her expected utilityis greater than or equal to prcw, where cw is a standard waitingcost per unit of PDT. Customers play a role in the decision processin which their independent decisions collectively dene thedemand level corresponding to the quoted retail price and PDT.Then, we can derive the demand model suggested by Liu et al.(2007), i.e.,

pr ; 0pr; 1where 0 is the size of the potential market, 40 is the pricesensitivity factor, and 40 is the PDT sensitivity factor. Further,we have cw =.

Now, let us dene the plays' objective functions. For theretailer, because the capacity of the supplier is nite and theremay exist supply and demand uncertainties in the DSC, the orderresponse time (ORT) may deviate from PDT, where the ORT is thetime period from the epoch when an order is received to the epochwhen the customer receives the order. One one hand, when latedelivery occurs (the realized ORT is longer than the PDT), custo-mers must be compensated. In practice, there are different ways tocompensate affected customers. For instance, a discount or partialrefund can be offered if a customer is willing to wait. Expediteddelivery, e.g., airfreight instead of ocean shipping, can also be usedwithout additional charge to the customer. Here, we assume thatcustomers' additional waiting cost and inconvenience from latedelivery will be covered exactly by the retailer with a genericpenalty cost per order per unit time given by b, and thus thepossible delivery delay and the associated compensation areknown to the customer. On the other hand, when an earlycompletion occurs (the realized ORT is shorter than the PDT),the rm may deliver the nished product to the customer or mayhold it until the promised delivery time. In the former case, therm may or may not incur an early delivery cost to compensatethe customer for the early delivery. In the latter case, the rmincurs a nished-goods holding cost. In either case, we can modelthis scenario by charging the rm an early completion cost h perorder per unit time. Let R be the distribution function of the ORTfor a given demand rate . Then, the retailer's operating cost isgiven by

C;R hZ 0t dRtb

Z 1

t dRt; 2

where C;R also represents the retailer's expected leadtimecost. The retailer's optimization problem (ROP) is given by

ROP : maxpr ;

rpr ;; ps; prpscrC;R

pr ;; 3

where cr is the administrative expenses, arising from communica-tion with customers, order handling and delivery. Note that rrepresents the prot rate of the retailer that is dened as themultiplication of the prot per unit of demand and the demandrate. Further, affects the probability distribution of the order

response time, which has an effect on the retailer's operating cost.If the early delivery is allowed without penalty, the holding cost hcan be set to zero and the above model is still completely valid andthe corresponding results hold.

For the supplier, the prot structure is relatively simpler, andincludes ps as the unit wholesale price, cs as variable cost per unit,and c0 as cost per unit capacity. Thus, the supplier's optimizationproblem (SOP) is given by

SOP : max;ps

spr ;; ps; pscspr ;c0; 4

where we assume that the capacity cost is a linear function interms of the capacity like So and Song (1998) and Ray and Jewkes(2004). In fact, we extend this assumption to a convex function of and our main results can still hold. The list of notation issummarized in Table 1.

4. Stackelberg equilibrium

In this section, we characterize both the supplier's and retailer'soptimal strategies. We rst present the retailer's best response tothe supplier's decisions. It is usually very difcult to derive theORT distribution for a real supply chain. Even though for somesimple systems we can derive the exact ORT distributions, theywould be too complicated to be useful in our optimization. Sincewe focus on understanding the impact of the pricing and PDTdecisions by the retailer and the capacity decision by the supplieron the realized delivery lead time through the demand rate, weassume that the supplier's production facility is modelled by anM/M/1 queue. The M/M/1 model has been often used in the supplychain literature, e.g., So and Song (1998) and So (2000). It providesa good approximation of light-tailed waiting times (Boyaci andRay, 2003). From standard queueing results, we have

Rt 1e t ; 5

where Rt represents the probability that the order responsetime is no longer than t. By substituting (5) into (2), we obtain

C;R hhbe h

:

Table 1Notations.

Decision variables The supplier's production capacityps The wholesale price per unitpr The retail price per unit The promised delivery time quoted by the retailer The demand rate

Model parameters0 The size of the potential market The price sensitivity factor The time sensitivity factorcw The waiting cost per unit timeh The inconvenience cost of early completing per order per unit timeb The penalty cost per order per unit timecr The administrative cost per unit timec0 Unit capacity cost

Other notationsR The probability distribution of the order response timeC;R The retailer's operating costs The supplier's prot rater The retailer's prot raten Denotes optimality

S.X. Zhu / Int. J. Production Economics 164 (2015) 142316

4.1. Retailer's best response

Using (1) to write pr in terms of and substituting (5) into (3),we can express the retailer's problem in terms of and , i.e.,

max;

r; pmaxr =cwC;Rcrps

; 6

where pmaxr 0= is the maximum feasible retail price.By (6), for any given , it is obvious that the retailer's best PDT is

uniquely given by

n 1 ln

hbhcw

: 7

Remark 1. From (7), if the retailer's penalty cost is less thancustomers' waiting cost, i.e., brcw, then it is optimal for theretailer to always quote a zero leadtime regardless of the demandrate. Thus, we assume b4cw to exclude this trivial solution.

