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Coordination by Option Contracts in a Retailer-Led Supply Chain with Demand Update
TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 22/22 pp570-5 80 Volume 13, Number 4, August 2008
Coordination by Option Contracts in a Retailer-Led Supply Chain with Demand Update* WANG Xiaolong (王小龙), LIU Liwen (刘丽文)** School of Economics and Management, Tsinghua University, Beijing 100084, China Abstract: This research examines how to use an option contract to coordinate a retailer-led supply chain where the market information can be updated. Based on Stackelberg game theory, we build a mode with one supplier and one retailer in which the retailer designs contracts to coordinate the supplier’s production in a two-mode production environment. This focuses on an option contract that consists of two option prices and one exercise price. By theoretical analysis and numerical example, we find that such a contract can coordinate the supplier and retailer to act in the best interest of the channel. The optimal pricing conditions are given as follows: First, option prices should be negatively correlated to the exercise price and should be in a relevant range. Second, the first-period option price should be no greater than the second-period price and should be linearly correlated to the second-period option price when the latter is beyond some threshold. The results show that such option contracts can arbitrarily allocate the extra system profit between the two parties so that each party is in a win-win situation. Key words: supply chain coordination; option contracts; retailer-led; demand update
Introduction Contemporary business puts more and more emphasis on channels, shifting power from supplier to retailer at the downstream end of a supply chain, which is the closest to the customers. Messinger and Narasimhan[1], Ailawadi[2], and Bloom and Perry[3] offer empirical evidence on how power has transferred in several international retailing markets. Such shifting enables the retailers to lead the supply chain in some sense. They then begin to raise stringent requirements on the suppliers’ production, compared with the traditional case where the supplier dominates the supply chain and coordinates the retailer’s ordering behavior. Another Received: 2006-10-24
* Supported by the National Natural Science Foundation of China (Nos. 70532004 and 70621061)
** To whom correspondence should be addressed. E-mail: [email protected] Tel: 86-10-62783553; Fax: 86-10-62785876
aspect is that the market situation often changes greatly long after the retailer has set an order. Some retailers, like Wal-Mart, deal with such problems by ordering less but more frequently. However, this volatile market puts great pressure on the suppliers who must make wise production/capacity building decisions to cope with the irregular orders which mirror a volatile market. Unfortunately, quite often suppliers fail to do that. In an effort to reduce mismatches between supply and demand, many industries, like the fashion industry (as reported by Fisher et al.[4]) and the consumer electronics industry, are moving to tighten coordination across their supply chains. One important initiative for suppliers is to develop faster, but typically more expensive, production modes that are capable of producing a second run of products closer to the selling season. This mode allows both sides to take advantage of updated demand forecasts. Also, the retailer can better manipulate the supplier’s production so that channel coordination is reached. This phenomenon has become more
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
common with advances in computer-based manufacturing technologies which significantly reduce production run times and product changeover times. However, the addition of another production mode is not enough to reach channel coordination due to double marginalization which refers to the fact that a party’s relative cost structure becomes distorted when a transfer price is introduced within a supply channel. For example, introducing a wholesale price into a single-stage newsvendor model causes the ratio of the buyer’s underage and overage costs to fall short of the channel ratio. As a result, the buyer orders less than the channel optimal quantity. In a two-stage production environment, double marginalization threatens to impact not only the total production quantity but also proper allocation of production between periods. The double marginalization concept was first introduced in the economics literature by Spengler[5], who raised the issue in terms of inappropriate prices rather than inappropriate quantities. Tirole[6] provided further details on the phenomenon. Many works in the operations management field, like those of Lariviere and Porteus[7], have shown that price-only contracts generally fail to coordinate the supply chain due to double marginalization. However, other contracts have proven to efficiently coordinate the supply chain. These contracts include buyback contracts[8], revenue sharing contracts[9], sales rebate contracts[10], quantity discount contracts[11], and quantity flexibility contracts[12]. The goal of this paper is to provide guidance in designing an option contract in a retailer-led supply chain, in which the supplier’s production needs coordinate with the existence of a market demand information update. The researches on option contracts in supply chain management are mainly divided into economic and operational areas. Typical economic works include Newbery[13] and Wu et al.[14] Typical operational works include Donohue[15] and Barnes-Schuster et al.[16] Our contract pricing scheme builds most closely upon the work of Barnes-Schuster et al.[16], which considers a combination of committed orders and options. Barnes-Schuster et al.[16] analyze a two-period model with correlated demand. The present analysis studies a two-period model with demand information update in which the market demand does not realize until the end of the second period. Specifically, consider one supplier and one retailer.
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At the beginning of the first period both parties anticipate a demand information update that will appear at the beginning of the second period and they take this into account in their first-period decisions. The coordinating contract is designed as follows: The option contract has three parameters for two option prices o1 and o2 and one exercise price e. The option prices serve as the unit compensation for the supplier’s production quantities. When the realized demand is beyond the retailer’s total order, the retailer will purchase the supplier’s excess inventory, if any, to meet the market demand at the unit price e. With this contract, the retailer’s decision is to determine orders at the beginning of each period and the supplier’s decision is to determine the production allocation between the two periods. This project aims to find an optimal set of contracts to coordinate the channel and to realize a proper profit allocation.
1
Two-Period Model
The two-period model depicts the dynamics of a retailer who purchases from one main supplier and markets these products to a number of retail outlets, reselling at retail price p for a single selling season. Unlike traditional coordination models, the retailer in the model leads the supply chain and designs a proper contract to coordinate the supplier’s production quantities. The specific contract form will be investigated. The retailer utilizes two periods of demand information to determine the order quantities. During the first period, the demand prediction is relatively less accurate with a distribution function denoted as F(D). At the beginning of the second period (market demand not yet realized), more current market information is available which helps the retailer fine-tune the forecast where an effective market signal will be observed depending on specific industry situations and the lengths of the two periods which may not necessarily be equal. This updated information is the market signal ε. Let G(·) denote its estimated cumulative distribution function and let F(·|ε) denote the demand distribution function conditional on the market signal. Assume that ε is expressed in the same units as the total demand and is also non-negative. Market demand is stochastically increasing in the market signal, i.e., given x, F ( x | ε h )< F ( x | ε l ) for all ε h > ε l (ε h and ε l are two arbitrary
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market signals). Assume that all distributions are continuous, invertible, doubly differentiable, and independent of the wholesale and retail prices. Furthermore, they are all common knowledge to both parties from the beginning of period 1. At the beginning of period 1, the retailer orders d1 units at a unit wholesale price w1 based on a rough knowledge of the demand distribution. Once ε is observed, the retailer will adjust the order quantities by ordering another d 2 units at a higher unit wholesale
Tsinghua Science and Technology, August 2008, 13(4): 570-580
q = q1 + q2 . 1.1
To provide a benchmark, first consider the problem where the supplier and the retailer are each owned by a risk-neutral entity. The owner aims to maximize his own expected profit by choosing first and second period production quantities (q1 , q2 ) which solve the following two-period optimization problem. ∞
max Π c (q1 ) = −c1q1 + ∫ Π c (ε , q1 , q2 )dG (ε )
price w2 . Note that the retailer’s second order is contingent on the specific market signal. If the signal exhibits a rather weak demand in the future, the retailer will not set another order, i.e., d 2 = 0 . The supplier produces over two time periods in response to the retailer’s orders, with orders filled in one shipment before the selling season begins. The supplier has two different production modes. One is the cheap mode which incurs a unit production cost c1 , and the other is the expensive mode which incurs a unit production cost c2 > c1 . Assume that production costs include the cost of holding inventory until delivery and the cost of delivery itself. Other cost parameters include a shortage penalty s paid by the retailer to the customers for each unit of unfilled demand, and a salvage value v received for each unit of inventory remaining at the end of the season no matter to which side it belongs. Suppose that all these prices and costs are exogenous and the analysis is limited to the most relevant and interesting case where p > w2 > w1 > c2 > c1 > v. Next assume that the lead time for the cheap
mode (and the timing of the market signal) is such that, to ensure completion before the start of the selling season, the supplier must begin production before observing the market signal. In contrast, the lead time for the expensive mode is short enough to allow production after the market signal is observed. Therefore, the supplier’s problem is to allocate production quantities between the two modes. During the first period, the supplier needs to determine how many products, denoted as q1 , would be produced among the cheap mode. In the second period after a market signal is observed, the supplier has to determine how many products, denoted as q2 , will be produced among the expensive mode, if necessary. Denote d = d1 + d 2 and
Centralized system performance: A benchmark
q1 .0
0
(1)
where Π c (ε , q1 , q2 ) = max π c (ε , q1 , q2 ) and q2 .0
π (ε , q1 , q2 ) = −c2 q2 + pE min{D, q1 + q2 } + v(q1 + q2 − E min{D, q1 + q2 }) − sE[ D − min{D, q1 + q2 }] (2) c
Here, Π c (ε , q1 , q2 ) represents the second-stage problem after the market signal is observed and the demand forecast is updated accordingly. To express the second-stage problem independently of q1 , substitute q for q1 + q2 and rearrange the terms. This new formulation yields the same objective solution as the problem in Eq. (1). ∞
max Π c (q1 ) = (c2 − c1 )q1 + ∫ Π c (ε , q1 )dG (ε ) q1 .0
0
(3)
where Π c (ε , q1 ) = max π c (ε , q) and q.q1
q
π (ε , q ) = ( p − c2 + s )q − ( p − v + s) ∫ F ( x | ε )dx − sED c
0
(4) Note that given q1 , the owner faces a newsvendor problem in period 2 but with an initial inventory and an adjusted demand distribution. Let (q1c , q c ) denote the optimal solution to the central problem. The solution structure is given by Lemma 1. Lemma 1 The argument of the central first period problem, Π c (q1 ) , is concave in the initial production quantity q1 . The optimal total production quantity is ⎧ q1c , if ε - ε (q1c ); ⎪ q c = ⎨ −1 ⎛ p − c2 + s ⎞ ⎪ F ⎜ p − v + s ε ⎟ , otherwise, ⎝ ⎠ ⎩
p − c2 + s ⎫ ⎧ where ε (q1c ) = sup ⎨ε : F (q1c | ε ) . ⎬ and the p−v+s ⎭ ⎩ first-period optimal production quantity is implied by
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s) F (q1c | ε )]dG (ε ) = 0 .
