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the replenishment period: the greater the replenishment period T, the larger the base-stock level, 0 )( > ∂ ∂ T Ts R nT . Proof: First note that neither J d nor its derivative, which is the left-hand side of equation (3.54), denoted by B, explicitly depends on s nT . Further- more, according to Proposition 3.3, no matter what base-stock level s nT we choose, the quantity that retailer n orders has the same distribution, which depends only on demand. Thus, given replenishment period T, f Q (.) does not depend on the base-stock policy s nT employed. This is to say that J d and B are independent on s nT . However, if B does not depend on s nT, then the distributor's best response T=T R (s nT ) does not depend on s nT , i.e, 0= ∂ ∂ nT R s T . The retailer's best response is determined with the standardized base- stock level, *)( nnn R nTnT sTTTss σµ +== (see Proposition 3.4), and thus, * 2 1 )( nnn R nT s T T Ts σµ += ∂ ∂ >0, as stated in the proposition. There are two important conclusions related to Proposition 3.5. The first conclusion is concerned with the supply chain's performance and thereby the corresponding centralized supply chain. If the supply chain is vertically integrated with one decision-maker responsible for setting both a replenish- ment period and base-stock level for each retailer, then the centralized objective function is a summation of all costs involved: ∑ += n n rd JJTJ )( . The distributor's cost J d is independent of the base-stock level, as shown in Proposition 3.5. Therefore, applying the first-order optimality condition to J(T) with respect to either n t q or nT s , we obtain equation (3.48). This implies that the condition for the Nash base-stock level is identical to the system-wide optimality condition. Next, to find the system-wide optimality condition for the replenishment period, we differentiate J(T) with respect to T, which, when taking into account (3.54) and (3.55), results in T TJ ∂ ∂ )( = ξξ ξξ df T TC TT TC Q )() ),(),( ( 2 − ∂ ∂ ∫ ∞ ∞− - ∫∫ ∞ − ∞− + Φ−−Φ−− * * ])()()()([ 2 ** 3 n n s nnn s n n dzzzshdzzzsh T σ =0. (3.56) 148 3 MODELING IN A MULTI-PERIOD FRAMEWORK 3.2 REPLENISHMENT GAME: CASE STUDIES 149 Comparing equations (3.56) and (3.54) we find the following property. Proposition 3.6. Let T B ∂ ∂ >0 and f nT (.) be the normal density function with mean Tµ n and standard deviation T  n . The system-wide optimal replenish- ment period and base-stock level are greater than the Nash replenishment period and base-stock level respectively. Proof: Let us substitute T in equation (3.56) with the Nash period T n = . Then the first term in (3.56) vanishes as it is identical to B from (3.54), while the second term is negative, i.e., T J ∂ ∂ )( β <0. Since both T B ∂ ∂ >0 and 0 2 2 > ∂ ∂ − T J n r (see Proposition 3.4), then T TJ ∂ ∂ )( increases if T increases and thus (3.56) holds only if the system-wide optimal period T*> . Finally, it is shown in Proposition 3.5, that 0> ∂ ∂ T s nT , i.e., if T>  , then s nT >s n . Proposition 3.6 sustains the fact that vertical competition causes the supply chain performance to deteriorate as discussed in Chapter 2. Similar to the double marginalization effect, this happens because the retailers ignore the distributor’s transportation cost by keeping lower, base-stock inventory levels. The distributor, on the other hand, ignores the retailers’ inventory costs when choosing the replenishment period. Figure 3.4 illustrates the effect of vertical competition on the supply chain. The second property, which is readily derived from Proposition 3.5, is related to the uniqueness of the Nash solution. Proposition 3.7. Let f nT (.) be the normal density function with mean Tµ n and standard deviation T  n . The Nash equilibrium (T n , s nT n ) determined by Theorem 3.2 is unique. Proof: The proof immediately follows from Proposition 3.5 and Theorem 