We use a sequential solution procedure to solve (6): for a given, we rst obtain n as a function of ; we then substitute ninto r and change the retailer's decision problem to a single-variable problem, i.e.,

r;n pmaxr =cwnCn;Rcrps: 8

Lemma 1. The retailer's prot rate (8) is concave in .

The proof of this lemma as well as all the other proofs in thispaper are given in the appendix. The following proposition givesthe retailer's best response to any given and ps.

Proposition 4.1. For given and ps, the retailer's best PDT andpricing strategies are given by (7) and

pnr pmaxr =cwn; 9where is uniquely determined by

02

pscr0

2 0; 10

and 0 hcw lnhb=hcwcw .

4.2. Supplier's optimal strategy

Using (10) to write ps in terms of and , we have

ps 02

0

2cr : 11

Then, substituting (11) into (4), we can express the supplier'sproblem in terms of and , i.e.,

max;

s; 02

0 2

crcs" #

c0: 12

Lemma 2. For a xed , the supplier's best capacity choice isuniquely given by

0 3

c0 0: 13

By (13), we can treat n as a function of . Then, we canrewrite the supplier's decision problem to a single-variable pro-blem, i.e.,

s; n 02

0n

n 2crcs

" #c0n: 14

We have the following lemma.

Lemma 3. The supplier's prot rate (14) is unimodal in .

The following theorem characterizes the Stackelberg equili-brium of the game.

Theorem 1. The unique Stackelberg equilibrium for the decentra-lized problem is given by n; pnr ;pns ; n, where n from (7), pnr from(9), pns from (11), and

n from (13). The optimal demand rate n is theunique solution to

n 1c04n= 3

1c04n= 1 c00; 15

where 1 0=crcs:The optimal prot rates of the retailer and the supplier are now

given, respectively, by

ns n2n

0

n

nn 2" #

;

nr n21 0

nn 2" #

:

The following two corollaries specify the range of the demandrate and compare the supplier's and retailer's prot rates underdifferent intervals.

Corollary 1. We have lonou, where u 1c0=4 andl 51c0

21c20141c0

q=24. u and l represent the

upper and lower bound of the demand rate in the DSC, respectively.

Corollary 2. If nA 1c0=5; u, nsZnr ; if nA l; 1c0=5, nsonr .

Corollary 2 indicates that on one hand, because of the capacityinvestment, the supplier needs to have a sufciently large marketshare to generate enough prot to cover the overhead cost. On theother hand, the retailer mainly tries to nd the optimal trade-offbetween the revenue and the cost for the maximal prot. The retailerprefers not to have a high-congested/high-utilization system.

5. Impact of capacity decision

As a long term decision, the supplier usually makes thecapacity decision based on historical demand information butignores the impact of the capacity decision on her supply chainpartner, which may eventually cause a negative effect on her own.To motivate the supplier to consider the appropriate capacitydecision, we rst investigate the prot gain by the integration ofthe capacity decision and comparing the current model (termedthe CPL model) with the same model without a capacity decision(termed the PL model). Next, we demonstrate that the incorpora-tion of the capacity decision may also signicantly reduce thenegative impact of double marginalization on the channel prot.

5.1. On players' prot

Here, we investigate the PL model with a pre-xed capacity. Inother words, the supplier does not treat the capacity as a decision.Therefore, we assume that the supplier only determines thewholesale price. To distinguish with the solutions to the CPLmodel, we use the symbol bar to indicate the correspondingsolution to the PL model. For example, means the demand ratefor the PL model.

Similar to Theorem 1, we can show that the unique Stackelbergequilibrium for the decentralized problem is given by n; pnr ; pns ,where n from (7), pnr from (9), p

n

s from (11). The optimal demand

S.X. Zhu / Int. J. Production Economics 164 (2015) 1423 17

rate n is uniquely given by

04n

0n

n 3 crcs 0: 16

The prot rates are given by

n

s n2 2

20

n 3

264

375c0;

n

r n2 1

0

n 2

264

375:

Proposition 5.1. n is increasing in , n is decreasing in , nr isincreasing in .

Proposition 5.1 indicates that when the supplier's capacityincreases, the retailer offers a shorter PDT to expand its marketshare. As a result, the retailer's prot rate increases.

