SUPPLY CHAIN GAMES: OPERATIONS MANAGEMENT AND RISK VALUATION phần 6 pps

Equilibrium Consider now the supplier’s problem. Applying the first-order optimality condition to the supplier’s objective function (4.136), we find that the opti- mal wholesale price w is defined by the equation: 0)0( )0( )( =+− X dw dX cw s . (4.168) Then, with respect to Proposition 4.27 and ∫ = 1 0 1 )()( t dttatA , this implies that X(0)=bA(t 1 ) and equation (4.168) transforms into 0)()()()( )( )( 11 1 1 1 =+−=+− tbAta dw bdt cwtbA dw tbdA cw ss , where t 1 is determined by c s + ]) )( )( ()[( 1 0 1 cdth tA tbA hh t =−Φ+ −−+ ∫ . (4.169) Using implicit differentiation of (4.169) and the fact that )( −+ − + =Φ hh h b , we find that ]) )( )( ()( )( )[( 1 1 0 1 1 1 dt tA tbA ta tA b hh dw dt t ϕ ∫ −+ + −= , (4.170) which implies that the greater the wholesale price, the earlier the manu- 0)()()( 11 1 =+− tbAta ds bdt cw s , is greater than c s . Let ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − b X At )0( 1 1 , then equation (4.170) takes the following form ]) )( )0( () )0( ( )( )[( 1 )0( 0 1 1 1 dt tA X b X Aa tA b hh dw dt b X A ϕ ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −= , (4.171) which by substituting into the first-order optimality condition results in 252 4 MODELING IN AN INTERTEMPORAL FRAMEWORK means that a solution, w, which satisfies the optimality condition, facturer will start using his in-house capacity. Moreover, this also 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 253 0)0( ]) )( )0( () )0( ( )( )[( ) )0( ()( )0( 0 1 1 1 =+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+ − − X dt tA X b X Aa tA b hh b X Abacw b X A s ϕ . We thus conclude with the following proposition. Proposition 4.31. Let all conditions of Proposition 4.27 be met, ds dt 1 be defined by (4.171), 0 )( > ∂ Φ∂ z z and,  and  satisfy the following equations  + ]) )( ()[( 1 0 cdth tA hh b A =−Φ+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+ ∫ − δ δ , (4.172) 0 ]) )( ()( )( )[( )()( 1 0 1 1 =+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+ − − δ δ ϕ δ δ λ δ dt tAb Aa tA b hh b Abac b A s . (4.173) If 0)()()()()( 11 1 1 1 1 2 1 2 <++ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ +− tbata dw bdt ta dw bdt ta dw tbd c s λ , (4.174) then the wholesale price w s = <c, the manufacture’s advance order X s (0)=  and production policy u s (t)=0 for 1 0 tt < ≤ ; u s (t)=ba(t) for 21 ttt <≤ ; u s (t)=0 for Ttt ≤≤ 2 constitute the unique Stackelberg equilibrium in the differential production balancing game. The following example illustrates the results for demands following a uniform distribution. The following example is based on a problem faced by a large supplier of fashion goods, where demand is quite steady, i.e., a(t)=1, but the amplitude d is a random parameter characterized by the uniform distribution, Proof: To prove the proposition, it is sufficient to verify the secondorder optimality condition, which immediately results in condition (4.174) stated in the proposition. Example 4.5. ⎪ ⎩ ⎪ ⎨ ⎧ ≤≤ = otherwise. 0, ;0for , 1 )( RD R D ϕ . Then R e e =Φ )( for Re ≤ and 1)( = Φ e for e>R and with respect to (4.152), −+ − + = hh h Rb . Given b<U, the input data for the supply chain are presented in Table 4.3. Table 4.3. System parameters. c S h + h - C R T b 3.0 1.0 4.0 6.0 20.0 100 16.0 We start by calculating the system-wide optimal solution. Since A(t)=t, )()( 1 1 tT cdt t e hh h T t − +Φ+ > ∫ −+ − and b ≤ U, we apply Proposition 4.27 to find all swit- ching points, 0<t 1 <t 2 <T as follows: (i) X(0)=bt 1 ; (ii) c s + ])()[( 1 0 1 cdth t bt hh t =−Φ+ −−+ ∫ , i.e., c s -c+ ]1)[( 1 0 dthhh R bt −−+ −⋅+ ∫ + dth Rt bt hh t R bt ])[( 1 1 1 −−+ −+ ∫ =0, and thus c s -c+ 1 1 1 1 )lnln1()( th R bt t R bt hh −−+ −−++ =0, which, by taking into account that −+ − + = hh h Rb , results in − −+ − + − = h hh h cc t s ln 1 ; (4.175) (iii) ])()[( 2 2 dth t bt hh T t −−+ −Φ+ ∫ -c=0, i.e., 254 4 MODELING IN AN INTERTEMPORAL FRAMEWORK 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 255 ])()[( 2 2 dth Rt bt hh T t −−+ −+ ∫ -c=0 , and thus 0)()ln(ln)( 22 2 =−−+−+− −−+ ctThtT R bt hh . (4.176) Using data from Table 4.3, equations (4.175) and (4.174) result in 4 41 ln4 36 1 + − =t =3.3610 06)100(4)ln100(ln4 222 = − − + − − ttt , (4.177) respectively. Solving