304 5 AN INTERTEMPORAL FRAMEWORK WITH PERIODIC REVIEW + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ ∫ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ ∫ = ∫∫ ∫∫∫ + − + ++ dtdDDftDdssuXh dDDftDdssuXhhuJ tU t dssuX t t dssuX p t t })()()0( )()()0({),( 1111 0 )()0( 0 1111 0 )()0( 0 0 0 τ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∫ T dttXhE τ ))(( (5.7) Using Propositions 5.1-5.2, the last term in (5.7) can be readily determined as a sum of J det obtained for each particular case multiplied by the correspon- ding probability. Specifically, from Proposition 5.1 for X() ≥ 0, we have J det = ))( 2 ))((( 2 2 2 τττ −−− + T D TXh , when D 2 τ τ − < T X )( and J det = 2 2 2 2 )( D X h τ + when 2 )( T X − τ τ ∫ ≥− τ τ 0 1 0)( ddssu occurs with probability ∫ ∫ + τ τ 0 )()0( 0 111 )( dssuX dDDf , we conclude = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ ≥ T X dttXCE τ τ ))(( 0)( 2122 2 2 )( 0 2 )()0( 0 11 )())( 2 ))((([)( 0 dDDDfT D TXhDf T X dssuX τττ τ τ τ τ −−− ∫ ∫∫ − + + + 1212 2 2 )( 2 ])( 2 )( dDdDDDf D X h U T X τ τ τ ∫ − + . Similarly, from Proposition 5.2 we have for X()<0: = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ < T X dttXCE τ τ ))(( 0)( 2122 2 2 )( 0 )()0( 11 )( )(2 )( [)( 0 dDDDf DU X hDf T X U U dssuX − ∫ ∫∫ − + − + τ τ τ τ τ - += τ )0()( XX≤ D . Taking into account equation (5.1), , which by definition of the demand dis tribution f(.), 5.1 TWO-PERIOD INVENTORY OUTSOURCING 305 12122 2 2 )( ])())( 2 ))((( dDdDDDfT DU TXh U T X U τττ τ τ − − +−− ∫ − + − . Consequently, = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ T dttXCE τ ))(( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ ≥ T X dttXCE τ τ ))(( 0)( + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ < T X dttXCE τ τ ))(( 0)( . (5.8) Let us introduce a new variable, Y(t): )()( tutY = & , Y(0)=X(0), τ ≤ ≤ t0 . (5.9) Then, by substituting ∫ −+= τ ττ 0 1 )()0()( DdssuXX =Y()-D 1  into (5.8) and taking into account (5.9), the objective function (5.7) takes the following form () () ))((})()()()({),( 1111 )( 0 1111 )( 0 1 τϕ τ YdtdDDftDtYhdDDftDtYhhuJ U t tY t tY p +−−−= ∫∫∫ −++ ,(5.10) where = ))(( τ ϕ Y 2122 2 2 1 )( 0 2 )( 0 11 )())( 2 ))()((([)( 1 dDDDfT D TDYhDf T DY Y ττττ τ ττ τ τ −−−− ∫∫ − − + + 12122 2 2 1 )( 2 ])( 2 ))(( 1 dDdDDDf D DY h U T DY ττ τ ττ − ∫ − − + + 2122 2 2 1 )( 0 )( 1 )( )(2 ))(( [)( 1 dDDDf DU DY hDf T DY U U Y − − ∫∫ − − + − ττ τ ττ τ τ 12122 2 2 1 )( ])())( 2 ))()((( 1 dDdDDDfT DU TDYh U T DY U ττττ τ ττ − − +−−− ∫ − − + − . (5.11) We thus proved the following theorem. Theorem 5.1. Control u(t), which is optimal for deterministic problem (5.5) when Tt ≤≤ τ and for deterministic problem (5.2),(5.9)-(5.11) when τ <≤ t0 is optimal for stochastic problem (1)-(4) for Tt ≤ ≤ 0 . Problem (5.9)-(5.11) and (5.2) is a canonical, deterministic, optimal con- trol problem which can be studied with the aid of the maximum principle. Since all constraints are linear, the maximum principle-based optimality con- ditions are not only necessary but also sufficient if the objective function 306 5 AN INTERTEMPORAL FRAMEWORK WITH PERIODIC REVIEW (5.10) is convex. Moreover, this problem has a unique solution if the objective function is strictly convex, which evidently holds if 0 )( 1 > ∂ ∂ D DF and 0 ))(( 2 2 > ∂ ∂ Y Y τϕ . Accordingly, we next use the maximum principle by first constructing the Hamiltonian () () )()()()()()( 111 )( 111 )( 0 1 tutdDDftDtYhdDDftDtYhH U t tY t tY ψ +−+−−= ∫∫ −+ , (5.13) where the co-state variable )(t ψ is determined by the co-state differential equation )( )( tY H t ∂ ∂ −= ψ & = −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + h t tY Fhh )( )( 11 (5.14) with boundary (transversality) condition )( ))(( )( τ τϕ ψ Y Y T ∂ ∂ −= . (5.15) According to the maximum principle, the optimal control maximizes the Hamiltonian, that is, ⎪ ⎩ ⎪ ⎨ ⎧ =∈ < > = .0)( if ],,0[ ;0)( if ,0 ;0)( if , )( tUb t tU tu ψ ψ ψ (5.16) We resolve the ambiguity of the third condition from (5.16) in the follow- ing proposition. () −+ − + = hh h F 1 1 β and 0)( = t ψ at a measurable inter- val, τ . If  ≤ U, then Y(t)=t and β = )(tu for τ ∈ t . Proof: Differentiating the condition 0)( = t ψ over τ and taking into account (5.14), we find −+ − + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ hh h t tY F 1 1 )( . (5.17) Thus t tY )( = β and, therefore, Y(t)=t and β == )()( tutY & for τ ∈ t . We next introduce two switching points t a and t b which satisfy dth t ttUt Fhh aa t a ] )( )[( 11 −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+ + ∫ β τ )()( )( ))(( aa tUtY Y Y −+= ∂ ∂ −= τβτ τ τϕ , (5.18) (5.12) Proposition 5. 3. Let 5.1 TWO-PERIOD INVENTORY OUTSOURCING 307 dth t t Fhh b t b ])[( 11 −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∫ β τ b tY Y Y βτ τ τϕ = ∂ ∂ −= )( )( ))(( (5.19) respectively and assume that the production system has sufficient capacity, i.e.,  ≤ U. Similar to the optimal solution for the second period, an optimal solu- tion for the first period can be structured into a number of cases depending on the parameters of the production system. We study first two general cases of a non-negative initial inventory level which are described in the follow- ing two propositions (see Figures 5.1 - 5.4). Proposition 5.4. Let X(0) ≥ 0 and β τ < )0(X . If β )0(X t a > and 0 )( ))(( )()( < ∂ ∂ −+= aa tUtY Y Y τβτ τ τϕ , then the optimal production control is u(t)=0 for β )0( 0 X t <≤ , u(t)= for a tt X <≤ β )0( , and Utu = )( for τ ≤ ≤ tt a . Proof: Consider the following solution for the state variables u(t)=0, Y(t)=X(0) for β )0( 0 X t <≤ ; u(t)=, Y(t)= t for a tt X <≤ β )0( ; Utu = )( , Y(t)= t a +U(t-t a ) for τ ≤ ≤ tt a and co-state variables ∫ −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−= β ψ )0( 11 ] )0( )[()( X t dth t X Fhht for β )0( 0 X t <≤ ; 0)( = t ψ for a tt X <≤ β )0( ; ∫ −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+ += t t aa a dth t ttUt Fhht ] )( )[()( 11 β ψ for τ ≤ ≤ tt a . If this solution is feasible and satisfies optimality conditions (5.16), then it is optimal. The feasibility, β )0(X t a > and 0 )( ))(( )()( < ∂ ∂ −+= aa tUtY Y Y τβτ τ τϕ , is imposed by the statement of this proposition. The optimality conditions are verified straightforwardly. Specifically, it is easy to observe that from β τ < )0(X and 308 5 AN INTERTEMPORAL FRAMEWORK WITH PERIODIC REVIEW () −+ − + = hh h F 1 1 β , we have 0 )0( )( 11 <− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −−+ h t X Fhh for β )0( 0 X t <≤ and thus ∫ −−+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−= β ψ )0( 11 ] )0( )[()( X t d t h t X Fhht >0, i.e., the second optimality condition, − a dth ] >0 for t a a tt X <≤ β )0( and therefore u(t)=. Note that the optimal control described in Proposition 5.4 consists of three different trajectories. If the feasibility requirement 0 )( ))(( )()( < ∂ ∂ −+= aa tUtY Y Y τβτ τ τϕ is not met, then the third trajectory implies no-production instead of pro- duction at maximum rate as stated in the