Next, we numerically explore the impact of the capacity decisionon pricing decisions and prot rates. We try to address the followingquestions: How are the optimal pricing strategies affected by thecapacity? Does the supplier's prot rate increase signicantly byincorporating the capacity decision? Does the retailer also benetfrom the incorporation of the capacity decision?

In the following numerical example, we set 0 30, 1, 4,cs 1:5, cr 0:8, h0.5, b6, and c0 2. Under this parametersetting, we rst compute the optimal solution to the CPL model, i.e., n 12:58, n 5:85, n 0:06, pnr 23:93, pns 15:93,ns 59:25, and nr 38:49. In Table 2, we compute the protgaps of the supplier and the retailer between the CPL model and

the PL mode, where s and r are the prot gaps of the supplierand the retailer, respectively, where

i ni

n

i

n

i

100%

and i s; r.From Fig. 1, we nd that for the PL model, both the retail price

and the wholesale price are decreasing in . When the supplier'scapacity is low, the retailer has to charge a high price and quote along PDT to customers, which deteriorates competitiveness of theDSC. When the capacity becomes larger, the retailer can offer alower the price and a shorter PDT to attract more customers,which benets both parties.

From Table 2, we observe the following phenomena: (i) theinappropriate capacity decision may cause the signicant protloss for the supplier. For example, when the capacity is wrongly setat 25, the supplier's prot could be decreased about 40%; (ii) thesupplier's prot is more sensitive on the change of the capacitywhen the capacity is overestimated compared with the optimalvalue; (iii) the retailer's prot is more sensitive on the change ofthe capacity when the capacity is underestimated compared withthe optimal value; (iv) when the pre-xed capacity is smaller thanthe optimal capacity, the retailer's prot increases with theincorporation of capacity decision. However, when the capacityis larger than the optimal capacity, the retailer's prot decreases.

According to the above results, we believe that there does exista strong incentive for the supplier as the leader to integrate thecapacity as a decision in order to improve the prot. In particular,the supplier should avoid the overcapacity situation. Further, fromthe retailer's perspective, the larger the capacity is, the higher theretailer' prot is. Thus, the retailer only benets from the capacitydecision when the current capacity of the PL model is insufcientcompared with the optimal capacity of the CPL model. However,the retailer suffers from the prot loss if the current capacity of thePL model is higher than the optimal one of the CPL model. In short,when the capacity is underestimated compared with the optimalvalue, the retailer's interest is aligned with that of the supplier inorder to make the right capacity decision. However, when thecapacity is overestimated compared with the optimal value, theretailer is not willing to participate in the coordination. Thus, it isnecessary to design an appropriate incentive mechanism to makeboth parties pull in the same direction in order to ensure that thesupply chain deliver goods and services quickly and cost-effec-tively, which will be addressed in Section 6.

5.2. On channel performance

It is well-known that the channel performance in the DSC isalways deteriorated by double marginalization (e.g., Lee et al.,2000), which reects the incentive conict between the supplierand the retailer. To explore the impact of the incorporation ofcapacity decision on the channel performance, we rst comparethe performance of the DSC with that of a corresponding centra-lized supply chain (CSC). Second, we demonstrate the value of theincorporation of capacity decision for the reduction of doublemarginalization by numerical examples.

For the CPL model, let us consider the CSC with a singledecision maker and the same cost structure as that of the DSC.The wholesale price, as an internal transfer in the CSC, is notconsidered. For convenience, we use a subscript c or d to indicate acentralized system or a decentralized system whenever necessary,for example, prc and c are the retail price and PDT for the CSC.

Performing a similar linear transformation on the demandfunction, we have

prc pmaxr c=cwc; 17

Table 2Impact of the capacity decision on the prot gap.

s (%) r (%) s (%) r (%)

8 14.10 50.63 17 6.02 11.839 7.59 32.58 18 8.62 13.00

10 3.52 19.51 19 11.61 13.9111 1.19 10.01 20 14.96 14.6312 0.15 3.11 21 18.68 15.2213 0.07 1.92 22 22.80 15.6914 0.75 5.59 23 27.34 16.0915 2.03 8.30 24 32.34 16.4216 3.80 10.31 25 37.84 16.70

5 10 15 20 25

price

14

16

18

20

22

24

26

28pspr

Fig. 1. Impact of on pr and ps for the PL model.