equation (4.177) in t 2 , we find that t 2 = 83.1862. Thus, the system-wide optimal advance order quantity is X*(0)=bt 1 =53.7770 and the system-wide optimal production rate (see Figure 4.12.) is u*(t)=0 for 36 1 .30 <≤t ; u*(t)=16 for 1862.83361.3 < ≤ t u*(t)=0 for 1001862.83 ≤ ≤ t . Next, to find the Stackelberg equilibrium, we employ Proposition 4.31. Specifically, we first solve equation (4.172), which results in − −+ − + − = h hh h wcb X s s ln )( )0( , (4.178) where −+ − + = hh h Rb . Next solving (4.173) we have 0)0( (0) )ln( )( )0( ]) 1 )[( )( s )0( 0 =+ + − −=+ + − − −+ −+ ∫ s s s s b X s s X b X hh Rcw X dt tR hh cw s . Substituting (4.178) into the last expression, we obtain the equation for the equilibrium wholesale price w s : () bR wc bRh wc cw s s s s ln ln ln − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − . (4.179) Consequently, plugging the data from Table 4.3 into (4.179) results in the Stackelberg wholesale price w s =4.769642<c=6. Substituting this value into (4.178) provides equilibrium advance order X s (0)= 22.055. Then t 1 =X(0)/b=1.378438, while t 2 = 83.1862 remains unchanged. Thus the equilibrium production rate is u s (t)=0 for 3784.10 < ≤ t ; u s (t)=16 for 1862.83378438.1 < ≤ t and u s (t)=0 for 1001862.83 ≤ ≤ t . Finally, the uniqueness of the found wholesale price (condition (4.174)) can be straightforwardly verified by differentiating expression () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − − − bRh wc cw bR wc s s s s ln ln ln from (4.179), which results in a negative expression. Comparing the system-wide optimal solution with the equilibrium solution, we observe that X s (0)< X * (0) and t 1 s < t 1 *, as stated in Proposition 4.30. Figure 4.12. System-wide optimal solution of Example 4.5 Coordination According to Proposition 4.30, system performance deteriorates if the firms are non-cooperative. This result is to be expected since balancing production with advance orders presents a case of vertical competition. bt 1 /R T t 2 t 1 c c b u ( t ) X ( 0 ) Y ( t ) )(t m ψ 256 4 MODELING IN AN INTERTEMPORAL FRAMEWORK 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 257 Comparing Propositions 4.27 and 4.31, it is readily seen that if the supplier sets the wholesale price equal to his marginal cost, then the advance order quantity becomes equal to the system-wide optimal quantity and the manu- facturer’s production rate converges to the system-wide optimal policy. Thus, the production balancing game is an example of when double marginalization not only decreases the order quantity but also affects production and invent- tory dynamics. Therefore, perfect coordination, is straightforwardly obtained by the two-part tariff. The supplier sets the wholesale price equal to his mar- ginal cost and makes a profit by choosing the appropriate fixed transforma- tion cost. In this section we consider a supply chain consisting of one producer (manu- facturer) and multiple suppliers of limited capacity. The suppliers (or service providers) are the leaders and the individual producer is a follower. To compensate for the suppliers’ power asymmetry and capacity restrictions, the producer may use a number of potential suppliers. Accordingly, in contrast to the balancing production game described in the previous section, this outsourcing game involves the decision to select a number of external suppliers (or service providers) for contingent future demands. Furthermore, the orders can be issued at any point of time rather than only once before the selling season starts. Similar to the production balancing problem and in contrast to the static outsourcing game considered in Chapter 2, there is no fixed or setup cost incurred when in-house production is launched. Nor are in-house production costs necessarily lower than the suppliers’ whole- sale prices. We assume that the producer maintains the contingent and bounded in- house capacity to produce, at a known cost, quantities over time, u(t). The demand consists of two components. One component reflects the regular, relatively low demand – of a known, steady level – which is traditionally met by in-house production and thus does not affect the optimization. As a result, this component (which was modeled in the production balancing problem) is not introduced explicitly