Stock Replenishment and Shipment Scheduling for Vendor-Managed Inventory Systems

1. Stock Replenishment and Shipment Scheduling for Vendor-Managed Inventory Systems Sila Cetinkaya , Chung-Yee Lee Industrial Engineering Department, Texas A&M University Management Science 2000, Vol46, No 2 발 표 자 : 정 성 원

2. Related article I • Designing and Managing the Supply Chain • David Simchi-Levi , Philp Kaminsky, Edith Simchi-Levi • Irwin McGraw-Hill • Contents • Introduction to Supply Chain Management • Logistics Network Configuration • Inventory Management and Risk Pooling • The Value of Information • Distribution Strategies • International Issues in Supply Chain Management • Coordinated Product and Supply Chain Design • Information Technology for Supply Chain Management • Decision-Support Systems for Supply Chain Management

3. Related article II • A Decision Support System for Vendor Managed Inventory • Dale D. Achabal, Shelby H. Mcintyre (Santa Clara University) • Journal 0f Retailing, Vol 76 pp. 430-454 • Contents • VMI Background and benefits (Vendor and Retailer) • Model Development • Assessing the benefits

4. Introduction I • VMI (Vendor Managed Inventory) • The vendor assumes responsibility for managing inventory • Inventory information at retailer accessible to the supplier Information Inventory Control What is the advantage of VMI ? Reduce the bullwhip effect

5. Introduction II • The advantage of VMI in view of this paper Traditional relationship Order Batch ordering – Transportation cost , service level Local Decision Transportation setup cost at each retailer Delivery VMI relationship Information Allocation – Transportation cost , service level Global Decision Transportation setup cost at each region Allocation Region2 Region1

6. Introduction III • Decision of vendor in VMI Model • When to dispatch ? • How large the dispatch quantity should be ? One decision determines the other • Consolidation policy (for scale economies) • A quantity-based policy ships : a delivery time is r.v. • The time-based policy ships : a quantity is r.v. • The problem interest in this study • The case where the vendor adopts a time-based policy • The optimal replenishment quantity for vendor • The shipment frequency T for retailer Common between 3PL and their partnering vendor

7. Problem Characteristics I R1 Lead time is negligible Model R3 Reorder point is zero at both side Retailer doesn’t want to hold inventory Retailer wants to hold inventory [1,0] R2 V M R5 M : Manufacturer V : Vendor Ri : Retailer i For the consolidation policy of vendor, Retailer inventory level will be [1,-k] R4 Demand from retailer is random Cost variable AR : Fixed cost of replenishment inventory CR : Unit procurement cost h : Inventory carrying cost per unit per unit time AD : Fixed cost of dispatching CD : Unit transportations cost W : Customer waiting cost per unit per unit time In classical inventory models, AD and CD are sunk costs and needed not be modeled

8. Problem Characteristics II • Assumption • Identical retailer • Lead time is negligible at both side (vendor, retailer) • Reorder point is zero at both side • Retailer doesn’t want to hold many inventories • Retailer wants to maintain inventory level [0,1] • Demand occurs randomly *이 논문의 경우 Inventory를 라면 한 박스 등으로 생각하면 이해하는데 도움이 됨

9. Problem Characteristics III • A realization of the Demand Process R1 R3 R2 V inventory information R5 R4 x1 Xn : inter arrival time between demands at vendor (Assume i.i.d) Sn : the time when the nth order occurs N(t) : a value that represent the number of demand orders placed by t N(t) = sup{n:Sn<t}

10. Problem Characteristics IV • A realization of the Inventory Process at vendor 1) I(t) and L(t) are observed 2) The vendor employs (s,S) policy : (0,Q) – lead time 0 3) When inventory level is 0 , vendor order Z(t) 4) Upon the receipt of Z(t), a load L(t) units is dispatched instantaneously I(t) : the inventory level at time t L(t): the size of the accumulated load (number of outstanding demands at time epoch t) Q : the inventory level immediately after a replenishment order arrivals Z(t) : the replenishment order quantity (t) : The units of inventory after a new shipment-consolidation cycle begins

11. Problem Characteristics V • A Example of the inventory process at vendor • we assume Q = 4 • L(T) in the first consolidation cycle is 3 • I(T)=4>L(T) SO Z(T)=0 • the second consolidation cycle begins with • L(T) in the second consolidation cycle is 4 • I(T)=1<L(T) SO Z(T)=Q+L(T)-I(T)=4+4+-1=7 • The third consolidation cycle begins with 4 T 2T

12. E[Replenishment Cycle Cost] C(Q,T) = E[Replenishment Cycle Length] Problem Formulation I • Objective function • Min C(Q,T) • C(Q,T) T V M Q

13. Problem Formulation II • Replenish Cycle length • K : a r.v representing the number of dispatch decisions within a given inventory cycle • E[Replenish Cycle length] = E[K]T • Replenish Cycle Cost • Inventory replenishment cost (to vendor , 1 during cycle) • Delivery cost (to vendor , 1 during cycle) • Inventory carrying costs(to retailer , k during cycle) • Customer waiting costs (to retailer, k during cycle)

14. Problem Formulation III • Expected Inventory replenishment cost per rep cycle • AR+CRE[K]E[N(t)] • Expected Inventory delivery cost per rep cycle • ADE[K] + CDE[K]E[N(t)] • Expected Inventory carrying cost at vendor per rep cycle • Expected customer Waiting costs per rep cycle • E[K]E[waiting costs per consolidation cycle] = W(T)

15. Analysis • Cost function • Assume N(T) follow a Poisson process Inventory carrying cost Customer waiting cost Inventory replenishment cost Inventory delivery cost

16. Solution I • The necessary condition for optimal solution • C(Q,T) is convex function in T but not in Q • Let C(Q,T) reduce to C(Q) • Put value T* to C(Q,T)

17. Solution II • The result of C(Q) • Let us define C1(Q), C2(Q) ,

18. Solution III • Theorem 1 • If there exits a solution for C(Q) over [1, ], then it is unique • Theorem 2 • Provided that (Q*,T*) is the unique solution, otherwise Q*=1 and T*=

19. Solution III • The meaning of solution • Provided that (Q*,T*) is the unique solution, otherwise Q*=1 and T*= • Unless h is extremely large compared to other cost parameters, the inequality in equation 1 holds • If h is extremely high, then a target inventory level of 0 1 Make sense!!

20. Numerical Illustration • The result of sensitivity analysis • As AR increases, the resulting Q* and cost values increase; • As increases, the corresponding Q* increases whereas the corresponding T* decreasing • As h increases, the resulting Q* and T* decreases • As AD increases, the corresponding T* value increases • As w increases T* decreases if

21. Discussion • Model • How do you think about the model? • The quantity based policy • Assumption • Lead time is negligible • The concept of safety stock