S.X. Zhu / Int. J. Production Economics 164 (2015) 142318

and thus the system-wide expected prot rate is

cc;c; c pmaxr c=cwccrcsCc;Rc

cc0c:By performing the sequential optimization as Section 4.2, we can showthat there exists a unique solution to maximizing cc;c; c. Thesolution can be found by the following procedure: compute nc from

nc 1c02nc= 2 c00; 18

then, compute nc by

0n

c

ncnc 2c0 0; 19then compute pnrc by (17), and

nc by

nc 1

ncncln

hbhcw

;

nally the maximal prot rate is given by

nc nc 3nc=1c0:To examine the impact of decision inefciency on the rms'

decisions and performance, we compare the optimal decisions andthe corresponding prot rates in the DSC with those in the CSC in thefollowing theorem. Denote nd ns nr as the maximal prot rate ofthe DSC.

Theorem 2. The following inequalities hold:

(i) nc42n

d;(ii) nc4

n

d. In particular, if 1r2c0, nc42nd;(iii) Denote n=n. c4d;(iv) nc4

n

d and pnrcopnrd;

(v) nc4n

d and nc42

ns .

Theorem 2 shows that for a market with characteristics asdened here, the desirable strategy for a centralized supply chainis to use a lower price to capture a higher market share and alonger PDT to accommodate the higher demand rate while main-taining the same service level (the probability of on-time delivery).In a decentralized setting, independent players deviate from thisstrategy, resulting in a lower system-wide prot. Moreover, wend that if the decision inefciency can be eliminated, then themarket share can at least double in size. By choosing an appro-priate capacity decision, the supplier can improve the systemutilization and increase the prot, which strongly motivates thesupplier to promote the channel coordination.

Next, we use the following example to illustrate the impact ofthe integration of capacity decision on the channel performance interms of double marginalization. The base parameter setting isgiven by 0 16, 0:5, 0:2, cs 0:5, cr 0:2, h0.03, b0.3,and c01.5. We perform sensitivity analysis with respect to thepotential market size (0), price-sensitive factor (), time-sensitivefactor (). We denote as the prot rate gap between the DSC andthe CSC for the PL model and as that for the CPL model, i.e., nc

n

d=n

d; ncnd=nd:From Figs. 24, we observe that (i) is increasing in ; (ii) is

increasing in ; (iii) is robust with respect to the change of ; (iv) is

8 10 12 14 16 18 20 22 24 26 280.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Prof

it R

ate

Gap

=0.4=0.6=0.8

Fig. 2. Sensitivity analysis for .

8 10 12 14 16 18 20 22 24 26 280.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Prof

it R

ate

Gap

=0.05=0.2=0.35

Fig. 3. Sensitivity analysis for .

8 10 12 14 16 18 20 22 24 26 280.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Prof

it R

ate

Gap

0=12

0=16

0=20

Fig. 4. Sensitivity analysis for 0.

Table 3Prot rate gap of the CPL model

0

0.4 0.360 0.05 0.367 12 0.3630.6 0.378 0.2 0.369 16 0.3660.8 0.397 0.35 0.361 18 0.361

S.X. Zhu / Int. J. Production Economics 164 (2015) 1423 19

decreasing in 0. The prot rate gap of the CPL model is shown inTable 3 and the range of is from 0.36 to 0.4. Comparing with , wend that the integration of capacity decision can signicantly reducethe prot loss caused by double marginalization. In particular, whenthe market is very price sensitive ( 0:8), at 28 is 1.348 andabout three times as large as 0:397. The above nding indicatesthat in the DSC, before pursuing the channel coordination, the suppliershould integrate the capacity decision into the decision process withher supply chain partner as a rst step in order to gain more prot.

6. Coordination of the decentralized channel

The previous discussion shows that the integration of capacitydecision may reduce the negative impact of the double margin-alization on the channel prot. Table 3 shows that there may stillexist a prot rate gap about 0.4 between the DSC and the CSC.Therefore, it is necessary to nd an incentive mechanism toachieve channel coordination and a winwin outcome for bothparties. Further, because of nc42

ns , the supplier, as the leader in

the DSC, should have a strong incentive to design the incentivemechanism to gain more prot. Different coordination mechan-isms have been proposed in the supply chain literature for priceand inventory decisions. In the following, we study and comparethree well-known contracts: a two-part tariff, a revenue-sharingcontract, and a franchise contract.

