An Analytical Model for a Two Echelon (S-1, S) Inventory System With Full Lostsales

1. An Analytical Model for a Two Echelon (S-1, S) Inventory System With Full Lostsales İsmail KIRCI ikirci@dho.edu.tr Assoc.Prof.Dr. Emre BERK, Bilkent University Asst.Prof.Dr. A.Özgür TOY, Turkish Naval Academy İsmail KIRCI, Turkish Naval Academy

2. Agenda • Introduction • Literature Review • The Model • Assumptions • Modelling Approach • Steady State Anaysis • Operating Characteristics of the System • Numerical Studies • Conclusion

3. Introduction Multi-echelon Inventory Model: Inventory models comprising of several layers or stages of stock keeping units (SKU) are often referred as multi-echelon inventory models, where an echelon is a stage in the supply chain. RETAILER WAREHOUSE CUSTOMER RETAILER FACTORY WAREHOUSE RETAILER

4. Introduction (S-1, S) Inventory Policy: (S-1, S) A special case of (s,S) continuous review inventory policy S : Order Up to Level • Each demand (customer) triggers an order • Optimal policy for “high item value and low demand rate” order L Demand (customer) Retailer (S-1,S) Warehouse item

5. Introduction Ample Supplier L1 L0 Warehouse Retailer (S1-1, S1) (S0-1, S0) Two echelon inventory system with full lostsales: Customer Example : A rarely requested and expensive medicine The problem is to determine optimal inventory parameters ( S0* and S1* ) for the warehouse and the retailer that minimize total cost.

6. Literature Review Single-echelon (S-1,S) Studies : Multi-echelon (S-1,S) Studies :

7. THE MODEL • Assumptions • Modelling Approach • Steady State Analysis • Approximation • Operating Characteristics of the System

8. Ample Supplier L1 L0 Warehouse Retailer (S1-1, S1) (S0-1, S0) The Model Assumptions: Customer (Arrivals ~ poisson (l)) • (S-1, S)continous review inventory policy at each echelon • Poisson Arrivals • Full backorder at the Warehouse • Full lost sale at the Retailer • FIFO discipline • Holding cost at the retailer and warehouse • Lost sale cost at the Retailer

9. The Model The objective is to determine optimal inventory parameters of the warehouse and the retailer, (S0* , S1* ), that minimize the overall cost of the system Total cost=Holding cost + Lostsale cost • Minimize long run average cost per unit time (expected cost rate) E[TC] = h0 ∙ E[OH0]+ h1 ∙E[OH1 ] + p∙ E[LS1] h1: Holding cost of the retailer (per unit / per unit time) h0: Holding cost of the warehouse(per unit / per unit time) p : Lostsales cost (per unit)

10. The Model B C D E F On hand On hand On order On order (S-1, S) Inventory Policy • Inventory position is always “S” inventory position = On hand + on order – backorders Example: Suppose “S=6” Before Demand: On hand=3, on order=3, inventory position =6 Inventory Position of the Retailer A G After Demand: On hand=2, on order=4, inventory position =6

11. The Model Modelling Approach L1 L0 Warehouse Retailer (S1-1, S1) (S0-1, S0) D E F G A B C Before Demand F G H E B C D After Demand On-hand On-order

12. The Model Modelling Approach Define (S1+S0) dimensional stochastic process; d(t) ={d1(t), d2(t),........, dS1(t), dS1+1(t),........... dS1+S0-1(t), dS1+ S0(t) } Where di(t) denotes the age of the itholdest item in the system at time t d1(t) ≥ d2(t) ≥ ............. ≥dS1+S0-1(t) ≥ dS1+ S0(t) . . . . . . dS1+2 . . . . . . dS1+S0-1 d1 d2 d3 dS1 dS1+1 dS1+S0 Items of retailer Items of warehouse d1(t)= the age of the oldest item at the retailer dS1(t)= the age of the youngest item at the retailer dS1+1(t)= the age of the oldest item at the warehouse dS1+ S0(t) = the age of the youngest item at the warehouse