following proposition. Proposition 5.5. Let X(0) ≥ 0 and β τ < )0(X . If β )0(X t b > and 0 )( ))(( )( > ∂ ∂ = b tY Y Y βτ τ τϕ , then the optimal production control is u(t)=0 for β )0( 0 X t <≤ , u(t)= for b tt X <≤ β )0( , and u(t)=0 for τ ≤ ≤ tt b . Similarly we can state the two general cases when the initial inventory level is negative as shown in the following two propositions (see Figures 5.3-5.4).Proofs for Proposition 5.5 as well as for the next two propositions are similar and therefore omitted. Of course, special cases readily emanate from Propositions 5.4 and 5.5 when one of the switching points or both vanish. Proposition 5.6. Let X(0)<0 and )()0( β τ − − > UX . If β − − > U X t a )0( and 0 )( ))(( )()( < ∂ ∂ −+= aa tUtY Y Y τβτ τ τϕ , then the optimal production control is ∫ −+ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+ += t t aa t ttUt Fhht )( )[()( 11 β ψ u(t)=0, from (5.16) holds. Similarly, < t ≤ τ and thus the first optimality condition, u(t)=U, from (5.16) holds. The third condition from (5.16) is explicit, ψ (t) = 0 for 5.1 TWO-PERIOD INVENTORY OUTSOURCING 309 u(t)=U for β − − <≤ U X t )0( 0 , u(t)= for a tt U X <≤ − − β )0( , and Utu = )( for τ ≤ ≤ tt a . Figure 5.1. Optimal control over the first period for X(0)>0 when 0 )( ))(( < ∂ ∂ τ τϕ Y Y Proposition 5.7. Let X(0)<0 and )()0( β τ − − > UX . If β − − > U X t b )0( and 0 )( ))(( )( > ∂ ∂ = b tY Y Y βτ τ τϕ , then the optimal production control is u(t)=U for β − − <≤ U X t )0( 0 , u(t)= for b tt U X <≤ − − β )0( , and u(t)=0 for τ ≤ ≤ tt b . U  t  X(0)/ t a u(t) Y(t) )(t ψ a 310 5 AN INTERTEMPORAL FRAMEWORK WITH PERIODIC REVIEW Figure 5.2. Optimal control over the first period for X(0)>0 when 0 )( ))(( > ∂ ∂ τ τϕ Y Y Figure 5.3. Optimal control over the first period for X(0)<0 when 0 )( ))(( < ∂ ∂ τ τϕ Y Y u(t) )(t ψ Y(t)  t b X(0)/ b U  t  -X(0)/(U-) t a u(t) Y(t) )(t ψ a 5.1 TWO-PERIOD INVENTORY OUTSOURCING 311 Figure 5.4. Optimal control over the first period for X(0)<0 when 0 )( ))(( > ∂ ∂ τ τϕ Y Y Example 5.1 In this example we derive explicit equations for )( ))(( )( τ τϕ ψ Y Y T ∂ ∂ −= and ))(( τ ϕ Y . The example is motivated by the goods which have a short life- cycle during which it is likely that the demand has a single realization, i.e., D 1 =D 2 , which is estimated by time  of inventory review. Therefore, after deriving a general optimal solution, we focus on the example which is based on the conditional density function f 2 (D 2 |D 1 )=(D 2 -D 1 ), where (D 2 -D 1 ) is a Dirac function. When substituting this conditional dis- tribution function into equation (5.11), ))(( τ ϕ Y simplifies to:  t b -X(0)/(U-)  u(t) )(t ψ Y(t) 312 5 AN INTERTEMPORAL FRAMEWORK WITH PERIODIC REVIEW =))(( τ ϕ Y 1 2 1 12 )( 0 11 ))( 2 ))()((()( dDT D TDYhDf T Y ττττ τ −−−− + ∫ + 1 1 2 1 )( )( 211 2 ))(( )( dD D DY hDf Y T Y ττ τ τ τ − ∫ + + 1 1 2 1 )( )( 11 )(2 ))(( )( dD DU DY hDf T T U T Y Y − − ∫ − + − ττ ττ τ τ 1 2 1 1 )( 11 ))( 2 ))()((()( dDT DU TDYhDf U T T U T Y ττττ τ τ − − +−−− ∫ − + − . (5.20) This affects only boundary condition (5.15) as follows )( ))(( )( τ τϕ ψ Y Y T ∂ ∂ −= =( 2 2 1 )( 2 )( ) )( ( τ τ τ − + T T Y T Y fh + 1 )( 0 11 )()( dDThDf T Y τ τ − + ∫ - - 2 2 1 )( 2 )( ) )( ( τ ττ − + T T Y T Y fh + 1 1 1 )( )( 11 ))(( )( dD D DY hDf Y T Y ττ τ τ τ − ∫ + + + ))(( 2 )( ) )( ( 2 2 1 ττ τ τ τ YU T T T T U T Y fh − − − + − + 1 1 1 )( )( 11 )( )( dD DU DY hDf T T U T Y Y − − ∫ − + − ττ ττ τ τ ))(( 2 )( ) )( ( 2 2 1 ττ τ τ τ YU T T T T U T Y fh − − − +− − 1 )( 11 )()( dDThDf U T T U T Y τ ττ −− ∫ − + − ). That is, ) )( ()()( 12 T Y FThT τ τψ −− + 1 1 1 )( )( 112 )( )( dD D DY Dfh Y T Y ττ τ τ τ − − ∫ + - 1 1 1 )( )( 11 )( )( dD DU DY Dfh T T U T Y Y − − − ∫ − + − ττ τ τ τ τ )) )( (1)(( 1 T T U T Y FTh τ τ τ − +−−+ − . (5.21) The distributor handles at each time point t all inventory-related operations. For each time unit of this service, the distributor charges the producer per Distributor’s Problem 5.1 TWO-PERIOD INVENTORY OUTSOURCING 313 item holding cost, h + (t).The distributor’s goal is to minimize his expected cost (or, which is the same in this case, to maximize his profit): ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −= ∫ +++ ++ T h d h dttXthmEhuJ 0 )())((min),(min (5.22) s.t. (5.1)-(5.2) Mthm ≤≤ + )( , (5.23) where m is the distributor’s marginal cost and M is the maximum inventory holding cost so that the producer will not explore the market for another competing distributor. Using the same approach as in the previous section, we first analyze the deterministic part of the problem, i.e., when there is only one period to go. Consider time interval [, T]. At this interval, problem (5.1)-(5.2), (5.22) and (5.23) takes the following deterministic form: min)()( 2det →−= ∫ ++ T dttXhmI τ (5.24) s.t. (5.1),(5.2) and (5.23). An optimal solution for problem (5.24) and thus the best distributor’s response to any producer’s production and inventory policy is straightfor- ward. Indeed, the dynamic equation (5.1) depends neither on h + (t) explic- itly, nor the switching points in Propositions 5.1 and 5.2. As a result, the optimal solution for this problem is trivial, which is to charge the producer for handling the inventories as much as possible, Mh = + 2 . Similar to Pro- positions 5.1 and 5.2, the objective function value is then I det =0 if 0)( ≤ τ X ; I det = ))( 2 ))(()(( 2 2 2 τττ −−−− + T D TXhm , if )()( 2 τ τ − > TDX , I det = 2 2 2 2 )( )( D X hm τ + − if )()(0 2 τ τ − ≤ < TDX . (5.25) In what follows, we summarize these observations with the aid of the fact that the optimal production policy, u(t), from Propositions 5.1 and 5.2 can be presented in a time-dependant (integral) feedback form, *(X(),t). Deterministic Component of the Problem [...]... Periodic Demand Update EJOR, to appear Kogan K, Lou S, Tapiero C (20 07) Supply Chain With Inventory Review and Dependent Demand Distributions: Dynamic Inventory Outsourcing, Working paper Kogan K, Shu C, Perkins J (2004) Optimal Finite-Horizon Production Control in a Defect-Prone environment IEEE Transactions on Automatic Control 49(10): 179 5-1800 6 SUSTAINABLE COLLABORATION IN SUPPLY CHAINS The intertemporal... COLLABORATION IN SUPPLY CHAINS The intertemporal supply chain models presented in Chapters 4 and 5 focus on inventory, production and pricing relationships between a supplier and a retailer according to different types of demands However, in reality, there are many other factors that affect members of a supply chain For example, uncertainty can be associated not only with demands but also with production yields... five-echelon supply chain system Parameters Manufacturer 1 Manufacturer 2 Manufacturer 3 Manufacturer 4 Manufacturer 5 Ui 3 2 4 7 6 ci 2.5 0.5 3 2 1 334 6 SUSTAINABLE COLLABORATION IN SUPPLY CHAINS X 5 (t ) 5 (t ) c5 c2 u 5 (t ) X 3 (t ), X 4 (t ) 3 (t ), 4 (t ) u 3 (t ), u 4 (t ) X 2 (t ) 2 (t ) u 2 (t ) X 1 (t ) 1 (t ) u1 (t ) t1 t2 Figure 6.1 Optimal control of the five-echelon supply chain t 6.2 