Let us start with the two-part tariff that is a price policyconsisting of a xed payment for a good or service payment aswell as a per-unit charge (see Waldman, 2004). Under the two-part tariff, the supplier charges a wholesale price equal to cs, and axed fee, F, which serves to allocate the prot between thesupplier and the retailer. As pointed out by Zusman and Etgar(1981) and Moorthy (1987), the two-part tariff can achievecoordination by motivating the retailer to set the channel-protmaximizing price in a dyadic channel. However, we demonstratethat the two-part tariff cannot coordinate the DSC in our model.Since ps is set to cs, the supplier prot rate function can be writtenas s c0F . Because F is constant, the supplier will choosethe minimal capacity setting in order to make the least invest-ment. Thus, nc cannot be obtained. We nd that the reason is thatunder the assumption by the above authors, the retailer can fullycontrol the quantity sold by selecting the retail price. However, forour model, although the demand rate is directly controlled byboth the retail price and the PDT determined by the retailer,the demand rate is restricted by the supplier's capacity since thedemand rate cannot exceed the capacity available. In short, thetwo-part tariff works well when the supplier only needs to makethe wholesale-price decision. If the supplier also needs to makeother decisions, such as the capacity, the two-part tariff fails. Fromthe retailer's perspective, since the retailer's prot rate is increas-ing in and the supplier sets the capacity at the minimal level, thetwo-part tariff causes the decrease of the retailer's prot rate.Therefore, the retailer will reject the two-part tariff.

Next, we consider a revenue-sharing contract by Cachon andLariviere (2005). Under such a contract, the retailer pays a wholesaleprice per unit plus a percentage of the revenue , where ps is set ascscrcr , and is the retailer's share of revenue generated fromeach demand unit. Then, we can rewrite the supplier's prot ratefunction as

s pscsc010

0

;

where is a function of and is given by

02

crcs0

2 0:

Since s is concave in , the optimal capacity decision is given bythe following rst order condition:

pscsdd

c0102

0

2

!dd

1 0 2

0:

By simplication, the above equation is rewritten as

1 0 2

c0 0: 20

To achieve nc , we require (20) identical to (19). Hence, mustbe equal to 0, which means that no revenue is allocated to theretailer. Clearly, the retailer is worse off and will not accept thecontract. Some reader may propose a franchise contract combiningrevenue sharing with a two-part tariff. In other words, besides therevenue-sharing contract mentioned above, the supplier also paysa xed fee F to the retailer. However, under the franchise contract,we can show that the channel coordination is still impossible.Based on the analysis, 0. Then, r pscrF . Clearly, theretailer prefers the smallest value of by choosing the retail priceand PDT so that nc cannot be reached. As Cachon and Lariviere(2005) have pointed out that there exist several limitations for therevenue-sharing contract, we add one more. That is, even for thenon-competitive chain, when the supplier needs to make multipledecisions, the revenue-sharing contract may not coordinatethe DSC.

Finally, we consider one franchise contract that can coordinatethe channel and achieve a winwin outcome. Based on thefranchising arrangement described by Lal (1990), the supplier isthe franchisor and the retailer is the franchisee. For the franchisecontract, the supplier declares the wholesale price per unit, thepercentage of revenue, and a contingent rebate paid to the retailer.In short, the contract combines revenue sharing with a contingentrebate given by the theorem below.

Theorem 3. Under a franchise contract with , where A0;1,

ps 0

ln

cscrcr ;

and a contingent rebate ; given by

; 0

ln

; 21

we have

(i) The franchise contract achieves coordination;(ii) The resulting prots to the supplier and the retailer are

nsc 1ncc 0

nc

ncnc 2

" #

0n

c 2ncnc ncnc 2 ;

nrc ncnc 0

nc

ncnc 2

" #;

(iii) If Anr n2c =0ncnc= ncnc 2h i1; ncns n2c =

0ncnc= ncnc 21, both parties are better off and a win

win outcome is achieved.

The key insight from Theorem 3 is that the contingent rebatedepends on the revenue-sharing percentage, the retailer's costparameters, , and . Note that integrates the pricing and PDTdecisions at the retailer's side and represents the decision at thesupplier's side. Moreover, we can rewrite (21) as

0 ln

1:

S.X. Zhu / Int. J. Production Economics 164 (2015) 142320

We can show that is strictly decreasing in . In particular, foro1=2, 40, which means that to induce the retailer tochoose the rst-best solution, the supplier has to offer a sidepayment to the retailer when the utilization is below 50%. Thereason is that when the potential market size is relatively small(o50%), the retailer has to charge a lower price in the DSC,which causes the decrease of the prot. To motivate the retailer toparticipate in the coordination, the supplier has to compensate theretailer's loss. For 41=2, o0, which means that the supplierhas to be compensated for the capacity investment when theutilization is above 50%. In short, the contingent rebate suggests anallocation mechanism of the prot gain in order to coordinatethe DSC.