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Proof First, prove the concavity of Π c (q1 ). This
only needs to prove that Π (ε , q1 ) = max π (ε , q ) is c
c
q.q1
concave in q1 . Note that ⎧⎪π c (ε , q* ), if q1 - q* ; Π (ε , q1 ) = ⎨ c ⎪⎩π (ε , q1 ), otherwise, * where q maximizes π c (ε , q ). Now π c (ε , q ) is obvic
∂ π (ε , q) = − ( p − v + s) ⋅ ∂q 2 2
ously concave in q because
c
based on the orders. Let di denote the retailer’s order in period i. The problem is to choose d1 and d 2 = d − d1 to maximize the expected profit. The problem structure is analogous to that in the centralized system. ∞
max Π R (d1 ) = − w1d1 + ∫ Π R (ε , d1 , d 2 ) dG (ε ), d1 .0
0
where Π R (ε , d1 , d 2 ) = max π R (ε , d1 , d 2 ) and d 2 .0
π R (ε , d1 , d 2 ) = − w2 d 2 + pE min{D, d1 + d 2 } + v(d1 + d 2 − E min{D, d1 + d 2 }) − sE[ D − min{D, d1 + d 2 }].
f (q | ε ) - 0. Therefore, Π (q1 ) is also concave in q1 . c
The optimal total production quantity follows immediately from the fact that the second-period problem is a simple newsvendor problem with initial inventory q1. Intuitively this quantity is positively related to the market signal. When the signal is not strong enough, the central planner will not produce a second sum. This idea is exhibited in Lemma 1 with a threshold value for ⎧ p − c2 + s ⎫ c the market signal: ε (q1 ) = sup ⎨ε : F (q1c | ε ). ⎬. p−v+s ⎭ ⎩ This is the strongest signal that will still result in no second-period production. With this notation, suppose that the central planner’s expected profit is maximized at q1c , and then rewrite the problem as
Π c (q1c ) = (c2 − c1 )q1c + ∫ ∞
∫ε
( q1c )
ε ( q1c ) 0
Rearranging, ∞
max Π R (d1 ) = ( w2 − w1 )d1 + ∫ Π R (ε , d1 ) dG (ε ) d1 .0
d .d1
d
π R (ε , d ) = ( p − w2 + s )d − ( p − v + s ) ∫ F ( x | ε )dx − sED. 0
Let (d1* , d * ) denote the optimal solution to the retailer’s problem. The solution structure is given in Lemma 2. Lemma 2 The argument of the retailer’s first period problem, Π R (d1 ) , is concave in the initial order quantity d1 . The optimal total order is ⎧d1* , if ε - ε (d1* ); ⎪ d * = ⎨ −1 ⎛ p − w2 + s ⎞ ⎪ F ⎜ p − v + s ε ⎟ , otherwise, ⎝ ⎠ ⎩
π c (ε , q1c )dG (ε ) +
π c (ε , q c )dG (ε ),
0
where Π R (ε , d1 ) = max π R (ε , d ) and
where ⎧
p − w2 + s ⎫ ⎬ and the firstp−v+s ⎭
⎛ p − c2 + s ⎞ ε ⎟ . Because Π c (q1 ) is where q c = F −1 ⎜ p − v + s ⎝ ⎠ concave in q1 , the first order condition works, i.e., an
⎩ period optimal order quantity is implied by
optimal first-period production quantity is implied by
w2 − w1 + ∫
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s ) F (q1c | ε )]dG (ε ) = 0 .
ε (d1* ) = sup ⎨ε : F (d1* | ε ) . ε ( d1* ) 0
[ p − w2 + s − ( p − v + s ) F (d1* | ε )]dG (ε ) = 0.
The proof is quite similar to that of Lemma 1. Here, ε (d1* ) also partitions the market signals into two sets.
□ Lemma 1 introduces the concept of ε (q1c ) which
If ε - ε (d1* ) , then the retailer will not set a second
partitions the market signals into two sets. If ε - ε (q1c ) , then the second-period production is not
order. Note the differences from the notations defined in Section 1.1. Furthermore, for a given x , ε ( x) < ε ( x),
necessarily positive. Otherwise, it is exactly positive.
which coincides with the phenomenon that double marginalization makes the buyer conservative. Compared with the central planner, given the same inventory level, the buyer in the decentralized system needs a stronger market signal, which induces a more optimistic demand estimate, to build up to that level. The supplier’s problem is to choose production
1.2
Decentralized system performance without coordination
Here model the decentralized system as a Stackelberg game. The retailer sets orders in each period as mentioned above. The supplier in turn arranges production
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quantities (q1 , q2 ) that maximize his own expected profit, subject to the retailer’s ordering behavior. His decision on the second-stage production is relatively straightforward, i.e., q2* = max{d * − q1* ,0} .
to the supplier to overproduce in each period. Recall that q1 . d1 and q2 . max{d − q1 ,0} . Thus d1 and max{d − q1 ,0} are the minimum production number
The difficulty lies in his decision on q1 . Obviously
for the supplier’s production in each period. The coordinating contract is defined as follows: The option contract has three parameters of two option prices o1 and
he should at least produce d1 , but should he produce
o2 and one exercise price e . The option prices are
more? The answer depends on his estimate on the likelihood of the retailer’s second order. For such a decision his problem is
the unit compensation for the supplier’s production quantities beyond the minimums in each period. When the realized demand exceeds the retailer’s total order, he will purchase the supplier’s excess inventory, if any, to meet the market demand at a unit price e . The sequence of events is summarized as follows: (1) At the beginning of period 1, the retailer sets a firm order d1 . The retailer also announces an option
max ΠS (q1 ) = w1d1 − c1q1 + ∫ q1.d1
ε ( q1 )
∫ε
( d1 ) ∞
∫ε
( q1 )
ε ( d1 ) 0
v(q1 − d1 ) dG (ε ) +
[ w2 d 2 + v(q1 − d )]dG (ε ) +
[ w2 d 2 − c2 (d − q1 )]dG (ε ).
The solution to this problem is given by Lemma 3. Lemma 3 Given the retailer’s ordering behavior, the supplier will maximize his expected profit by setting q1* = max{d1* , q1′} and q2* = max{d * − q1* ,0}, c −c where q1′ is implied by G (ε (q1′)) = 2 1 . c2 − v Because this problem has a newsvendor structure, the first order condition works. The proof is straightforward so the details are omitted. These problems have reviewed the benchmark and decentralized uncoordinated system performance. Although the precise relationships between d1* , q1*, and q1c depending on factors like the ratio of the two production costs, the ratio of the two wholesale prices, and the estimate of the market signal cannot be determined, the channel is certainly not coordinated. This is implied by Lemma 2 which states that for a given market signal the retailer’s total order quantity is less than the optimal total inventory required in a centralized system. The next section focuses on how an option mechanism coordinates the supply chain and realizes profit allocation.
2
Channel Coordination
To coordinate the channel, the retailer must choose the proper contract parameters so that the supplier’s production quantities in each period are equal to the optimal solutions in the centralized system, i.e., q1* = q1c and q* = q c . To do so the retailer must give incentives
contract with option prices o1 and o2 and exercise price e . The supplier, in turn, decides to produce q1 . d1 of goods in the cheap mode which will produce an income of o1 (q1 − d1 ) . (2) At the beginning of period 2, after the market signal is observed, the retailer orders d 2 . The supplier selects a second-period production, q2 . max{d − q1 ,0}, that involves the expensive mode and produces an income of o2 (q2 − max{d − q1 ,0}) . (3) At the beginning of the selling season, after the demand is realized, the retailer purchases additionally m = min{max{D − d ,0}, q − d } number of goods at unit price e . The supplier delivers all that the retailer wants. The problem assumes that the supplier is obligated to fill the retailer’s entire order and can neither sell directly to the final customer (i.e., the retailer’s customer) nor supply inventory in excess of the retailer’s total order quantity. With the option contract, the supplier’s problem becomes max ΠS (q1 ) = − (c1 − o1 )q1 + ( w1 − o1 )d1 + q1 . d1
∫ where ΠS (ε , q1 , q2 ) =
∞
0
ΠS (ε , q1 , q2 ) dG (ε ), max
q2 .max{d − q1 ,0}
π S (ε , q1 , q2 ) and
π S (ε , q1 , q2 ) = −c2 q2 + o2 (q2 − max{d − q1 ,0}) + w2 d 2 + ∫ v(q1 + q2 − d ) dF ( x | ε ) + d
0
∫
q1 + q2 d
[e( x − d ) + v(q − x)]dF ( x | ε ) +
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∫
∞
e(q1 + q2 − d )dF ( x | ε ) .
