Coordination by Option Contracts in a Retailer-Led Supply Chain with

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WANG Xiaolong (王小龙) et al:Coordination by Option Contracts in a Retailer-Led … N * c with ε ( q1) because d d < q . A final interesting N issue is that d is not continuous in ε for a given contract and q 1 is a kink point of the retailer’s expected profit function. No matter what the realized market signal is, the optimal total order never equals 1 . q This may occur with the mechanism which sets the second-period compensation income to be max{ d− q1, 0}.Thus, the derivative of π R ( ε, d) will never be equal to zero at q 1 . 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 be- c havior), the retailer coordinates it to be equal to q 1 by modifying o 1 . 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 1 o , o 2 , and e . p− v+ s e= p+ s− o2and c2−v ⎡ c2 − c1⎤ o1 = ⎢ G( εT ( o2)) − o2 c2−v ⎥ , ⎣ ⎦ where T ( 2) o ε is determined by −1 πR( εT( o2), F (max{ α,0}| εT( o2))) = −1 πR( εT( o2), F ( β | εT( o2))) when β > 0 and −1 { εT( o2) : πR( εT( o2), F (max{ α,0}| εT( o2))) = πR( εT( o2), −1 F ( β | εT( o2)))} ≠∅. Otherwise, define T ( 2) max o ε = ε . 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, ⎡ c2 − c1⎤ o1 = ⎢ G( εT ( o2)) − o2 c2−v ⎥ . ⎣ ⎦ Recall that in this model coordination means N c N c q = q and q1 = q1 . Based on the corollary in Proposition 2, write the first-order condition for the optimal N q 1 as 2 1 2 1 ε ( q1 ) 0 2 2 1 ∫ 577 c − c − o + o + [ e− c + o −( e− v) F( q | ε )]d G( ε ) + ∫ ∞ o ( 2d G( ε ) = 0. εT o2 ) p− v+ s Substitute e= p+ s− o2into this equation c2−v and rearrange ⎛ o ( 1) 2 ⎞ ε q c2 − c1− o2 + o1+ ⎜1 − [ p− c 0 2 + s− c2−v ⎟ ⎝ ⎠ ∫ ∞ ( p− v+ s) F( q | ε)]d G( ε) + o d G( ε) = 0. Comparing with 1 T ( 2 ) 2 o ε c ε ( q1 ) 2 1 0 2 ∫ ∫ c − c + [ p− c + s−( p− v+ c N c sFq ) ( 1 | ε)]d G( ε ) = 0, to let q1 = q1 , set o 1 so that the following equality holds: c o ε ( q1 ) 2 − o2+ o1− [ p− c 0 2+ s−( p− v+ s) F( q1| ε )]d G( ε ) + c −v∫ Note that 2 ∫ ∞ o d G( ε ) = 0. o ε ( 2 T 2 ) c ε ( q1 ) c 2− 1+ − 2+ − − + 1 0 c c ∫ [ p c s ( p v s) F( q | ε )]d G( ε ) = 0 and rearranging gives ⎡ c2 − c1⎤ o1 = ⎢ G( εT ( o2)) − 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 o 2 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
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 ), f2( ε )), then f2( ε) − f1( ε) ≡b−a and f1 ( ε ) and f2 ( ε ) are both increasing in ε . (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(0.5 t−ε), b−4(0.5 t− 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 o 2 , p− v+ s e= p+ s− o . (1) Calculate e according to (2) Solve for T ( 2) c2−v 2 o ε . If β > 0 , εT( o2) = min{150, ε′ T}, where T ε′ is determined by −1 −1 π R( ε′ T, F (max{ α,0}| ε′ T)) = πR( ε′ T, F ( β | ε′ T)) ; If β 0 , T ( 2) 150. o ε = ⎡ (3) Calculate o 1 according to o1 = ⎢G( εT ( o2)) − ⎣ c2 − c1 ⎤ o2 c2−v ⎥ . ⎦ N (4) Solve for d 1 . N If e+ o1 w1, d1 ≡ 0 ; If e+ o1 > w1, substitute N N d1−[ a− 2 t+ ( b−a)max{ α,0}] ˆ( ε d1 ) = (which is 4 N N derived from Fd ( | ˆ ε( d )) = max{ α,0} ) into 1 1 N ˆ( ε d1 ) N 2 1 1 ∫ 0 2 1 w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0. Tsinghua Science and Technology, August 2008, 13(4): 570-580 If ˆ( ε d ) < 150, the results can be directly used. If ˆ( ε d ) 150, then N 1 N 1 N d 1 is implied by 150 N 2 1 1 0 2 1 ∫ w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0. (5) Calculate each party’s expected profit. Figure 1 shows how the retailer’s first-period order changes as o 2 varies. As expected, the retailer sets N d ≡ 0 when 1 1 , e o w o 1 p− w1+ s c2v p− c1+ s + or equivalently, 2 ( − ) = 39.375. Such an ordering behavior directly influences each party’s expected profit curve shown in Fig. 2. 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 c profit Π = 221 209. Before coordination, however, Π R = 175 960 and Π S = 16 437. This leaves an improvement of 28 812 which is almost twice the supplier’s expected profit before coordination. Fig. 2 Expected profit for each party before and after coordination
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