3.2. Indeed the two best response curves T =T R (s nT ) and )(Tss R nTnT = can intersect only once if 0= ∂ ∂ nT R s T and 0 )( > ∂ ∂ T Ts R nT , i.e, a solution determined by Theorem 3.2 is unique. The transportation costs were obtained from a sample of 16 pharmacies which are being exclusively supplied every 14 days on a regular basis by Clalit's primary distribution center. The base-stock policy was determined according to service level definition and demand forecasts. Pharmacists place their orders using software that computes replenishment quantities for every item with respect to the base-stock level. The pharmacist electro- nically sends the completed order to the distribution center for packing and dispatching. If there is a shortage or expected shortage before the next planned delivery, the pharmacist can send an urgent order to be delivered not later than two working days from the time of the order. An external subcontractor (according to outsourcing agreement) delivers the orders to the pharmacies. The contractor schedules the appropriate vehicle (trucks in case of regular orders and mini-trucks for urgent orders) according to the supply plans for the following day. Delivery costs depend on the type of the vehicle used (track or mini-track) and the number of pharmacies to be supplied with the specific transport. To estimate the influence of a periodic review cycle on the transportation costs (planned and urgent deliveries) the replenishment period for the 16 pharmacies was changed from the original two weeks to three and four weeks. This resulted in a total of 18 replenishment cycles representing 34 working weeks. Monthly sales of the selected pharmacies varied from $50,000 to $136,000. Each order that was sent from a pharmacy was reported, and each transport, with every delivery on it, including invoices that were paid to the vehicle contractor, was reported. The data, processed with SPSS non-linear regression analysis, indicate that the resultant parameters of the transportation cost function are a=4463, b=0.0000163 while the average estimation error is less than 5%. Numerical Analysis The goal of our numerical analysis is to check whether this supply chain is predictable using equilibria and how it is affected by the distributor’s leader- ship. In other words, we compare the objective functions (3.43) and (3.45), as well as the effect on the overall supply chain (the sum of (3.43) and (3.45)). Specifically, with distributor leadership, its expected cost equation (3.52), is = 1d J ξξξα α dfC Q )(),( 1 ∫ ∞ ∞− , while without leadership it 3.2.3 EMPIRICAL RESULTS AND NUMERICAL ANALYSIS 150 3 MODELING IN A MULTI-PERIOD FRAMEWORK Empirical Results 3.2 REPLENISHMENT GAME: CASE STUDIES 151 is = 2d J ξξξβ β dfC Q )(),( 1 ∫ ∞ ∞− . Since  is found by minimizing the entire objective function J d1 , while  assumes the normal probability function independent on the period T, the distributor obviously is better off if he is the leader and therefore decides first rather than when the decision is made simultaneously (no leaders). Similarly, retailer n expected cost under the distributor leadership is = n r J 1 ∫∫ ∞ − ∞− + −−− α α αααα α n n s nnnn s nn dDDfDshdDDfDsh ])()()()([ 1 , while under no leadership it is = n r J 2 ∫∫ ∞ − ∞− + −−− β β ββββ β n n s nnnnn s n dDDfDshdDDfDsh ])()()()([ 1 . The numerical results of our empirical studies show that the current equilibrium of Clalit’s supply chain, which is an outcome of many adjust- ments it has undergone during many years of operations, is close to and positioned in between both the Stackelberg and Nash equilibria. This is in contrast to the skepticism of many practitioners who believe that a theoretical equilibrium is hardly attainable in real life. Specifically, the equilibrium replenishment period under equal competition is about 16 days; the current replenishment period is 14 days; and the equilibrium under the