q1 + q2
Here, ΠS (ε , q1 , q2 ) represents his second-stage problem. After making the supplier’s second-stage problem independent of q1 and after some algebra, the sup-
∞
max Π R (d1 ) = −o1 (q1 − d1 ) − w1d1 + ∫ Π R (ε , d1 ) dG (ε ), d1.0
0
where Π R (ε , d1 ) = max π R (ε , d ) and d .d1
π R (ε , d ) = − w2 d 2 − o2 (q2 − max{d − q1 ,0}) + pE min{D, q} + v(d − E min{D, d }) − eE min{max{D − d ,0}, q − d } −
plier’s problem is max ΠS (q1 ) = (c2 − c1 − o2 + o1 )q1 + ( w1 − o1 )d1 + q1.d1
∫
∞ 0
ΠS (ε , q1 ) dG (ε ),
where ΠS (ε , q1 ) = max π S (ε , q ) and q.max{d ,q1 }
s ( ED − E min{D, q}) . Rearranging, max Π R (d1 ) = (o2 − o1 )q1 + ( w2 − w1 + o1 )d1 + d1.0
π S (ε , q) = (e − c2 + o2 )q + w2 d 2 − ed − o2 max{d − q1 ,0} − q
( e − v ) ∫ F ( x | ε ) dx . d
N
N 1
Let (q , q ) denote the solution to this problem, and then we have Proposition 1. Proposition 1 Given an option contract specified by the retailer, the supplier’s optimal total production quantity is ⎧ ⎛ e − c2 + o2 ⎞ ⎫ q N = max ⎨q1N , F −1 ⎜ ε ⎟⎬ . ⎝ e−v ⎠⎭ ⎩ The proof is relatively straight forward since the supplier’s problem follows a newsvendor structure. Proposition 1 shows clearly that to coordinate q N = e − c2 + o2 p − c2 + s q c , it is necessary to set = , i.e., e−v p−v+s p−v+s (5) e= p+s− o2 c2 − v Based on Eq. (5), e > v and e + o2 > c2 to make ⎛ e − c2 + o2 ⎞ | ε ⎟ meaningful. Economically these two F −1 ⎜ ⎝ e−v ⎠ conditions prevent the supplier from suffering losses. e + o2 is all the revenue that the supplier can get from
one unit of product if it can be sold after the market demand is realized. If e + o2 < c2 , the supplier has no incentive to overproduce in the second period, and thus channel coordination is never reached. Since e = p + s − p−v+s o2 , the two conditions are equivalent with c2 − v
o2 < (c2 − v) . Therefore, o2 < (c2 − v) from now on. Next, consider the retailer’s problem. In all the folp−v+s o2 holds lowing parts, assume that e = p + s − c2 − v except where specified. The retailer’s problem is
∫
∞
0
Π R (ε , d1 ) dG (ε ),
where Π R (ε , d1 ) = max π R (ε , d ) and d .d1
d
π R (ε , d ) = (e − w2 )d − (e − v) ∫ F ( x | ε ) dx + 0
o2 max{d − q1 , 0} + ( p + s − e − o2 )q − q
( p + s − e) ∫ F ( x | ε ) dx − sED . 0
Let ( d1N , d N ) denote the optimal solution to the retailer’s problem. Proposition 2 describes the retailer’s ordering behavior. e − w2 e − w2 + o2 Proposition 2 Denote α = , β= , e−v e−v and γ (ε ) = max{d1N , F −1 (max{α ,0}| ε )}. Define εˆ (d1N) = sup{ε : F (d1N | ε ) . max{α ,0}} and allow εˆ (d1N ) = +∞
if ε max is infinite. For a given option contract, the optimal ordering behavior for the retailer has the following structure: (1) After the market signal ε has appeared, the optimal total order quantity is determined as follows: (i) If F (q1 | ε ) . β , d N = γ (ε ) ;
(ii) Otherwise, ⎧γ (ε ), if π R (ε , F −1 (max{α ,0}| ε )) > ⎪ dN = ⎨ π R (ε , F −1 ( β | ε )); ⎪ −1 ⎩ F ( β | ε ), otherwise. (2) The optimal first-period order quantity is given by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0
when e + o1 > w1 . Otherwise, d1N ≡ 0 . Corollary For some second-period option price(s) o2 such that β > 0 and set A = {ε : π R (ε , F −1 (max{α , 0}| ε )) - π R (ε , F −1 ( β | ε ))} is not empty, there exists a
Tsinghua Science and Technology, August 2008, 13(4): 570-580
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signal ε T (o2 ) such that d N = F −1 ( β | ε ) > q1 if and only if ε . ε T (o2 ) . Otherwise, d
N
will never be
greater than q1 and ε T (o2 ) = ε max for such cases. Proof First, prove part (1). Omitting all the irrelevant parts, the retailer’s second-period problem can be rewritten as ⎧(e − w )d − (e − v) d F ( x | ε )dx, 2 ∫0 ⎪ ⎪ if d - q1 ; ⎪ max π R (ε , d ) = ⎨ d d .d1 ⎪(e − w2 + o2 )d − (e − v) ∫ F ( x | ε )dx, 0 ⎪ ⎪⎩ otherwise. Then, the unconditional optimal d N may not be unique but can be either F −1 (max{α ,0}| ε ) or F −1 (max{β , 0}| ε ) or both. However, if d N = F −1 ( β | ε ),
there must be F −1 ( β | ε ) > q1 , i.e., 0 < F (q1 | ε ) < β , so that max{d − q1 ,0} = d − q1 makes sense. This completes the proof of part (i) when F (q1 | ε ) . β and d N = γ (ε ) .
Now consider 0 < F (q1 | ε ) < β . The criterion is simple but clear. Whether d N = γ (ε ) or d N = F −1 ( β | ε ) depends on which can bring more expected profit, i.e., ⎧γ (ε ), if π R (ε , F −1 (max{α ,0}| ε )) > ⎪ dN = ⎨ π R (ε , F −1 ( β | ε )); ⎪ −1 ⎩ F ( β | ε ), otherwise. This completes the proof of part (1) in Proposition 2. Next, show the link between the retailer’s second-period expected profit and the market signal to estimate the optimal total order from the beginning of period 1 based on the market signal conditions, which is the core idea of the corollary. When β > 0, let π R (ε , F −1 (max{α , 0}| ε )) = π R (ε , F −1 ( β | ε )) . Note that d N ≡ d1N when β - 0 because
β . α . Suppose that there exists a solution ε T (o2 ) , and then for ε T (o2 ) the retailer is indifferent towards
Next consider part (2) which depicts the first-period ordering behavior. If o2=0, then the retailer’s second-period problem is given by the following: For any α , similar to the reasoning in proof of Lemma 1, d1N is implied by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Particularly, when α - 0 , i.e., e - w2 , inequality F (d1N | ε ) . 0 holds for any signal ε , then εˆ (d1N )
must be equal to ε max , which is infinite in our model. Under such cases, we can rewrite the above equation as w2 − w1 + o1 + ∫
+∞ 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Rearranging, e − w1 + o1 − (e − v) ∫
+∞ 0
F (d1N | ε ) dG (ε ) = 0 .
∞
Note that 0 - ∫ F (d1N | ε ) dG (ε ) - 1 . There are some 0
special cases when e + o1 - w1 , then d N = d1N = 0. Note that e − w1 + o1 . e − v cannot succeed because it requires o1 . w1 − v , which would incentivize the supplier to produce infinitely in period 1 and, thus, is ridiculous. In summary, the optimal first-period order quantity d1N is given by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0
when e + o1 > w1 . Otherwise, d1N ≡ 0 .
□
Proposition 2 shows that the retailer’s ordering behavior is closely linked to the option price o2 and the market signal ε . Different option prices result in different ordering behavior. Under some circumstances, the optimal total order may not even be unique. Signals such as εˆ and ε T can be used to categorize the re-
q1 . Otherwise, if such a ε T (o2 ) does not exist, or
tailer’s orders. These marking signals are important in models concerned with demand information update. The conclusion in the corollary is meaningful for disclosing the relationship between the retailer’s second-period profit curve and the market signal that optimal total order more than q1 is possible if and only
equivalently, π R (ε , F −1 (max{α ,0}| ε )) >π R (ε , F −1 ( β | ε ))
if ε T (o2 ) < ε max exists. ε T (o2 ) must be always greater
the choice of F −1 (max{α ,0}| ε ) and F −1 ( β | ε ) . Obviously, for stronger signals ε > ε T (o2 ) , d N= F −1 ( β |ε ) >
for any signals, define ε T (o2 ) = ε max . Thus, d =
than ε (q1 ) for a given q1 because the conservative
F ( β | ε ) > q1 is impossible because the market signal
retailer has incentive to choose an optimal total order greater than q1 with ε T (o2 ) that he will never do
N
−1
cannot be greater than its maximum.