7. Conclusions and future research

This paper studies the integration of three critical decisions(capacity, price, and lead time) in the DSC with one supplier andone retailer. The dynamic decision process is modelled by theStackelberg game, where the supplier is the leader who deter-mines the capacity decision and the retailer is a follower whodetermines the price and lead time decision. We rst obtain theoptimal strategies for both the supplier and the retailer. Then, weexplore the impact of the incorporation of capacity decision on theplayers' optimal strategies and prots. Comparing with the bench-mark model without capacity decision, we nd that when thecapacity decision is underestimated in the benchmark model, theincorporation of capacity decision can yield a shorter PDT, a higherretail price, and a larger market share. As a result, the prots ofboth rms increase greatly. However, when the capacity decisionis overestimated, only the supplier's prot is increased while theretailer's prot is decreased. Further, we demonstrate that theincorporation of capacity decision can signicantly reduce the protloss caused by double marginalization. Finally, to achieve channelcoordination and a winwin outcome, we propose a franchisecontract with revenue-sharing and contingent rebate to align theplayers' interests so that the decision inefciency disappears.

There exist several extensions for the current model. First, weassume that the ORT distribution is given by the sojourn timedistribution of theM/M/1 queue. It is natural to extend to a generalORT distribution. However, due to the complexity of the ORTdistribution, the model may not be tractable. One possible way isto use the combination model by Liu et al. (2007) as an approx-imation. Second, it may be interesting to investigate the model inwhich the retailer acts a leader and the supplier acts as a follower.Can the incorporation of capacity decision still result in a sig-nicant benet? Third, if the information of system parameters isnot shared between the supplier and the retailer, how should thesupplier and retailer make their decisions under the scenario withasymmetric information? According to the knowledge of micro-economic theory, the mechanism of signaling and screening couldbe applied.

Acknowledgments

The author is grateful to the referees and the editor for theirconstructive suggestions that signicantly improved this study.The research was partly supported by Netherlands Organisationfor Scientic Research under Grant 040.02.009, The EuropeanRegional Development Fund for the ITRACT extension, and RoyalNetherlands Academy of Arts and Sciences under Grant 530-4CDI18.

Appendix

Proof of Lemma 1. Differentiating r;n with respect to twice, we have

d2r;nd2

20 3

:

Clearly, r is concave in since the second derivative isnegative.

Proof of Proposition 4.1. By Lemma 1, because of the concavity, is uniquely determined by the rst order condition, i.e., (10). Then,substituting into (7), we obtain n. By (1), (9) is derived.

Proof of Lemma 2. For a xed , by (12), we can show that s isconcave in . Thus, the supplier's best capacity decision is given bythe rst-order condition (13).

Proof of Lemma 3. Differentiating s with respect to , we have

ds; nd

04

0nnn 3

crcs;

04

c0n

crcs; 22

where the second equality is obtained by substituting (13) into therst equality.

Denote f 04=c0n=crcs. Differentiatingf twice with respect to , we have

df d

4c0

n2222n2;

df 2d2

3c0n44n32n224n334

43 n2 3:

Since n4, df 2=d2o0, which indicates that f is concave in .By (22), we have o1c0=4. If Z1c0=4, then the

optimal demand rate becomes zero, which makes the problemtrivial and uninteresting. Thus, we have 0oo1c0=4. More-over, because of the concavity of f , lim-0 f 040, andlim-1 c0=4 f

0o0 (note that when -1c0=4), weobtain that s; is unimodal in .

Proof of Theorem 1. By Lemma 3, there exists a unique globaloptimal n that satises the following rst order condition:

04n

0nnnnnnnn 3 crcs 0: 23

Substituting (13) into (23), we have 04n=c0nn=ncrcs 0. We can rewrite it as

nn c0

04n

crcs

: 24

Substituting (24) into (23), we obtain (15).By Lemma 2, n is uniquely determined by n. By (7), since n is

increasing in , n is uniquely determined by n. Furthermore, by(11), ps is decreasing in , hence the optimal pns is uniquelydetermined by n. Finally, by (9), pnr is also uniquely determinedby n. It is obvious that changing the solution pnr ;n; pns ; n by anyrm unilaterally will reduce the prot rate of that rm. Thus, it isindeed the unique equilibrium of the game.

Proof of Corollary 1. By Lemma 3, we have ou to make surethat n40.

S.X. Zhu / Int. J. Production Economics 164 (2015) 1423 21

To show that lon, let us consider s. To participate the game,it is clear that ns40, which requires

2n

0

n

nn 24032n

1c04

n=14n=1c04n=

40;

where the second inequality is obtained by substituting

n n

c014n=;

nn n

c01c04n=:

By the above second inequality, lon.