distributor’s leadership is 11 days. Figure 3.3 presents the equilibria over the distributor’s transportation cost function. Figure 3.3. The transportation cost as a function of T along with the Stackelberg, The Stackelberg equilibrium demonstrates the power the distributor can harness as a leader. The economic implication of harnessing the distributor’s C T current T  17 11  500 1000 Nash and current equilibrium replenishment periods power is about 20 NIS per day ($ 4 per day) for the sampled supply volumes. The annual significance, in terms of the overall supply chain, is 1.4 million NIS, or 14% of the total delivery costs. Interestingly enough, the current equilibrium is closer to the Nash replenishment period rather than to the Stackelberg which sustains Clalit’s managerial intuition that its distribution centers do not succeed in taking full advantage of their power over the pharmacies. Figure 3.4. Overall supply chain cost, total retailers cost, and distributor’s cost Figure 3.4 presents the results of the calculation for the supply chain as a whole, i.e., including the retailers’ inventory management costs and the distributor’s transportation costs. In Figure 3.4, the total costs for the Stackel- berg, current and Nash strategies as well as the system-wide optimal (global) solution appear as dots on the total cost curve. From this diagram it is easy to observe the effect of the total inventory-related cost on the entire system performance. Specifically, we can see that if the supply chain is vertically integrated or fully centralized and thus has a single decision-maker who is in charge of all managerial aspects, the system-wide optimal replenishment period is 18 days versus the current equilibrium of 14 days. The significance of this gap (which agrees with Proposition 3.6) is that more than 3 million NIS could be saved if the system were vertically integrated. If the distri- butor attempts to locally optimize (the Stackelberg strategy) this would lead to annual savings in transportation costs of only 1.4 million NIS. However, the significance of such an optimization for the supply chain as a whole is a loss of 8 million NIS. This is the price to be paid if the System-wide T Distributor’s Cost Retailers Cost C T Current T  17 11  500 1000 Total Cost 152 3 MODELING IN A MULTI-PERIOD FRAMEWORK REFERENCES 153 supply chain is either decentralized or operates as a decentralized system. Coordination This case study was motivated by increasingly high transportation costs incurred by a large health service provider which is part of a supply chain consisting of multiple retailers (pharmacies) and a distribution center. The costs are attributed to unlimited urgent orders that the retailers could place in the system. Management’s approach to handling this problem was to reduce the replenishment period or even transform the policy from periodic to continuous-time review. The latter option in the current conditions would simply imply daily (regular) product deliveries. As shown in Proposition 3.6, such an approach would only lead to further deterioration in supply chain performance due to the double marginalization effect inherent in vertical supply chains. This is also sustained by a numerical analysis of the equili- brium solutions for the case of a normal demand distribution. The analysis shows that if a distributor imposes his leadership on the supply chain, i.e., acts as the Stackelberg leader, then the replenishment equilibrium period is reduced. This makes it possible to cut high transportation costs. However, if instead of an imposed leadership on the supply chain, it is vertically integrated or the parties cooperate, then the potential savings in overall costs are much greater. In such a case, the system-wide optimal replenishment period must increase rather than decrease or transform into a continuous- review policy. Thus, in the short run, imposing leadership