577
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
with ε (q1 ) because d N - d * < q c . A final interesting
optimal q1N as
issue is that d N is not continuous in ε for a given contract and q1 is a kink point of the retailer’s ex-
c2 − c1 − o2 + o1 + ∫
0}.Thus, the derivative of π R (ε , d ) will never be equal
∞
modifying o1 . This is the standard Stackelberg game. Combining Eq. (5), the overall findings for coordination are summarized in Proposition 3. e − w2 Proposition 3 Denote α = and β = e−v e − w2 + o2 . To coordinate the channel, ensure that e−v o2 < c2 − v and maintain some specific functional relationships between the contract parameters of o1 , o2 , and e .
p−v+s o2 and c2 − v c −c ⎤ ⎡ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 , c2 − v ⎦ ⎣ e= p+s−
where ε T (o2 ) is determined by
π R (ε T (o2 ), F −1 (max{α ,0} | ε T (o2 ))) = π R (ε T (o2 ), F −1 ( β | ε T (o2 )))
when β > 0 and
{ε T (o2 ) : π R (ε T (o2 ), F −1 (max{α ,0} | ε T (o2 ))) = π R (ε T (o2 ),
F −1 ( β | ε T (o2 )))} ≠ ∅. Otherwise, define ε T (o2 ) = ε max . Proof The necessity of equation e = p + s − p−v+s o2 was proved in Proposition 1. Next, show c2 − v that coordination still needs another condition, c −c ⎤ ⎡ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 . c2 − v ⎦ ⎣ Recall that in this model coordination means N q = q c and q1N = q1c . Based on the corollary in Proposition 2, write the first-order condition for the
T
( o2 )
o2dG (ε ) = 0 .
Substitute e = p + s −
p−v+s o2 into this equation c2 − v
and rearrange o ⎞ ε ( q1 ) ⎛ [ p − c2 + s − c2 − c1 − o2 + o1 + ⎜1 − 2 ⎟ ∫ ⎝ c2 − v ⎠ 0
to zero at q1 . Now, consider the supplier’s problem. Knowing the retailer’s ordering behavior, the supplier will set his first-period production quantity accordingly. When the retailer knows the supplier’s first-period production decision (which is related to the retailer’s ordering behavior), the retailer coordinates it to be equal to q1c by
[e − c2 + o2 − (e − v) F (q1 | ε )]dG (ε ) +
∫ε
pected profit function. No matter what the realized market signal is, the optimal total order never equals q1. This may occur with the mechanism which sets the second-period compensation income to be max{d − q1 ,
ε ( q1 ) 0
( p − v + s ) F (q1 | ε )]dG (ε ) + ∫
c2 − c1 + ∫
Comparing with
∞
ε T ( o2 )
ε ( q1c ) 0
o2dG (ε ) = 0.
[ p − c2 + s − ( p − v +
s ) F (q1c | ε )]dG (ε ) = 0 , to let q1N = q1c , set o1 so that
the following equality holds: ε ( q1c ) o −o2 + o1 − 2 ∫ [ p − c2 + s − ( p − v + s ) F (q1 | ε )]dG (ε ) + c2 − v 0 ∞
∫ε
T
( o2 )
o2dG (ε ) = 0 .
Note that
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s ) F (q1c | ε )]dG (ε ) = 0
and rearranging gives c −c ⎤ ⎡ □ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 . c2 − v ⎦ ⎣ There are the pricing equations which assure channel profit maximization. However, an important issue to be considered before implementing any new pricing scheme is whether the scheme is Pareto improving with respect to an existing policy. In other words, will both the retailer and the supplier be better off with the proposed policy? To shed light on this question, examine how the retailer’s and supplier’s expected profits change as the contract parameters vary. Unfortunately, due to the model complexity, closed form solutions are not possible for some of the key variables; thus, quantitative comparisons are almost impossible. We cannot even prove that the monotonicity of each side’s expected profit function as o2 varies because it is related to the specific forms of the distribution functions.
3
Example
Suppose that F and G are both uniform distribution functions with F ~ U (a, b) and G ~ U (0, t ) . The critical
Tsinghua Science and Technology, August 2008, 13(4): 570-580
578
factor in models considering demand update is how the market signal updates the demand information, i.e., how to define F ( x | ε ) . Make the following assumptions related to the market signal: (1) The market signal does not change the demand distribution pattern, i.e., F ( x | ε ) is still uniform. (2) The market signal only affects the mean of the demand distribution, not the variance. As the signal becomes stronger the mean shifts up accordingly. For uniform distributions, this implies that if F ( x | ε ) ~ U ( f1 (ε ), f 2 (ε )), then f 2 (ε ) − f1 (ε ) ≡ b − a and f1 (ε ) and f 2 (ε ) are both increasing in ε .
If εˆ (d1N ) < 150 , the results can be directly used. If εˆ (d1N ) . 150 , then d1N is implied by w2 − w1 + o1 + ∫
150 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
(5) Calculate each party’s expected profit. Figure 1 shows how the retailer’s first-period order changes as o2 varies. As expected, the retailer sets d1N ≡ 0 when e + o1 - w1 , or equivalently, o2 . p − w1 + s (c2 − v) = 39.375 . Such an ordering behavior p − c1 + s
directly influences each party’s expected profit curve shown in Fig. 2.
(3) A market signal as mathematically expected maintains the original demand distribution function. That is, if ε = 0.5t , then F ( x | ε ) = F ( x) . Moreover, assume F ( x | ε ) ~ U (a − 4(0.5t − ε ), b − 4(0.5t − ε )) . To avoid uninteresting cases, define a > 2t . Consider the following numerical example. Suppose the original demand distribution is uniform with a = 1000 and b = 2000 . The market signal is uniformly distributed between zero and t = 150. The cost parameters are c1 = 50, c2 = 65, w1 = 60, w2 = 80, v = 20, s = 30, and p = 100 . For any o2 , (1) Calculate e according to e = p + s −
p−v+ s o2 . c2 − v
(2) Solve for ε T (o2 ) . If β > 0 , ε T (o2 ) = min{150, ε T′ }, where ε T′ is determined by π R (ε T′ , F −1 (max{α ,0} | ε T′ )) = π R (ε T′ , F −1 ( β | ε T′ )) ; If β - 0 , ε T (o2 ) = 150. (3) Calculate o1
⎡ according to o1 = ⎢G (ε T (o2 )) − ⎣
Fig. 1
Retailer’s optimal first-period order quantity
Figure 2 compares each party’s expected profit after coordination with that before coordination which is shown by the benchmark curve. The powerful retailer, who acts as a contract designer, will never be worse off, but the results are not so optimistic. The supplier’s expected profit even becomes negative for some cases! Notably, with these settings, the first-best expected profit Π c = 221 209 . Before coordination, however,
Π R = 175 960 and ΠS = 16 437. This leaves an improvement of 28 812 which is almost twice the supplier’s expected profit before coordination.
c2 − c1 ⎤ o2 . c2 − v ⎥⎦
(4) Solve for d1N . If e + o1 - w1 , d1N ≡ 0 ; If e + o1 > w1 , substitute d1N − [a − 2t + (b − a ) max{α ,0}] (which is 4 derived from F (d1N | εˆ (d1N )) = max{α ,0} ) into
εˆ (d1N ) =
w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Fig. 2 Expected profit for each party before and after coordination
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
Table 1 illustrates the profit allocation issues in detail. For these calculations, the new contract is feasible only when 34.8 - o2 - 43.8 . For low o2 in this range, each party becomes strictly better off. In particular, consider the best profit allocation situations for which each side gets a double-digit share of the extra profit. When o2 is around 40, the supplier will get more than 60% of the extra system profit which is double what he earned before coordination. Such a rosy profit improvement surely offers a strong incentive for the supplier to accept the new contract. Most interestingly, such profit allocations correspond to o2 . 39.6 which implies e + o1 - w1 and d N = d1N = 0. Thus, order postponement (until the demand uncertainty is resolved) can be beneficial to each party. This is contrast to the traditional thought that suppliers always welcome as-early-as-possible orders. Our opinion is that suppliers hate late orders mostly because they make the work of arranging production schedules more difficult. However, a properly designed incentive mechanism, which both clarifies the quantity that the supplier should produce and brings extra profit, eliminates these negative concerns. Such a result, however, needs further proof from theory and practice. Table 1
Profit allocations between the two parties
Second-period option price 0
Extra profit allocation on supplier (% of improvement scope) 100
34.8
1
…
…
39.6
67
39.9
63
40.2
58
40.5
53
40.8
48
41.1
44
41.4
39
41.7
35
42.0
30
42.3
26
…
…
43.8
3
Note: For simplicity, we state here only the allocation status for the supplier. Its counterpart for the retailer can be calculated since the sum is equal to 1.
4
579
Conclusions
This analysis considers the coordination of option contracts in a retailer-led supply chain where the retailer designs the contract and the supplier’s production needs to be coordinated with the contract. Consider the case where the supplier operates two modes of production and the retailer has the ability to update the demand forecast as the selling season approaches. Within this contract class, a set of pricing conditions are defined that align the supplier and retailer to act in the best interests of the channel. The analysis leads to some interesting results. First, option prices should be negatively correlated to the price and should be in a relevant range. This gives a trade-off between the higher unit compensation the supplier requires and the lower repurchasing price he will get. To assure that the option mechanism is an effective incentive, the option prices should be within a relevant range. The second-period option price should be no greater than the difference between the secondperiod unit production costs and the salvage value. Otherwise, the supplier will earn little due to the rapidly decreasing exercise price and the coordination will fail. Second, the first-period option price should be no greater than the second-period price and should be linearly correlated to the second-period option price when the latter is beyond some threshold. This result is true regardless of the specific functional form of the market demand or market signal. Third, the retailer’s ordering behavior is influenced by the dual effect of the predictive power of the new market information and option prices. Some extreme pricing cases may result in the ordering behavior being independent of the new market information. The influence pattern is so complex that there is no closed form solution to the retailer’s first-period optimal order quantity. Fortunately, this does not prevent exploring the optimal pricing conditions which are stated in Proposition 3. A numerical example is given for the profit allocation issues. An interesting result is that order postponement may be beneficial to both parties, though this needs further theoretical proof.