in our model. It is accounted for by reduced (with respect to the regular demand consumption) in-house maxi- mum capacity U. The other demand component represents peak demands (a new feature compared to the production balancing problem) and is intro- duced explicitly as a random variable. For example, oil and gas contracts are often negotiated and in some cases implemented well before energy demands are revealed. In such cases, home-heating firms may tend to build-up supplies for the winter, preventing problems associated with high 4.4.2 THE DIFFERENTIAL OUTSOURCING GAME demands (whether expected or not). Similarly, universities build up Internet server capacities by entering early into contractual agreements with Internet suppliers, building thereby an optional capacity to meet potential and future demand for services. In some cases, the firms, in addition to their own limited capacity, use external suppliers, relying thereby on outside capacities to meet future demands for products and service. Some extreme cases involve, of course, an outsourcing problem which consists in transferring activities that were previously in-house to a third party (Gattorna 1988; La Londe and Cooper 1989; Razzaque and Sheng 1998). Since supply chain management frequently relies on sequential trans- missions of information (Malone 2002), the problem of production-supply outsourcing is set as a hierarchical game where suppliers are sequential leaders while the producer is a follower. In this framework, the producer uses a demand estimate for some future date T (for example, the demand for oil at the beginning of the winter), selects time-sensitive production and a supply policy which is time-consistent with the firm’s cost-minimizing objectives. The supply policy implies that given N potential suppliers, a subset of them is selected by the producer. Based on the producer’s rational outsourcing decisions, each supplier selects a wholesale price to offer while the producer orders a certain product quantity v n (t) from the nth, n=1,2,…,N supplier at the stated price. The producer’s problem Assume a firm producing a single product-type (commodity or service) to satisfy an exogenous demand, d, for the product-type at the end of a plan- ning horizon, T. Inventories (or service capacity) are stored until the selling season starts, i.e., until t=T: .)0(),()()( 0 XXtvtutX n n =+= ∑ & (4.180) where )(tX is the surplus level of inventories by time t; u(t) is the pro- ducer in-house production rate at time t; v n (t) is the supply rate of ordered and received from supplier n products; and 0 X is a constant. Both self- production and supplier capacity are bounded: 0 ≤ u(t) ≤ U, (4.181) 0 ≤ v n (t) ≤ V n , n=1, ,N (4.182) where U is the producer’s capacity and V n is the capacity of supplier n. The demand d at the end of planning period T is a random variable given by probability density and cumulative distribution )(D ϕ and 258 4 MODELING IN AN INTERTEMPORAL FRAMEWORK 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 259 ∫ =Φ a dDDa 0 )()( ϕ functions respectively. For each planning horizon T, there will be a realization D of d, which is known only at time T. Equation (4.180) presents the flow of products determined by production and supply rates from all engaged suppliers. The difference between the cumulative supply and production of the product and its demand, X(T)-D, is a surplus. If the demand exceeds the cumulative production and supplies, a penalty is paid. On the other hand, if X(T)-D>0, an overproduction cost is incurred at the end of the planning horizon. Furthermore, production costs are incurred at time t when the producer is not idle; holding costs are incurred when inven- tory levels are positive, 0)( >tX . Note that (4.180) implies that 0)( ≥tX always holds. The producer’s objective is to find such a production program, u(t), and supply schedule rates, v n (t) (outsourcing program), that satisfy constraints (4.180)- (4.182) while minimizing the following expected cost over the planning horizon T: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++ = ∫ ∑ T n nn vvu NNm vvu DTXPdtthXtcutvtwE wwuvvJ N N 0 , ,, 11 , ,, ))(()()()()(min ), ,,,, ,(min 1 1 , (4.183) where w n (t) is the supplier n unit wholesale price at time t; h is the invent- tory holding cost of one product per time unit; and a piece-wise linear cost function is used for the surplus/shortage costs, −−++ += ZpZpZP )( , (4.184) where },0max{ ZZ = + , },0max{ ZZ −= − , + p and − p are the costs of one product surplus and shortage respectively. Substituting (4. 