by reducing the replenishment period may cut high transportation costs. However, in the long-run, greater savings are possible if, for example, the vendor-managed inventory (VMI) approach is adopted by the retailers or imposed on the retailers by the health provider. In such a case, a distribution center will decide when and how to replenish inventories and the system will become vertically integrated with respect to transportation and inventory considera- tions. This illustrates the economic potential in cooperation and a total view of the whole supply chain. REFERENCES Anupinidi R, Bassok Y (1998) Approximations for multiproduct contracts with stochastic demands and business volume discounts: Single-supplier case. IIE Transactions 30: 723-734. Ballou RH (1992) Business Logistics Management, Englewood Cliffs, NJ, Prentice Hall. Bylka S (2005) Turnpike policies for periodic review inventory model with emergency orders. International Journal of Production Economics, Cachon GP (2001a) Managing a retailer’s shelf space, inventory, and transportation. Manufacturing and Service Operations Management 3: 211-229. Cachon GP (2001b) Stock wars: inventory competition in a two-echelon supply chain with multiple retailers. Operations Research 49: 658-674. Cachon G, Netessine S (2004) Game theory in Supply Chain Analysis in Handbook of Quantitative Supply Chain Analysis: Modeling in the eBusiness Era. edited by Simchi-Levi D, Wu SD, Shen Z-J, Kluwer. Chiang C (2003) Optimal replenishment for a periodic review inventory system with two supply modes. European journal of Operational Research 149: 229-244. Chiang C, Gutierrez GJ (1996) A periodic review inventory system with two supply modes. European journal of Operational Research 94: 389- 403. Debreu D (1952) A social equilibrium existence theorem, Proceedings of the National Academy of Science 38: 886-893. Flynn J, Garstka S (1997) The optimal review period in a dynamic inventory model. Operations Research 45: 736-750. Hause RG, Jamie KD (1981) Measuring the impact of alternative market classification systems in distribution planning, Journal of Business Logis- tics 2: 1-31. Kogan K, Hovav S, Perlman Y (2007) Equilibrium Replenishment in a Supply Chain with a Single Distributor and Multiple Retailers. Working paper, Bar-Ilan University. Leng M, Parlar M (2005) Game theoretic applications in supply chain management: a review. INFOR 43: 187-220. Simchi-Levi SD, Wu, Shen Z (2004) Handbook of quantitative Supply chain analysis: Modeling in the E-Business era, pp. 13-66. Sethi, SP, Yan H, Zhang H (2005) Inventory and Supply Chain Manage- ment with Forecast Updates. International Series in Operations Research & Management Science, Vol. 81, Springer Rao US (2003) Properties of the Periodic Review (R, T) Inventory Control Policy for Stationary, Stochastic Demand. MSOMS 5: 37-53. Teunter R, Vlachos D (2001) An inventory system with periodic regular review and flexible emergency review. IIE Transactions 33: 625-635. Veinott Jr.AF (1966) The status of mathematical inventory theory. Mana- 154 3 MODELING IN A MULTI-PERIOD FRAMEWORK 93: 357-373. gement Science 12: 745-777. REFERENCES 155 Wang H, Guo M, Efstathiou J (2004) A game theoretical cooperative mecha- nism design for two-echelon decentralized supply chain. European journal of operational research 157: 372-388. Whittmore AS, Saunders S (1977) Optimal inventory under stochastic demand with two supply options. SIAM Journal of Applied Mathematics, 32: 293–305. PART II INTERTEMPORAL SUPPLY CHAIN MANAGEMENT [...]... c(t))q(w(t) m(t),t)dt m, w (4. 7) 0 s.t (4. 2)- (4. 3) and (4. 5)- (4. 6) We henceforth omit independent variable t wherever the dependence on time is obvious System-wide optimal solution To evaluate the best possible performance of the supply chain, we first study the centralized problem by employing the maximum principle Specifically, the Hamiltonian for the problem (4. 