580
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Coordination by Option Contracts in a Retailer-Led Supply Chain with Demand Update* WANG Xiaolong (王小龙), LIU Liwen (刘丽文)** School of Economics and Management, Tsinghua University, Beijing 100084, China Abstract: This research examines how to use an option contract to coordinate a retailer-led supply chain where the market information can be updated. Based on Stackelberg game theory, we build a mode with one supplier and one retailer in which the retailer designs contracts to coordinate the supplier’s production in a two-mode production environment. This focuses on an option contract that consists of two option prices and one exercise price. By theoretical analysis and numerical example, we find that such a contract can coordinate the supplier and retailer to act in the best interest of the channel. The optimal pricing conditions are given as follows: First, option prices should be negatively correlated to the exercise price and should be in a relevant range. Second, the first-period option price should be no greater than the second-period price and should be linearly correlated to the second-period option price when the latter is beyond some threshold. The results show that such option contracts can arbitrarily allocate the extra system profit between the two parties so that each party is in a win-win situation. Key words: supply chain coordination; option contracts; retailer-led; demand update
Introduction Contemporary business puts more and more emphasis on channels, shifting power from supplier to retailer at the downstream end of a supply chain, which is the closest to the customers. Messinger and Narasimhan[1], Ailawadi[2], and Bloom and Perry[3] offer empirical evidence on how power has transferred in several international retailing markets. Such shifting enables the retailers to lead the supply chain in some sense. They then begin to raise stringent requirements on the suppliers’ production, compared with the traditional case where the supplier dominates the supply chain and coordinates the retailer’s ordering behavior. Another Received: 2006-10-24
* Supported by the National Natural Science Foundation of China (Nos. 70532004 and 70621061)
** To whom correspondence should be addressed. E-mail: [email protected] Tel: 86-10-62783553; Fax: 86-10-62785876
aspect is that the market situation often changes greatly long after the retailer has set an order. Some retailers, like Wal-Mart, deal with such problems by ordering less but more frequently. However, this volatile market puts great pressure on the suppliers who must make wise production/capacity building decisions to cope with the irregular orders which mirror a volatile market. Unfortunately, quite often suppliers fail to do that. In an effort to reduce mismatches between supply and demand, many industries, like the fashion industry (as reported by Fisher et al.[4]) and the consumer electronics industry, are moving to tighten coordination across their supply chains. One important initiative for suppliers is to develop faster, but typically more expensive, production modes that are capable of producing a second run of products closer to the selling season. This mode allows both sides to take advantage of updated demand forecasts. Also, the retailer can better manipulate the supplier’s production so that channel coordination is reached. This phenomenon has become more
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
common with advances in computer-based manufacturing technologies which significantly reduce production run times and product changeover times. However, the addition of another production mode is not enough to reach channel coordination due to double marginalization which refers to the fact that a party’s relative cost structure becomes distorted when a transfer price is introduced within a supply channel. For example, introducing a wholesale price into a single-stage newsvendor model causes the ratio of the buyer’s underage and overage costs to fall short of the channel ratio. As a result, the buyer orders less than the channel optimal quantity. In a two-stage production environment, double marginalization threatens to impact not only the total production quantity but also proper allocation of production between periods. The double marginalization concept was first introduced in the economics literature by Spengler[5], who raised the issue in terms of inappropriate prices rather than inappropriate quantities. Tirole[6] provided further details on the phenomenon. Many works in the operations management field, like those of Lariviere and Porteus[7], have shown that price-only contracts generally fail to coordinate the supply chain due to double marginalization. However, other contracts have proven to efficiently coordinate the supply chain. These contracts include buyback contracts[8], revenue sharing contracts[9], sales rebate contracts[10], quantity discount contracts[11], and quantity flexibility contracts[12]. The goal of this paper is to provide guidance in designing an option contract in a retailer-led supply chain, in which the supplier’s production needs coordinate with the existence of a market demand information update. The researches on option contracts in supply chain management are mainly divided into economic and operational areas. Typical economic works include Newbery[13] and Wu et al.[14] Typical operational works include Donohue[15] and Barnes-Schuster et al.[16] Our contract pricing scheme builds most closely upon the work of Barnes-Schuster et al.[16], which considers a combination of committed orders and options. Barnes-Schuster et al.[16] analyze a two-period model with correlated demand. The present analysis studies a two-period model with demand information update in which the market demand does not realize until the end of the second period. Specifically, consider one supplier and one retailer.
571
At the beginning of the first period both parties anticipate a demand information update that will appear at the beginning of the second period and they take this into account in their first-period decisions. The coordinating contract is designed as follows: The option contract has three parameters for two option prices o1 and o2 and one exercise price e. The option prices serve as the unit compensation for the supplier’s production quantities. When the realized demand is beyond the retailer’s total order, the retailer will purchase the supplier’s excess inventory, if any, to meet the market demand at the unit price e. With this contract, the retailer’s decision is to determine orders at the beginning of each period and the supplier’s decision is to determine the production allocation between the two periods. This project aims to find an optimal set of contracts to coordinate the channel and to realize a proper profit allocation.
1
Two-Period Model
The two-period model depicts the dynamics of a retailer who purchases from one main supplier and markets these products to a number of retail outlets, reselling at retail price p for a single selling season. Unlike traditional coordination models, the retailer in the model leads the supply chain and designs a proper contract to coordinate the supplier’s production quantities. The specific contract form will be investigated. The retailer utilizes two periods of demand information to determine the order quantities. During the first period, the demand prediction is relatively less accurate with a distribution function denoted as F(D). At the beginning of the second period (market demand not yet realized), more current market information is available which helps the retailer fine-tune the forecast where an effective market signal will be observed depending on specific industry situations and the lengths of the two periods which may not necessarily be equal. This updated information is the market signal ε. Let G(·) denote its estimated cumulative distribution function and let F(·|ε) denote the demand distribution function conditional on the market signal. Assume that ε is expressed in the same units as the total demand and is also non-negative. Market demand is stochastically increasing in the market signal, i.e., given x, F ( x | ε h )< F ( x | ε l ) for all ε h > ε l (ε h and ε l are two arbitrary
572
market signals). Assume that all distributions are continuous, invertible, doubly differentiable, and independent of the wholesale and retail prices. Furthermore, they are all common knowledge to both parties from the beginning of period 1. At the beginning of period 1, the retailer orders d1 units at a unit wholesale price w1 based on a rough knowledge of the demand distribution. Once ε is observed, the retailer will adjust the order quantities by ordering another d 2 units at a higher unit wholesale
Tsinghua Science and Technology, August 2008, 13(4): 570-580
q = q1 + q2 . 1.1
To provide a benchmark, first consider the problem where the supplier and the retailer are each owned by a risk-neutral entity. The owner aims to maximize his own expected profit by choosing first and second period production quantities (q1 , q2 ) which solve the following two-period optimization problem. ∞
max Π c (q1 ) = −c1q1 + ∫ Π c (ε , q1 , q2 )dG (ε )
price w2 . Note that the retailer’s second order is contingent on the specific market signal. If the signal exhibits a rather weak demand in the future, the retailer will not set another order, i.e., d 2 = 0 . The supplier produces over two time periods in response to the retailer’s orders, with orders filled in one shipment before the selling season begins. The supplier has two different production modes. One is the cheap mode which incurs a unit production cost c1 , and the other is the expensive mode which incurs a unit production cost c2 > c1 . Assume that production costs include the cost of holding inventory until delivery and the cost of delivery itself. Other cost parameters include a shortage penalty s paid by the retailer to the customers for each unit of unfilled demand, and a salvage value v received for each unit of inventory remaining at the end of the season no matter to which side it belongs. Suppose that all these prices and costs are exogenous and the analysis is limited to the most relevant and interesting case where p > w2 > w1 > c2 > c1 > v. Next assume that the lead time for the cheap
mode (and the timing of the market signal) is such that, to ensure completion before the start of the selling season, the supplier must begin production before observing the market signal. In contrast, the lead time for the expensive mode is short enough to allow production after the market signal is observed. Therefore, the supplier’s problem is to allocate production quantities between the two modes. During the first period, the supplier needs to determine how many products, denoted as q1 , would be produced among the cheap mode. In the second period after a market signal is observed, the supplier has to determine how many products, denoted as q2 , will be produced among the expensive mode, if necessary. Denote d = d1 + d 2 and
Centralized system performance: A benchmark
q1 .0
0
(1)
where Π c (ε , q1 , q2 ) = max π c (ε , q1 , q2 ) and q2 .0
π (ε , q1 , q2 ) = −c2 q2 + pE min{D, q1 + q2 } + v(q1 + q2 − E min{D, q1 + q2 }) − sE[ D − min{D, q1 + q2 }] (2) c
Here, Π c (ε , q1 , q2 ) represents the second-stage problem after the market signal is observed and the demand forecast is updated accordingly. To express the second-stage problem independently of q1 , substitute q for q1 + q2 and rearrange the terms. This new formulation yields the same objective solution as the problem in Eq. (1). ∞
max Π c (q1 ) = (c2 − c1 )q1 + ∫ Π c (ε , q1 )dG (ε ) q1 .0
0
(3)
where Π c (ε , q1 ) = max π c (ε , q) and q.q1
q
π (ε , q ) = ( p − c2 + s )q − ( p − v + s) ∫ F ( x | ε )dx − sED c
0
(4) Note that given q1 , the owner faces a newsvendor problem in period 2 but with an initial inventory and an adjusted demand distribution. Let (q1c , q c ) denote the optimal solution to the central problem. The solution structure is given by Lemma 1. Lemma 1 The argument of the central first period problem, Π c (q1 ) , is concave in the initial production quantity q1 . The optimal total production quantity is ⎧ q1c , if ε - ε (q1c ); ⎪ q c = ⎨ −1 ⎛ p − c2 + s ⎞ ⎪ F ⎜ p − v + s ε ⎟ , otherwise, ⎝ ⎠ ⎩
p − c2 + s ⎫ ⎧ where ε (q1c ) = sup ⎨ε : F (q1c | ε ) . ⎬ and the p−v+s ⎭ ⎩ first-period optimal production quantity is implied by
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s) F (q1c | ε )]dG (ε ) = 0 .