Proof of Corollary 2. By comparing ns and nr , we have

ns nr 21 0

nn 2n

n1

" #

1 0

nn 21c4n=

c

1c0

1c4n=2 1c4n= 2

11c4

n=

n:

Proof of Proposition 5.1. Differentiating (16), (7), and nr withrespect to , we obtain

dn

d 0

2n24n4 n 4

042n;

dn

d 0

1dn=dn 2 ;

dnrd

2n 1 0

n 3

264

375dnd

20n2

n 3:

It is clear that dn=d40. Substituting dn=d into the expressionsof dn=d and d nr =d, we have d

n=do0, and dnr =d40.

Proof of Theorem 2. For (i), by (15) and (18), the relationshipbetween nc and

n

d must be either n

c42n

d or n

cond. Now, we showthat ncond is impossible. The logic is as follows. To guarantee thatnc40, it is clear that

n

c4 1c0=3. If ncond, by Corollary 1,we have nco1c0=4, which makes nco0. Thus, nc42nd.

For (ii), we prove it by contradiction. We have

nd ndc0

04nd

crcs

; 25

nc ncc0

02nc

crcs

: 26

By (25), ndr21=16c0. Suppose that ncond. Then,nco21=16c0 for all the possible values of nc . Denote f x x12x==c0. We can show that f 1c0=3421= 16c0. Toguarantee a positive nc , we require

n

c41c0=3. Because ofthe continuity of f(x), we can always nd a nc so thatnc4

21=16c0, which contradicts with the assumption. Thus,

nc4n

d. Further, if 1r2c0, by (25) and (26), we can write nd andnc as a function of

n

d and nc , respectively. That is

nc 1

2218c0nc

q4

;

nd 1

22116c0nd

q8

:

Since nc42n

d, it is obvious that 2n

donc .For (iii), based on the rst order conditions, we have

14nd=c0nd=nd 0;12nc=c0nc=nc 0:Since nc42

n

d, it is clear that nc=

n

cond=nd. By denition of , wehave c4d.

For (iv), we have

ncnc ncc01c02nc=

01c02nc=

o 01c04nd=

o01c04n

d=1c04nd= 2 ndnd:

By (7), since ndnd4ncnc , nc4nd. By (1), since nc42nd andnc4

n

d, pnrcopnrd.

For (v), it is clear that nc4n

d since the CSC simultaneouslyoptimizes all the decisions. By comparing nc with

ns , we have

nc n2c 0

n

c

ncnc;

ns 2n2d

0n

dn

d

ndnd 2;

nc2ns n2c 4n2d

20

n

dn

d

ndnd 2 0ncncnc ;4

20ndn

d

ndnd 2 0ncncnc ;

2c0n

dndndndnd

c0ncnc :

Since ndnd4ncnc and nd4nd, we have nc2ns40.

Proof of Theorem 3. For (i), to achieve the coordination, the keyis how to induce the retailer to make the rst-best decision. underthis franchise contract, the retailer's prot rate function is given by

rpr ; prC;Rpscr

pr ;; :By Proposition 4.1, the optimal demand rate is given by

02

0

2

" #pscr

;

0:

To make equal to nc , we require that ps ; =cscrcr .Substituting ps into the supplier's prot rate function, we have

s ;

cscrcscr

1prC;Rc0; :

To make equal to nc , we nd the expression of ; given by (21).For (ii), it is straightforward to show the results by substituting

the optimal solutions into the prot rate functions.For (iii), to achieve a winwin outcome, we require that nsc4

ns

and nrc4nr . From (ii), the results immediately follow.

References

Boyaci, T., Ray, S., 2003. Production differentiation and capacity cost interaction intime and price sensitive demand. Manuf. Serv. Oper. Manag. 5, 1836.

Boyaci, T., Ray, S., 2006. The impact of capacity costs on product differentiation indelivery time, delivery reliability, and price. Prod. Oper. Manag. 15, 179197.

Braham, J., 1987. The marriage of marketing and manufacturing. Ind. Week 233,4144.

Cachon, G.P., 2003. Supply chain coordination with contracts. In: de Kok, A.G.,Graves, S. (Eds.), Supply Chain Management: Design, Coordination and Opera-tion. Elsevier, Amsterdam, The Netherlands.

Cachon, P., Lariviere, M., 2005. Supply chain coordination with revenue-sharingcontracts: strengths and limitations. Manag. Sci. 51, 3044.

Celik, S., Malglaras, C., 2008. Dynamic pricing and lead-time quotation for a multi-class make-to-order queue. Manag. Sci. 54, 11321146.

Crittenden, V.L., 1992. Close the marketing/manufacturing gap. Sloan Manag. Rev.33, 4152.