184) into the objective (4.183), we have: ∫ ∑ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++= T n nnm dtthXtcutvtwJ 0 )()()()( dDDTXDpdDDDTXp TX TX )())(()())(( )( )( 0 ϕϕ ∫∫ ∞ −+ −+−+ min→ . (4.185) Objective (4.185) is subject to constraints (4.180)- (4.182), which together constitute a deterministic problem equivalent to the stochastic problem (4.180)-( 4.183). The supplier’s problem Let the suppliers be ranked and then numbered with respect to their marginal costs, s n , so that s n-1 <s n for n=2, ,N. The information on their wholesale prices, w n , is obtained sequentially, starting from the highest rank supplier n=N (see Kubler and Muller, 2004 for known examples and experimental evidence of sequential price setting). We assume that wholesale prices depend on the marginal costs and that the rank of a supplier, n (1<n<N), is not reconsidered if w n-1 ≤ w n ≤ w n+1 holds. Each supplier operates without inventories, supplying just-in-time at maximum rate V n . Therefore the supplier’s inventory dynamics is trivial. The nth supplier objective is: ∫ −= T nnnn w NN n s w dttvtwtvswwuvvJ nn 0 11 ))()()((min), ,,, ,(min , n=1,2, ,N, (4.186) where s n v n (t) – the supplier n expenditure rate and w n (t)v n (t) – the supplier n revenue rate from wholesales at time t. Naturally, for the supplier to be sustainable and maintain his ranking, we require that w n (t) ≥ s n , n=1,2, ,N and w n (t) ≤ w n+1 (t), n=1,2, ,N-1. (4.187) The centralized problem The centralized formulation excludes vertical competition by replacing the wholesale (transfer) prices with the corresponding marginal costs: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++ = ∫ ∑ T n nn vvu NN vvu DTXPdtthXtcutvtsE wwuvvJ N N 0 , ,, 11 , ,, ))(()()()()(min ), ,,,, ,(min 1 1 (4.188) subject to constraints (4.180)- (4.182). Similar to the producer’s problem, substituting (4.184) into the objective (4.188), we have: ∫ ∑ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++= T n nn dtthXtcutvtsJ 0 )()()()( dDDTXDpdDDDTXp TX TX )())(()())(( )( )( 0 ϕϕ ∫∫ ∞ −+ −+−+ min→ . (4.189) Objective (4.189) is subject to constraints (4.180)- (4.182), which together constitute a deterministic problem equivalent to the stochastic centralized problem (4.180) - (4.182) and (4.188). 260 4 MODELING IN AN INTERTEMPORAL FRAMEWORK 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 261 System-wide optimal solution The Hamiltonian for problem (4.189), (4.180)- (4.182) is as follows ∑ ∑ ++−−−= n n n nn tvtutthXtcutvstH ))()()(()()()()( ψ . (4.190) The co-state variable )(t ψ is the shadow price or margin gained by producing/outsourcing one more product unit at time t. According to the maximum principle, )(t ψ satisfies the following co-state equation: )( )( )( tX tH t ∂ ∂ −= ψ & , with transversality (boundary) condition: )( )())(()())(( )( )( )( 0 TX dDDTXDpdDDDTXp T TX TX ∂ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −+−∂ −= ∫∫ ∞ −+ ϕϕ ψ dDDpdDDp TX TX )()( )( )( 0 ϕϕ ∫∫ ∞ −+ +−= . That is, )( ht = ψ & , (4.191) )))((1())(()( TXpTXpT Φ−+Φ−= −+ ψ . (4.192) Rearranging only the decision variable-dependent terms of the Hamil- tonian, we obtain: )())(()( tucttH u − = ψ , (4.192) ( ) ∑ −= n nnv tvsttH )()()( ψ . (4.192) Thus, the optimal production and supply rates that maximize the Hamil- tonian are: ⎪ ⎩ ⎪ ⎨ ⎧ < =∈ > = .)( if ,0 ;)( if ],[0, ;)( if , )( ct ctUa ctU tu ψ ψ ψ (4.193) ⎪ ⎩ ⎪ ⎨ ⎧ < =∈ > = .)()( if ,0 ;)( if ],[0, ;)( if , )( tst stVb stV tv n n nn n ψ ψ ψ (4.194) An immediate insight from equations (4.193) and (4.194) is: (i) it is optimal to either not produce or produce only at maximum rate U, (ii) if it [...]... unit and time unit; U=10, V1=5, V2=15 product units per time unit; s1=0, s2=$0.8, T=10 time units; and A=200 product units This implies that the sequence of suppliers is n=1 (s1=0), in-house production (c=0.7) and n=2 (s2=0.8) Inserting these in the last equations we have: t0 274 4 MODELING IN AN INTERTEMPORAL FRAMEWORK 7.06w1 – (10-t1)=0, 0.48(w2-0.8) – (10-t1)+10(w2 – w1)=0; t1 =6. 