573
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
Proof First, prove the concavity of Π c (q1 ). This
only needs to prove that Π (ε , q1 ) = max π (ε , q ) is c
c
q.q1
concave in q1 . Note that ⎧⎪π c (ε , q* ), if q1 - q* ; Π (ε , q1 ) = ⎨ c ⎪⎩π (ε , q1 ), otherwise, * where q maximizes π c (ε , q ). Now π c (ε , q ) is obvic
∂ π (ε , q) = − ( p − v + s) ⋅ ∂q 2 2
ously concave in q because
c
based on the orders. Let di denote the retailer’s order in period i. The problem is to choose d1 and d 2 = d − d1 to maximize the expected profit. The problem structure is analogous to that in the centralized system. ∞
max Π R (d1 ) = − w1d1 + ∫ Π R (ε , d1 , d 2 ) dG (ε ), d1 .0
0
where Π R (ε , d1 , d 2 ) = max π R (ε , d1 , d 2 ) and d 2 .0
π R (ε , d1 , d 2 ) = − w2 d 2 + pE min{D, d1 + d 2 } + v(d1 + d 2 − E min{D, d1 + d 2 }) − sE[ D − min{D, d1 + d 2 }].
f (q | ε ) - 0. Therefore, Π (q1 ) is also concave in q1 . c
The optimal total production quantity follows immediately from the fact that the second-period problem is a simple newsvendor problem with initial inventory q1. Intuitively this quantity is positively related to the market signal. When the signal is not strong enough, the central planner will not produce a second sum. This idea is exhibited in Lemma 1 with a threshold value for ⎧ p − c2 + s ⎫ c the market signal: ε (q1 ) = sup ⎨ε : F (q1c | ε ). ⎬. p−v+s ⎭ ⎩ This is the strongest signal that will still result in no second-period production. With this notation, suppose that the central planner’s expected profit is maximized at q1c , and then rewrite the problem as
Π c (q1c ) = (c2 − c1 )q1c + ∫ ∞
∫ε
( q1c )
ε ( q1c ) 0
Rearranging, ∞
max Π R (d1 ) = ( w2 − w1 )d1 + ∫ Π R (ε , d1 ) dG (ε ) d1 .0
d .d1
d
π R (ε , d ) = ( p − w2 + s )d − ( p − v + s ) ∫ F ( x | ε )dx − sED. 0
Let (d1* , d * ) denote the optimal solution to the retailer’s problem. The solution structure is given in Lemma 2. Lemma 2 The argument of the retailer’s first period problem, Π R (d1 ) , is concave in the initial order quantity d1 . The optimal total order is ⎧d1* , if ε - ε (d1* ); ⎪ d * = ⎨ −1 ⎛ p − w2 + s ⎞ ⎪ F ⎜ p − v + s ε ⎟ , otherwise, ⎝ ⎠ ⎩
π c (ε , q1c )dG (ε ) +
π c (ε , q c )dG (ε ),
0
where Π R (ε , d1 ) = max π R (ε , d ) and
where ⎧
p − w2 + s ⎫ ⎬ and the firstp−v+s ⎭
⎛ p − c2 + s ⎞ ε ⎟ . Because Π c (q1 ) is where q c = F −1 ⎜ p − v + s ⎝ ⎠ concave in q1 , the first order condition works, i.e., an
⎩ period optimal order quantity is implied by
optimal first-period production quantity is implied by
w2 − w1 + ∫
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s ) F (q1c | ε )]dG (ε ) = 0 .
ε (d1* ) = sup ⎨ε : F (d1* | ε ) . ε ( d1* ) 0
[ p − w2 + s − ( p − v + s ) F (d1* | ε )]dG (ε ) = 0.
The proof is quite similar to that of Lemma 1. Here, ε (d1* ) also partitions the market signals into two sets.
□ Lemma 1 introduces the concept of ε (q1c ) which
If ε - ε (d1* ) , then the retailer will not set a second
partitions the market signals into two sets. If ε - ε (q1c ) , then the second-period production is not
order. Note the differences from the notations defined in Section 1.1. Furthermore, for a given x , ε ( x) < ε ( x),
necessarily positive. Otherwise, it is exactly positive.
which coincides with the phenomenon that double marginalization makes the buyer conservative. Compared with the central planner, given the same inventory level, the buyer in the decentralized system needs a stronger market signal, which induces a more optimistic demand estimate, to build up to that level. The supplier’s problem is to choose production
1.2
Decentralized system performance without coordination
Here model the decentralized system as a Stackelberg game. The retailer sets orders in each period as mentioned above. The supplier in turn arranges production
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quantities (q1 , q2 ) that maximize his own expected profit, subject to the retailer’s ordering behavior. His decision on the second-stage production is relatively straightforward, i.e., q2* = max{d * − q1* ,0} .
to the supplier to overproduce in each period. Recall that q1 . d1 and q2 . max{d − q1 ,0} . Thus d1 and max{d − q1 ,0} are the minimum production number
The difficulty lies in his decision on q1 . Obviously
for the supplier’s production in each period. The coordinating contract is defined as follows: The option contract has three parameters of two option prices o1 and
he should at least produce d1 , but should he produce
o2 and one exercise price e . The option prices are
more? The answer depends on his estimate on the likelihood of the retailer’s second order. For such a decision his problem is
the unit compensation for the supplier’s production quantities beyond the minimums in each period. When the realized demand exceeds the retailer’s total order, he will purchase the supplier’s excess inventory, if any, to meet the market demand at a unit price e . The sequence of events is summarized as follows: (1) At the beginning of period 1, the retailer sets a firm order d1 . The retailer also announces an option
max ΠS (q1 ) = w1d1 − c1q1 + ∫ q1.d1
ε ( q1 )
∫ε
( d1 ) ∞
∫ε
( q1 )
ε ( d1 ) 0
v(q1 − d1 ) dG (ε ) +
[ w2 d 2 + v(q1 − d )]dG (ε ) +
[ w2 d 2 − c2 (d − q1 )]dG (ε ).
The solution to this problem is given by Lemma 3. Lemma 3 Given the retailer’s ordering behavior, the supplier will maximize his expected profit by setting q1* = max{d1* , q1′} and q2* = max{d * − q1* ,0}, c −c where q1′ is implied by G (ε (q1′)) = 2 1 . c2 − v Because this problem has a newsvendor structure, the first order condition works. The proof is straightforward so the details are omitted. These problems have reviewed the benchmark and decentralized uncoordinated system performance. Although the precise relationships between d1* , q1*, and q1c depending on factors like the ratio of the two production costs, the ratio of the two wholesale prices, and the estimate of the market signal cannot be determined, the channel is certainly not coordinated. This is implied by Lemma 2 which states that for a given market signal the retailer’s total order quantity is less than the optimal total inventory required in a centralized system. The next section focuses on how an option mechanism coordinates the supply chain and realizes profit allocation.
2
Channel Coordination
To coordinate the channel, the retailer must choose the proper contract parameters so that the supplier’s production quantities in each period are equal to the optimal solutions in the centralized system, i.e., q1* = q1c and q* = q c . To do so the retailer must give incentives
contract with option prices o1 and o2 and exercise price e . The supplier, in turn, decides to produce q1 . d1 of goods in the cheap mode which will produce an income of o1 (q1 − d1 ) . (2) At the beginning of period 2, after the market signal is observed, the retailer orders d 2 . The supplier selects a second-period production, q2 . max{d − q1 ,0}, that involves the expensive mode and produces an income of o2 (q2 − max{d − q1 ,0}) . (3) At the beginning of the selling season, after the demand is realized, the retailer purchases additionally m = min{max{D − d ,0}, q − d } number of goods at unit price e . The supplier delivers all that the retailer wants. The problem assumes that the supplier is obligated to fill the retailer’s entire order and can neither sell directly to the final customer (i.e., the retailer’s customer) nor supply inventory in excess of the retailer’s total order quantity. With the option contract, the supplier’s problem becomes max ΠS (q1 ) = − (c1 − o1 )q1 + ( w1 − o1 )d1 + q1 . d1
∫ where ΠS (ε , q1 , q2 ) =
∞
0
ΠS (ε , q1 , q2 ) dG (ε ), max
q2 .max{d − q1 ,0}
π S (ε , q1 , q2 ) and
π S (ε , q1 , q2 ) = −c2 q2 + o2 (q2 − max{d − q1 ,0}) + w2 d 2 + ∫ v(q1 + q2 − d ) dF ( x | ε ) + d
0
∫
q1 + q2 d
[e( x − d ) + v(q − x)]dF ( x | ε ) +
575
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
∫
∞
e(q1 + q2 − d )dF ( x | ε ) .