13. The Model L1 L0 Warehouse Retailer Demand (Arrivals ~ poisson (l) (S0-1, S0) (S1-1, S1) At a Demand Arrival: Customer New order . . . . . . . . . . . . ds1+1 ds1+2 ds1+3 ds1+s0 d1 d2 d3 ds1

14. The Model Steady State Analysis: p(t, x1, x2,...... , xS1+ S0-1 , xS1+ S0) The probability that system is in state {x1, x2,... , xS1+ S0-1 , xS1+ S0} at time t. Step 1:Find probability that system is in state {x1, x2,.....,xS1+ S0-1, xS1+ S0} at steady state (at any instance of time) → Stationary distribution Step 2: Use stationary distribution to find steady state expected cost rate.

15. The Model Steady State Analysis: The items in the system age uniformly in (t, t+dt ), as long as no effective arrival occurs Effective arrival rate: • Letwdenote waiting time of a customer for the next available item Effective arrival rate is a function of w

16. The Model Steady State Analysis: Effective arrival rate: Since w depends on states of x1 and xS0+1, we conclude that effective arrival rate is a function of (x1, xS0+1 )

17. The Model Exact Effective Arrival Rate: {x1 ≥ x2 ≥...... ≥xS0 ≥xS0+1 ≥ xS0+2 ≥ ....... ≥ xS1+ S0 } Approximation: • Define effective arrival rate as a function of x1 {x1 ≥ x2 ≥...... ≥xS0 ≥xS0+1 ≥ xS0+2 ≥ ....... ≥ xS1+ S0 }

18. The Model Steady State Analysis: The probability that the system is in state {x1, x2,... , xS1+ S0-1 , xS1+Ss0} at steady state

19. The Model Step 1:Find probability that system is in state {x1, x2,.....,xS1+ S0-1, xS1+ S0} at steady state (at any instance of time) → Stationary distribution Step 2: Use stationary distribution to find steady state expected cost rate. Expected Cost Rate: Total cost=Holding cost + Lostsale cost E[TC] = h0 ∙ E[OH0]+ h1 ∙E[OH1]+ p∙ E[LS1] E[OH0]: Expected on hand inventory at the warehouse E[OH1]:Expected on hand inventory at the retailer E[LS1] :Expected Lostsale

20. The Model Expressions For Cost Rate Components: Litttle’s law E[Length of queue] = λ.E[Wait of a costomer]

21. The Model As a result: E[TC] = h0 ∙ E[OH0]+ h1 ∙E[OH1]+ p∙ E[LS1] Step 1:Find probability that system is in state {x1, x2,.....,xS1+ S0-1, xS1+ S0} at steady state (at any instance of time) → Stationary distribution Step 2: Use stationary distribution to find steady state expected cost rate.

22. Numerical Study We compare the results of our analytical model with simulation results and evaluate the effectiveness of our approximation. Parameter Set: • Performance of the approximation when two echelon system reduces to single echelon • Optimization Study

23. Numerical Study 1) Performance of the approximation when two echelon system reduces to single echelon • Customers see only lead time between warehouse and ample supplier • Single echelon model where lead time is L0and inventory level is (S1 + S0) L1=0 K.Moinzadeh (1989) “ Operating characteristics of the (S - 1,S) inventory system with partial backorders and constant resupply times” L1=0 l=2 t =0

24. Numerical Study 2) Optimization Study • Effectiveness of the approximation to evaluate optimum stocking levels (S1* and S0*) • Comparison with simulation results Search Method S0 S1 Exhaustive Golden Section

25. 2) Optimization Study • Performance of the approximation with respect to h1/p % D=(Aprx. Cost – Sim Cost)*100 / Sim Cost

26. Numerical Study 2) Optimization Study Performance of the approximation with respect to l∙(L1+L0)

27. Numerical Study 2) Optimization Study Comparison of the approximation with Fullbackorder Aproach: % D1=(Aprx.-Sim)*100/Sim % D2=(F.B.Aprc.-Sim)*100/Sim

28. Conclusion As a result: • The approximation gives effective results with acceptable errors. • The approximation error increases as l increases. *→No Consistency • Future Study: • Considering multiple retailers • Considering recoverable or perishable items

29. QUESTIONS?