65+5 .63 w1-4 .68 w2,... added supply capacity as well as goods that may be stored to meet prospective demands We assume that the producer lacks the capacity to meet peak demands at known specific times and therefore depends on suppliers This results in vertical outsourcing competition with a Stackelberg equilibrium solution different from that for the corresponding centralized supply chain The deterioration of the supply chain. .. best response does not change (see Proposition 4.38) This implies that the overall supply chain profit does not change as well Therefore the performance of the supply chain does not 4.5 INTERTEMPORAL CO-INVESTMENT IN SUPPLY CHAINS 277 deteriorate and the amendments in the wholesale prices are just internal transfers of the chain Thus, the two-part tariff in this intertemporal system has a “third part”... cost 4.5 INTERTEMPORAL CO-INVESTMENT IN SUPPLY CHAINS This section considers investment in a supply chain infrastructure using an inter-temporal model We assume that firms’ capital is essentially the supply chain s infrastructure As a result, firms’ policies consist in selecting an optimal level of employment as well as the level of co-investment in the supply chain infrastructure So far we have mainly... capital, I1n= F1 ( 1 ( K )) and I2n= F2 ( 2 ( K )) 4.5 INTERTEMPORAL CO-INVESTMENT IN SUPPLY CHAINS 287 1 (K ) The resultant feedback policies of the two firms, I1n= and 2c I 1 (1 ) 2 (K ) , is illustrated graphically in Figure 4. 16 The corresI2n= 2c I 2 (1 ) ponding evolution in time of the capital and investments for the case of K(0)=0.2< K =69 .91217939 are depicted in Figures 3 and 4, respectively Figure... t0, and then seek supplies at a maximum rate beginning with the least costly supplier, say n=1, starting from a point in time, t1 Next, he will seek supplies from the second less costly supplier and so on This type of supply is advantageous when the producer’s own capacity is relatively low while the expected demands are high and thus can be dealt with by just-in-time supply deliveries On the other hand,... equal to 0.1 Coordination Building-up a supply capacity to meet future and uncertain demands for products and services is a costly strategic issue which involves decisions being made in the course of time with the sole purpose of meeting a demand in real-time that may outstrip an available capacity Of course, firms may build-up their self-capacity and thereby meet demands when they occur, but such an approach... competing firms This section presents both openloop and feedback solutions for non-cooperating firms, as well as, longand short-run investment cooperation 4.5.1 THE DIFFERENTIAL INVESTMENT AND LABOR GAME Consider N-firms operating in a supply chain, each characterized by its output price pj(t) at time t, labor force Lj(t), investment policy Ij(t) and an f f aggregate production function Q=f(K,Lj),... current supply chain infrastructure capital, deteriorating at the rate The process of capital accumulation is then given by: 278 4 MODELING IN AN INTERTEMPORAL FRAMEWORK N dK (t ) K (t ) I j (t ) , K(0)=K0, I j (t ) 0, j 1, ; ; ; , N (4.210) dt j 1 The firms’ objective consists in maximizing the discounted profit by selecting an optimal employment policy on the one hand and a co-investment in supply chain. .. (t ) Ij e r jt rjt , (4.2 16) 0, otherwise with ij(t) determined by j (t ) C Ij ((1 )i j (t )) Ij e (4.217) Since the objective function (4.211) is concave and constraints (4.210) are linear, conditions (4.214)-(4.217) are necessary and sufficient for optimality and will be considered next in detail, providing specific insights regarding the investment process in supply chain infrastructure The N-Firms . production and supply rates from all engaged suppliers. The difference between the cumulative supply and production of the product and its demand, X(T)-D, is a surplus. If the demand exceeds. Londe and Cooper 1989; Razzaque and Sheng 1998). Since supply chain management frequently relies on sequential trans- missions of information (Malone 2002), the problem of production -supply. ])()[( 2 2 dth Rt bt hh T t −−+ −+ ∫ -c=0 , and thus 0)()ln(ln)( 22 2 =−−+−+− −−+ ctThtT R bt hh . (4.1 76) Using data from Table 4.3, equations (4.175) and (4.174) result in 4 41 ln4 36 1 + − =t =3. 361 0 06) 100(4)ln100(ln4 222 = − − + − −