q1 + q2
Here, ΠS (ε , q1 , q2 ) represents his second-stage problem. After making the supplier’s second-stage problem independent of q1 and after some algebra, the sup-
∞
max Π R (d1 ) = −o1 (q1 − d1 ) − w1d1 + ∫ Π R (ε , d1 ) dG (ε ), d1.0
0
where Π R (ε , d1 ) = max π R (ε , d ) and d .d1
π R (ε , d ) = − w2 d 2 − o2 (q2 − max{d − q1 ,0}) + pE min{D, q} + v(d − E min{D, d }) − eE min{max{D − d ,0}, q − d } −
plier’s problem is max ΠS (q1 ) = (c2 − c1 − o2 + o1 )q1 + ( w1 − o1 )d1 + q1.d1
∫
∞ 0
ΠS (ε , q1 ) dG (ε ),
where ΠS (ε , q1 ) = max π S (ε , q ) and q.max{d ,q1 }
s ( ED − E min{D, q}) . Rearranging, max Π R (d1 ) = (o2 − o1 )q1 + ( w2 − w1 + o1 )d1 + d1.0
π S (ε , q) = (e − c2 + o2 )q + w2 d 2 − ed − o2 max{d − q1 ,0} − q
( e − v ) ∫ F ( x | ε ) dx . d
N
N 1
Let (q , q ) denote the solution to this problem, and then we have Proposition 1. Proposition 1 Given an option contract specified by the retailer, the supplier’s optimal total production quantity is ⎧ ⎛ e − c2 + o2 ⎞ ⎫ q N = max ⎨q1N , F −1 ⎜ ε ⎟⎬ . ⎝ e−v ⎠⎭ ⎩ The proof is relatively straight forward since the supplier’s problem follows a newsvendor structure. Proposition 1 shows clearly that to coordinate q N = e − c2 + o2 p − c2 + s q c , it is necessary to set = , i.e., e−v p−v+s p−v+s (5) e= p+s− o2 c2 − v Based on Eq. (5), e > v and e + o2 > c2 to make ⎛ e − c2 + o2 ⎞ | ε ⎟ meaningful. Economically these two F −1 ⎜ ⎝ e−v ⎠ conditions prevent the supplier from suffering losses. e + o2 is all the revenue that the supplier can get from
one unit of product if it can be sold after the market demand is realized. If e + o2 < c2 , the supplier has no incentive to overproduce in the second period, and thus channel coordination is never reached. Since e = p + s − p−v+s o2 , the two conditions are equivalent with c2 − v
o2 < (c2 − v) . Therefore, o2 < (c2 − v) from now on. Next, consider the retailer’s problem. In all the folp−v+s o2 holds lowing parts, assume that e = p + s − c2 − v except where specified. The retailer’s problem is
∫
∞
0
Π R (ε , d1 ) dG (ε ),
where Π R (ε , d1 ) = max π R (ε , d ) and d .d1
d
π R (ε , d ) = (e − w2 )d − (e − v) ∫ F ( x | ε ) dx + 0
o2 max{d − q1 , 0} + ( p + s − e − o2 )q − q
( p + s − e) ∫ F ( x | ε ) dx − sED . 0
Let ( d1N , d N ) denote the optimal solution to the retailer’s problem. Proposition 2 describes the retailer’s ordering behavior. e − w2 e − w2 + o2 Proposition 2 Denote α = , β= , e−v e−v and γ (ε ) = max{d1N , F −1 (max{α ,0}| ε )}. Define εˆ (d1N) = sup{ε : F (d1N | ε ) . max{α ,0}} and allow εˆ (d1N ) = +∞
if ε max is infinite. For a given option contract, the optimal ordering behavior for the retailer has the following structure: (1) After the market signal ε has appeared, the optimal total order quantity is determined as follows: (i) If F (q1 | ε ) . β , d N = γ (ε ) ;
(ii) Otherwise, ⎧γ (ε ), if π R (ε , F −1 (max{α ,0}| ε )) > ⎪ dN = ⎨ π R (ε , F −1 ( β | ε )); ⎪ −1 ⎩ F ( β | ε ), otherwise. (2) The optimal first-period order quantity is given by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0
when e + o1 > w1 . Otherwise, d1N ≡ 0 . Corollary For some second-period option price(s) o2 such that β > 0 and set A = {ε : π R (ε , F −1 (max{α , 0}| ε )) - π R (ε , F −1 ( β | ε ))} is not empty, there exists a
Tsinghua Science and Technology, August 2008, 13(4): 570-580
576
signal ε T (o2 ) such that d N = F −1 ( β | ε ) > q1 if and only if ε . ε T (o2 ) . Otherwise, d
N
will never be
greater than q1 and ε T (o2 ) = ε max for such cases. Proof First, prove part (1). Omitting all the irrelevant parts, the retailer’s second-period problem can be rewritten as ⎧(e − w )d − (e − v) d F ( x | ε )dx, 2 ∫0 ⎪ ⎪ if d - q1 ; ⎪ max π R (ε , d ) = ⎨ d d .d1 ⎪(e − w2 + o2 )d − (e − v) ∫ F ( x | ε )dx, 0 ⎪ ⎪⎩ otherwise. Then, the unconditional optimal d N may not be unique but can be either F −1 (max{α ,0}| ε ) or F −1 (max{β , 0}| ε ) or both. However, if d N = F −1 ( β | ε ),
there must be F −1 ( β | ε ) > q1 , i.e., 0 < F (q1 | ε ) < β , so that max{d − q1 ,0} = d − q1 makes sense. This completes the proof of part (i) when F (q1 | ε ) . β and d N = γ (ε ) .
Now consider 0 < F (q1 | ε ) < β . The criterion is simple but clear. Whether d N = γ (ε ) or d N = F −1 ( β | ε ) depends on which can bring more expected profit, i.e., ⎧γ (ε ), if π R (ε , F −1 (max{α ,0}| ε )) > ⎪ dN = ⎨ π R (ε , F −1 ( β | ε )); ⎪ −1 ⎩ F ( β | ε ), otherwise. This completes the proof of part (1) in Proposition 2. Next, show the link between the retailer’s second-period expected profit and the market signal to estimate the optimal total order from the beginning of period 1 based on the market signal conditions, which is the core idea of the corollary. When β > 0, let π R (ε , F −1 (max{α , 0}| ε )) = π R (ε , F −1 ( β | ε )) . Note that d N ≡ d1N when β - 0 because
β . α . Suppose that there exists a solution ε T (o2 ) , and then for ε T (o2 ) the retailer is indifferent towards
Next consider part (2) which depicts the first-period ordering behavior. If o2=0, then the retailer’s second-period problem is given by the following: For any α , similar to the reasoning in proof of Lemma 1, d1N is implied by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Particularly, when α - 0 , i.e., e - w2 , inequality F (d1N | ε ) . 0 holds for any signal ε , then εˆ (d1N )
must be equal to ε max , which is infinite in our model. Under such cases, we can rewrite the above equation as w2 − w1 + o1 + ∫
+∞ 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Rearranging, e − w1 + o1 − (e − v) ∫
+∞ 0
F (d1N | ε ) dG (ε ) = 0 .
∞
Note that 0 - ∫ F (d1N | ε ) dG (ε ) - 1 . There are some 0
special cases when e + o1 - w1 , then d N = d1N = 0. Note that e − w1 + o1 . e − v cannot succeed because it requires o1 . w1 − v , which would incentivize the supplier to produce infinitely in period 1 and, thus, is ridiculous. In summary, the optimal first-period order quantity d1N is given by w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0
when e + o1 > w1 . Otherwise, d1N ≡ 0 .
□
Proposition 2 shows that the retailer’s ordering behavior is closely linked to the option price o2 and the market signal ε . Different option prices result in different ordering behavior. Under some circumstances, the optimal total order may not even be unique. Signals such as εˆ and ε T can be used to categorize the re-
q1 . Otherwise, if such a ε T (o2 ) does not exist, or
tailer’s orders. These marking signals are important in models concerned with demand information update. The conclusion in the corollary is meaningful for disclosing the relationship between the retailer’s second-period profit curve and the market signal that optimal total order more than q1 is possible if and only
equivalently, π R (ε , F −1 (max{α ,0}| ε )) >π R (ε , F −1 ( β | ε ))
if ε T (o2 ) < ε max exists. ε T (o2 ) must be always greater
the choice of F −1 (max{α ,0}| ε ) and F −1 ( β | ε ) . Obviously, for stronger signals ε > ε T (o2 ) , d N= F −1 ( β |ε ) >
for any signals, define ε T (o2 ) = ε max . Thus, d =
than ε (q1 ) for a given q1 because the conservative
F ( β | ε ) > q1 is impossible because the market signal
retailer has incentive to choose an optimal total order greater than q1 with ε T (o2 ) that he will never do
N
−1
cannot be greater than its maximum.
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WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
with ε (q1 ) because d N - d * < q c . A final interesting
optimal q1N as
issue is that d N is not continuous in ε for a given contract and q1 is a kink point of the retailer’s ex-
c2 − c1 − o2 + o1 + ∫
0}.Thus, the derivative of π R (ε , d ) will never be equal
∞
modifying o1 . This is the standard Stackelberg game. Combining Eq. (5), the overall findings for coordination are summarized in Proposition 3. e − w2 Proposition 3 Denote α = and β = e−v e − w2 + o2 . To coordinate the channel, ensure that e−v o2 < c2 − v and maintain some specific functional relationships between the contract parameters of o1 , o2 , and e .
p−v+s o2 and c2 − v c −c ⎤ ⎡ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 , c2 − v ⎦ ⎣ e= p+s−
where ε T (o2 ) is determined by
π R (ε T (o2 ), F −1 (max{α ,0} | ε T (o2 ))) = π R (ε T (o2 ), F −1 ( β | ε T (o2 )))
when β > 0 and
{ε T (o2 ) : π R (ε T (o2 ), F −1 (max{α ,0} | ε T (o2 ))) = π R (ε T (o2 ),
F −1 ( β | ε T (o2 )))} ≠ ∅. Otherwise, define ε T (o2 ) = ε max . Proof The necessity of equation e = p + s − p−v+s o2 was proved in Proposition 1. Next, show c2 − v that coordination still needs another condition, c −c ⎤ ⎡ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 . c2 − v ⎦ ⎣ Recall that in this model coordination means N q = q c and q1N = q1c . Based on the corollary in Proposition 2, write the first-order condition for the
T
( o2 )
o2dG (ε ) = 0 .