S.X. Zhu / Int. J. Production Economics 164 (2015) 142322

Duenyas, I., Hopp, W.J., 1995. Quoting customer lead times. Manag. Sci. 41, 4357.Feng, J., Liu, L., Liu, X., 2011. An optimal policy for joint dynamic price and lead-time

quotation. Oper. Res. 59, 15231527.Hu, Y., Guan, Y., Liu, T., 2011. Lead-time hedging and coordination between

manufacturing and sales departments using Nash and Stackelberg games.Eur. J. Oper. Res. 210, 231240.

Lal, R., 1990. Improving channel coordination through franchising. Mark. Sci. 9,299318.

Lee, H., 2004. The triple-A supply chain. Harv. Bus. Rev. 82, 103112.Lee, H., Padmanabhan, V., Taylor, A., Whang, S., 2000. Price protection in the

personal computer industry. Manag. Sci. 46, 467482.Leng, M., Parlar, M., 2009. Lead-time reduction in a two-level supply chain: non-

cooperative equilibria vs. coordination with a prot-sharing contract. Int. J.Prod. Econ. 118, 521544.

Liu, L., Parlar, M., Zhu, S., 2007. Pricing and lead time decisions in decentralizedsupply chains. Manag. Sci. 53, 713725.

Mendelson, H., Whang, S., 1990. Optimal incentive compatible priority pricing forthe M/M/1 queue. Oper. Res. 38, 870883.

Moorthy, K., 1987. Managing channel prots: comment. Market. Sci. 6, 375379.Narayanan, V.G., Raman, A., 2004. Aligning incentive in supply chains. Harv. Bus.

Rev. 82, 94102.O'Leary-Kelly, S.W., Flores, B.E., 2002. The integration of manufacturing and

marketing/sales decisions: impact on organizational performance. J. Oper.Manag. 20, 221240.

Palaka, K., Erlebacher, S., Kropp, D.H., 1998. Lead time setting, capacity utilization,and pricing decisions under lead time dependent demand. IIE Trans. 30,151163.

Pasternack, B.A., 1985. Optimal pricing and return policies for perishable commod-ities. Market. Sci. 4, 166176.

Pekgn, P., Grifn, P.M., Keskinocak, P., 2008. Coordination of marketing andproduction for price and leadtime decisions. IIE Trans. 40, 1230.

Plambeck, E.L., 2004. Optimal leadtime differentiation via diffusion approximations.Oper. Res. 52, 213228.

Ray, S., Jewkes, E.M., 2004. Customer leadtime management when both demandand price are leadtime sensitive. Eur. J. Oper. Res. 153, 769781.

Ray, S., Li, S., Song, Y., 2005. Tailored supply chain decision making underpricesensitive stochastic demand and delivery uncertainty. Manag. Sci. 51,18731891.

Shapiro, B.P., 1977. Can marketing and manufacturing coexist? Harv. Bus. Rev. 55,104114.

So, K.C., Song, J., 1998. Price, delivery time guarantees and capacity selection. Eur. J.Oper. Res. 111, 2849.

So, K.C., 2000. Price and time competition for service delivery. Manuf. Serv. Oper.Manag. 2, 392409.

Shang, W., Liu, L., 2011. Promised delivery time and capacity games in timecompetition. Manag. Sci. 57, 599610.

Stidham, S., 1992. Pricing and capacity decisions for a service facility: stability andmultiple local optima. Manag. Sci. 38, 11211139.

Taylor, T.A., 2002. Supply chain coordination under channel rebates with saleseffort effects. Manag. Sci. 48, 9921007.

Tsay, A.A., 1999. The quantity exibility contract and suppliercustomer incentives.Manag. Sci. 45, 13391358.

Waldman, D.E. 2004. Microeconomics. Pearson Addison Wesley, Boston.Wang, Y., Gerchak, Y., 2003. Capacity games in assembly systems under uncertain

demand. Manuf. Serv. Oper. Manag. 5, 252267.Wang, Y., Jiang, L., Shen, Z., 2004. Channel performance under consignment

contract with revenue sharing. Manag. Sci. 50, 3447.Xiao, T., Qi, X., 2012. A two-stage supply chain with demand sensitive to price,

delivery time, and reliability of delivery. Ann. Oper. Res. http://dx.doi.org/10.1007/s10479-012-1085-6.

Zusman, P., Etgar, M., 1981. The marketing channel as an equilibrium set ofcontracts. Manag. Sci. 27, 284290.

S.X. Zhu / Int. J. Production Economics 164 (2015) 1423 23

Integration of capacity, pricing, and lead-time decisions in a decentralized supply chainIntroductionLiterature reviewDecision modelsStackelberg equilibriumRetailer's best responseSupplier's optimal strategy

Impact of capacity decisionOn players' profitOn channel performance

Coordination of the decentralized channelConclusions and future researchAcknowledgmentsAppendixReferences