SUPPLY CHAINS... multi-echelon supply chain is reduced to determining optimal production order quantity; ranking manufacturers and warehouses; calculating a limited number of switching time points; and assigning production rates over the switching points with respect to the manufacturer and warehouse ranks 6.2 SUPPLY CHAINS WITH LIMITED RESOURCES Sharing resources is common in industrial applications and can involve energy and. .. two-part tariff, the supply chain becomes perfectly coordinated during the first period as well REFERENCES Fisher M, Raman A, McClelland A (2000) Rocket science retailing is almost here-Are you ready? Harvard Business Review 78 : 115-124 Fisher M, Raman A (1996) Reducing the cost of demand uncertainty through accurate response to early sales Operations Research 44: 87- 99 Kogan K, Herbon A (20 07) Optimal Inventory... decision-making We are thus interested in reexamining system-wide optimal production and inventory policies to account for additional constraints and conditions imposed on supply chains Special attention is paid here to production control of multiple manufacturers sharing limited supply chain resources 6.1 MULTI-ECHELON SUPPLY CHAINS WITH UNCERTAINTY This section addresses a multi-echelon, continuous-time... overall supply chain is unaffected However, during the maintenance service of the resource, the production of the manufacturers is interrupted which means that the system dynamics are affected 6.2.1 PRODUCTION CONTROL OF PARALLEL PRODUCERS WITH RANDOM DEMANDS FOR PRODUCTS In this section we consider a horizontal supply chain operating under uncertain demands subject to a renewable resource The demand for... as energy, raw materials, budget and logistics infrastructure, which can be limited or delivered by a supplier of bounded capacity Furthermore, the firms may choose to expand their outsourcing activities to include repair and maintenance operations rather than just production or inventory In this chapter we extend our attention to broader issues and consider supply chains in which the parties collaborate... the leader, the optimal distributor’s response is again to charge the producer as much as possible for handling his inventories, h1 M 2 F ( D) (Y ( )) 0, 1 0, 2 D Y and that the supply chain has no leader or the producer is the leader, then the production policy determined by Propositions 5.4-5 .7 and inventory holding price h1 M constitute a unique Nash equilibrium as well as a Stackelberg equilibrium... (t) 1 (t) i 1 6.1 MULTI-ECHELON SUPPLY CHAINS WITH UNCERTAINTY 3 27 Since this term is linear in u i (t ) , it can be easily verified that the optimal production rate that maximizes H(t) is Ui , if ui (t) i (t) w [0,Ui ],if 0, if i (t) i 1 i i 1 (t) (t) 0, i >1 and i (t) >0, i =1(production regime-PR); i 1 (t) 0, i >1 and i (t) =0, i =1(singular regime-SR) (t) 0, i >1 and i (t) 0, i =1 (idle regime-IR) . 49(10): 179 5-1800. Coordination 6 SUSTAINABLE COLLABORATION IN SUPPLY CHAINS The intertemporal supply chain models presented in Chapters 4 and 5 focus on inventory, production and pricing. Inventory Control under Periodic Demand Update. EJOR, to appear. Kogan K, Lou S, Tapiero C (20 07) Supply Chain With Inventory Review and Dependent Demand Distributions: Dynamic Inventory Outsourcing,. con- straints and conditions imposed on supply chains. Special attention is paid here to production control of multiple manufacturers sharing limited sup- ply chain resources. 6.1 MULTI-ECHELON SUPPLY CHAINS