Substitute e = p + s −
p−v+s o2 into this equation c2 − v
and rearrange o ⎞ ε ( q1 ) ⎛ [ p − c2 + s − c2 − c1 − o2 + o1 + ⎜1 − 2 ⎟ ∫ ⎝ c2 − v ⎠ 0
to zero at q1 . Now, consider the supplier’s problem. Knowing the retailer’s ordering behavior, the supplier will set his first-period production quantity accordingly. When the retailer knows the supplier’s first-period production decision (which is related to the retailer’s ordering behavior), the retailer coordinates it to be equal to q1c by
[e − c2 + o2 − (e − v) F (q1 | ε )]dG (ε ) +
∫ε
pected profit function. No matter what the realized market signal is, the optimal total order never equals q1. This may occur with the mechanism which sets the second-period compensation income to be max{d − q1 ,
ε ( q1 ) 0
( p − v + s ) F (q1 | ε )]dG (ε ) + ∫
c2 − c1 + ∫
Comparing with
∞
ε T ( o2 )
ε ( q1c ) 0
o2dG (ε ) = 0.
[ p − c2 + s − ( p − v +
s ) F (q1c | ε )]dG (ε ) = 0 , to let q1N = q1c , set o1 so that
the following equality holds: ε ( q1c ) o −o2 + o1 − 2 ∫ [ p − c2 + s − ( p − v + s ) F (q1 | ε )]dG (ε ) + c2 − v 0 ∞
∫ε
T
( o2 )
o2dG (ε ) = 0 .
Note that
c2 − c1 + ∫
ε ( q1c ) 0
[ p − c2 + s − ( p − v + s ) F (q1c | ε )]dG (ε ) = 0
and rearranging gives c −c ⎤ ⎡ □ o1 = ⎢G (ε T (o2 )) − 2 1 ⎥ o2 . c2 − v ⎦ ⎣ There are the pricing equations which assure channel profit maximization. However, an important issue to be considered before implementing any new pricing scheme is whether the scheme is Pareto improving with respect to an existing policy. In other words, will both the retailer and the supplier be better off with the proposed policy? To shed light on this question, examine how the retailer’s and supplier’s expected profits change as the contract parameters vary. Unfortunately, due to the model complexity, closed form solutions are not possible for some of the key variables; thus, quantitative comparisons are almost impossible. We cannot even prove that the monotonicity of each side’s expected profit function as o2 varies because it is related to the specific forms of the distribution functions.
3
Example
Suppose that F and G are both uniform distribution functions with F ~ U (a, b) and G ~ U (0, t ) . The critical
Tsinghua Science and Technology, August 2008, 13(4): 570-580
578
factor in models considering demand update is how the market signal updates the demand information, i.e., how to define F ( x | ε ) . Make the following assumptions related to the market signal: (1) The market signal does not change the demand distribution pattern, i.e., F ( x | ε ) is still uniform. (2) The market signal only affects the mean of the demand distribution, not the variance. As the signal becomes stronger the mean shifts up accordingly. For uniform distributions, this implies that if F ( x | ε ) ~ U ( f1 (ε ), f 2 (ε )), then f 2 (ε ) − f1 (ε ) ≡ b − a and f1 (ε ) and f 2 (ε ) are both increasing in ε .
If εˆ (d1N ) < 150 , the results can be directly used. If εˆ (d1N ) . 150 , then d1N is implied by w2 − w1 + o1 + ∫
150 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
(5) Calculate each party’s expected profit. Figure 1 shows how the retailer’s first-period order changes as o2 varies. As expected, the retailer sets d1N ≡ 0 when e + o1 - w1 , or equivalently, o2 . p − w1 + s (c2 − v) = 39.375 . Such an ordering behavior p − c1 + s
directly influences each party’s expected profit curve shown in Fig. 2.
(3) A market signal as mathematically expected maintains the original demand distribution function. That is, if ε = 0.5t , then F ( x | ε ) = F ( x) . Moreover, assume F ( x | ε ) ~ U (a − 4(0.5t − ε ), b − 4(0.5t − ε )) . To avoid uninteresting cases, define a > 2t . Consider the following numerical example. Suppose the original demand distribution is uniform with a = 1000 and b = 2000 . The market signal is uniformly distributed between zero and t = 150. The cost parameters are c1 = 50, c2 = 65, w1 = 60, w2 = 80, v = 20, s = 30, and p = 100 . For any o2 , (1) Calculate e according to e = p + s −
p−v+ s o2 . c2 − v
(2) Solve for ε T (o2 ) . If β > 0 , ε T (o2 ) = min{150, ε T′ }, where ε T′ is determined by π R (ε T′ , F −1 (max{α ,0} | ε T′ )) = π R (ε T′ , F −1 ( β | ε T′ )) ; If β - 0 , ε T (o2 ) = 150. (3) Calculate o1
⎡ according to o1 = ⎢G (ε T (o2 )) − ⎣
Fig. 1
Retailer’s optimal first-period order quantity
Figure 2 compares each party’s expected profit after coordination with that before coordination which is shown by the benchmark curve. The powerful retailer, who acts as a contract designer, will never be worse off, but the results are not so optimistic. The supplier’s expected profit even becomes negative for some cases! Notably, with these settings, the first-best expected profit Π c = 221 209 . Before coordination, however,
Π R = 175 960 and ΠS = 16 437. This leaves an improvement of 28 812 which is almost twice the supplier’s expected profit before coordination.
c2 − c1 ⎤ o2 . c2 − v ⎥⎦
(4) Solve for d1N . If e + o1 - w1 , d1N ≡ 0 ; If e + o1 > w1 , substitute d1N − [a − 2t + (b − a ) max{α ,0}] (which is 4 derived from F (d1N | εˆ (d1N )) = max{α ,0} ) into
εˆ (d1N ) =
w2 − w1 + o1 + ∫
εˆ( d1N ) 0
[e − w2 − (e − v) F (d1N | ε )]dG (ε ) = 0 .
Fig. 2 Expected profit for each party before and after coordination
WANG Xiaolong (王小龙) et al：Coordination by Option Contracts in a Retailer-Led …
Table 1 illustrates the profit allocation issues in detail. For these calculations, the new contract is feasible only when 34.8 - o2 - 43.8 . For low o2 in this range, each party becomes strictly better off. In particular, consider the best profit allocation situations for which each side gets a double-digit share of the extra profit. When o2 is around 40, the supplier will get more than 60% of the extra system profit which is double what he earned before coordination. Such a rosy profit improvement surely offers a strong incentive for the supplier to accept the new contract. Most interestingly, such profit allocations correspond to o2 . 39.6 which implies e + o1 - w1 and d N = d1N = 0. Thus, order postponement (until the demand uncertainty is resolved) can be beneficial to each party. This is contrast to the traditional thought that suppliers always welcome as-early-as-possible orders. Our opinion is that suppliers hate late orders mostly because they make the work of arranging production schedules more difficult. However, a properly designed incentive mechanism, which both clarifies the quantity that the supplier should produce and brings extra profit, eliminates these negative concerns. Such a result, however, needs further proof from theory and practice. Table 1
Profit allocations between the two parties
Second-period option price 0
Extra profit allocation on supplier (% of improvement scope) 100
34.8
1
…
…
39.6
67
39.9
63
40.2
58
40.5
53
40.8
48
41.1
44
41.4
39
41.7
35
42.0
30
42.3
26
…
…
43.8
3
Note: For simplicity, we state here only the allocation status for the supplier. Its counterpart for the retailer can be calculated since the sum is equal to 1.
4
579
Conclusions
This analysis considers the coordination of option contracts in a retailer-led supply chain where the retailer designs the contract and the supplier’s production needs to be coordinated with the contract. Consider the case where the supplier operates two modes of production and the retailer has the ability to update the demand forecast as the selling season approaches. Within this contract class, a set of pricing conditions are defined that align the supplier and retailer to act in the best interests of the channel. The analysis leads to some interesting results. First, option prices should be negatively correlated to the price and should be in a relevant range. This gives a trade-off between the higher unit compensation the supplier requires and the lower repurchasing price he will get. To assure that the option mechanism is an effective incentive, the option prices should be within a relevant range. The second-period option price should be no greater than the difference between the secondperiod unit production costs and the salvage value. Otherwise, the supplier will earn little due to the rapidly decreasing exercise price and the coordination will fail. Second, the first-period option price should be no greater than the second-period price and should be linearly correlated to the second-period option price when the latter is beyond some threshold. This result is true regardless of the specific functional form of the market demand or market signal. Third, the retailer’s ordering behavior is influenced by the dual effect of the predictive power of the new market information and option prices. Some extreme pricing cases may result in the ordering behavior being independent of the new market information. The influence pattern is so complex that there is no closed form solution to the retailer’s first-period optimal order quantity. Fortunately, this does not prevent exploring the optimal pricing conditions which are stated in Proposition 3. A numerical example is given for the profit allocation issues. An interesting result is that order postponement may be beneficial to both parties, though this needs further theoretical proof.
580
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