2)- (4. 3), (4. 5)- (4. 6) and (4. 7) is H (t ) (w(t... w and m represent internal transfers of the supply chain Thus, the proper notation for the payoff function is J(p) rather than J(m,w) and the only optimality condition is, q( p*,t) (4. 12) q( p*,t) ( p * c ) 0 p More exactly, p* is the unique optimal price if it satisfies equation (4. 12) and p*(t) c(t), where c and are determined by (4. 11) and (4. 10) respectively Otherwise p*(t)=c(t) and the supply chain. .. (4. 2)- (4. 3), (4. 5) (4. 6) and (4. 7), if the supply chain is profitable, i.e., P>p>c, the myopic retail price will be greater and the myopic retailer’s order less than the system-wide optimal (centralized) price and order quantity respectively for 0 t T Proof: Comparing (2.7) and (4. 12) and employing superscript M for myopic solution we observe that q( p*, t ) q( p M , t ) q( p M , t ) ( p M c) =0, (4. 14) ... solve system (4. 20), which, for the linear demand, takes the following form: (4. 23) a b (c 2 m n ) bm n =0 , n n w =c+m + (4. 24) Using equation (4. 22) or, equivalently, by differentiating (4. 23) and (4. 24) we have & a & & wn m n 3b and a (T ) a (t ) a (T ) a(t ) m n (t ) m n (T ) , w n (t ) w n (T ) 3b 3b 3b 3b 4. 2 INTERTEMPORAL PRODUCTION/PRICING COMPETITION 173 In addition from (4. 23) we obtain,... possible performance of the supply chain, we first study the centralized problem by employing the maximum principle Specifically, the Hamiltonian for problem (4. 43), (4. 38) - (4. 42) is [q1 (t )( p(q1 (t ) q2 (t ), t ) c1 (t )) + q2 (t )( p(q1 (t ) q2 (t ), t ) c2 (t ))] (4. 44) 1 1 q1 (t ) 2 2 q 2 (t ) , where it is assumed that since the supply chain is profitable at each point of time and constraints p(q1+q2,t)... of equations (4. 47) (4. 48), q*= q1*=q2*, 1 2 , c1=c2=c, satisfies the following equation p(2q*,t) (4. 50) [ p(2q*,t) c] 2q * 0 Q 4. 2 INTERTEMPORAL PRODUCTION/PRICING COMPETITION 183 Define the maximum order quantity, Q' (t), so that the supply chain is sustainable, p(Q',t)=c(t) Then, differentiating the left-hand side of equation (4. 50) with respect to q, we obtain 2 2 p H p 4 4 q p>c, and there is a q ( p,... conditions of Propositions 4. 2 and 4. 4 hold Proposition 4. 5 In the differential pricing game, if the supply chain is proq ( p, t ) fitable, and there is a demand time pattern q(p,t) such that exists, t then the supplier’s wholesale price and the retailer’s margin 172 4 MODELING IN AN INTERTEMPORAL FRAMEWORK monotonically increase at the same rate as long as q ( p, t ) t 0 , and they q ( p, t ) q ( p,... differentiating (4. 44) with respect to q1(t) and q2(t), we obtain the following equations for an interior solution p(q1 (t) q2 (t),t) 0 , (4. 47) [ p(q1 (t) q2 (t),t) c1 (t)] (q1 (t) q2 (t)) 1 (t ) 1 q1 (t) p(q1 (t) q2 (t),t) 0 , (4. 48) [ p(q1 (t) q2 (t),t) c2 (t)] (q1 (t) q2 (t)) 2 (t) 2 q2 (t) where the production cost (state variable) for each supplier is found from (4. 38) and (4. 41) t c1 (t ) C t... price demand, q(p,t)=a(t)-bp, and the demand potential a(t) first being an arbitrary function of time Then we plot the solutions for specific supply chain parameters Example 4. 1 Let the demand be linear in price with time-dependent customer demand potential a(t), q(p,t)=a(t)-bp, a>bC Since the demand requirements, 2 q q b . m max () ∫ + T dtttmtwqtm 0 ),()()( (4. 4) s.t. m(t) ≥ 0, (4. 5) q(w(t)+m(t),t) ≥ 0. (4. 6) Formulations (4. 1)- (4. 6) assume non-cooperative behavior of the supply chain members which affects the overall supply chain performance Proposition 4. 1. In intertemporal centralized pricing (4. 2)- (4. 3), (4. 5)- (4. 6) and (4. 7), if the supply chain is profitable, i.e., P>p>c, the myopic retail price will be greater and the myopic. 0 ),( 2 ≤ ∂∂ ∂ tp tpq and if 0 ),( > ∂ ∂ t tpq , then 0 ),( 2 ≥ ∂∂ ∂ tp tpq . Proposition 4. 2. In intertemporal centralized pricing (4. 2)- (4. 3), (4. 5)- (4. 6) and (4. 7), if the supply chain is profitable,