A DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION

Transcription

1 A DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION A Class Project for MATH 768: Applied Stochastic Processes Fall 2014 By Chudamani Poudyal & Dol Nath Khanal Department of Mathematical Sciences UW-Milwaukee Supervised By Prof. Dr. Richard Stockbridge

2 Abstract. This project studies the optimal design of an inventory system with one way substitution in which a high quality items fulfills its own demand and low quality items acts as backup safety stock. We use a discrete time Markov chain model to analyze the effect of one way substitution in periodic inventory system with (R, s, S) order policy assuming back-orders, zero restocking leadtime, and correlated demand. In more details, the optimal inventory control parameter (S and s) are determined in view of minimizing the expected total cost period (i.e., sum of inventory holding cost, purchasing cost, back-order cost, and adjustment cost). Key: Inventory management, one way substitution. 1. Introduction In the this project, a discrete time Markov chain (DTMC) is developed to investigate the optimal design of a two different products inventory system with one-way substation in multi-period setting. The goal of this project is to study the effect of one-way substitution in (R, s, S) inventory system assuming zero restocking lead time, back orders and correlated demand. More specifically, we would like to determine the optimal inventory control parameters for three strategies (one-way substitution, no pooling, and full pooling), in view of minimizing the expected total cost per review period. 2. Problem Description Consider two different product types (Product 1 and Product 2) as shown in the figure above. The demand d i for a specific product type 1

3 i is preferable satisfied by means of corresponding (product specific or dedicated) inventory which has indicated by the solid arrows. Only when that inventory is out of stock, demand can be satisfied by a substitution. In figure, it is assumed that demand for Product 1 can be satisfied by inventory of Product 2 (as indicated by the dashed arrow), and not vice versa. This is the essence of one-way substitution. This situation often arises naturally in supermarkets (when product 2 has higher quality or wider functional than product 1). We assumed that the inventories of both product are managed according to (R, s, S) policy. At the end of every review period R, the decision maker allocates the available inventory to the observed demand according to the rules described above. Note that, R, S) policy is a special case of the (R, s, S) policy with s = S 1, an order is placed at the end of every review period provided the inventory position is smaller than S. Demands for both products are assumed to be discrete and finite random variables with joint probability density function P D (d 1, d 2 ) with d i the demand realization of product i. Cost Parameters: c i : purchasing cost per unit of product i p i : shortage cost of unsatisfied demand of product i at the end of review period h i : holding cost per unit of product i left over at the end of the review period a: adjustment cost per unit of demand for product 1 satisfied by product 2 Random Variables: Q i : restocking order size of product i B i : amount of backorders of product i incurred at the end of review period I i : left over inventory or safty stock of product i at the end of review period Z: amount of demand rerouted to substitute the product at the end of a review period d i : single period demand of product i Decision Parameters: S i : order up to level of product i 2

4 s i : reorder point of product i 3. DTMC Approach We present a discrete -time Markov chain model that allows to determine the expected total cost per period for the one way substitution strategy by evaluating E[Q i ]), E[I i ] and E[Z] from steady-state probabilities. With one way substitution, the expected total cost per period E[T C] is given by: 2 E[T C] = (c i E[Q i ] + p i E[B i ] + h i E[I i ]) + ae[z] i=1 The expected values E[Q i ], E[I i ], E[B i ], and E[Z] can be evaluated using a DTMC approach where the state of inventory system is defined by two dimensional state vector (j, k). The first dimension j represent the net inventory (i.e. on hand inventory - number of back-orders) of product 1 at the end of the review period. The second dimension k represents the net inventory of product 2 in an analogous way. Since the demand is discrete and finite, i.e. the discrete set of possible sets is finite. The net inventory for product i (i = 1, 2) has an inherent upper bound (UB i ) equal to the order up to level S i for the three strategies. Since demand is finite, the net inventory of product i is also limited by a lower bound (LB i ). We would like to show that lower bounds are influenced by the reorder point s i and maximum demands max(d i ) s max(d 1 ) s < max(d 1 ) s max(d 2 ) s < max(d 2 ) LB 1 s max(d 1 ) s max(d 1 ) LB 2 s max(d 2 ) s max(d 2 ) LB 1 s max(d 1 ) + s max(d 2 ) s max(d 1 ) LB 2 0 s max(d 2 ) 4. DTMC for the (R, s, S) Policy The transition probabilities from state (j, k) to state (l, m) for an (R, s, S) policy with one way substitution. The transition probabilities depend on the current and next state (j, k) and (l, m) respectively, on the reorder points s i and order up to level S i, and on the joint probability mass function P D. Solving the Markov chain model throughout a system of linear equations, we obtain π j,k ; the steady-state probability of state (j, k) (for all j and k). The expected inventory of 3

5 product 1 at the end of the review period can then be evaluated as: E[I 1 ] = S 1 j=1 (j S 2 k=lb 1 π j,k ). Note that, S 2 k=lb 02 π j,k is the probability that the next inventory at the end of the period of product 1 is equal to j units. Multiplying with j and adding over all strictly positive values of j results in the expected inventory of product 1 at the end of the review period. The expression for E[I 2 ] is similar E[I 2 ] = S 2 k=1 (k S 1 π j,k ). Again, E[B 1 ] = 1 j=lb 01 ( j S 2 k=lb 2 π j,k ) E[B 2 ] = 1 k=lb 02 ( k S 1 π j,k ) E[Q 1 ] = S 1 S2 k=lb 2 (S 1 j)π j,k E[Q 2 ] = S 1 S2 k=lb 2 (S 2 k)π j,k Now, E[Z] = E[d 1 ] E[Q 1 ] = E[Q 2 ] E[d 2 ]. Therefore, the transition probabilities for an (R, s, S) policy with one way substitution are in the following table. From To Transition Probability For D (j l, k l) l > 0 j > s 1, k > s D (j l, k m) l 0, m < 0 (j, k) (0, m) k m w=0 P D(j + w, k m w) m 0 (j, k) (l, 0) k w=0 P D(j l + w, k m) l < 0 D (S 1 l, k m) l > 0 j > s 1, k > s D (S 1 l, k m) l 0, m < 0 (j, k) (0, m) k m w=0 P D(S 1 + w, k m w) m 0 (j, k) (l, 0) k w=0 P D(S 1 l + w, k w) l < 0 D (j 1, S 2 m) l > 0 j > s 1, k > s D (j 1, S 2 m) l 0, m < 0 (j, k) (0, m) S2 m w=0 P D (j + w, S 2 m w) m 0 (j, k) (l, 0) S2 w=0 P D(j 1 + w, S 2 w) l < 0 D (S 1 l, S 2 m) l > 0 j > s 1, k > s D (S 1 l, S 2 m) l 0, m < 0 (j, k) (0, m) S2 m w=0 P D (S 1 + w, S 2 m w) m 0 (j, k) (l, 0) S2 w=0 P D(S 1 l + w, k m) l < 0 5. Numerical Example Definition 5.1. It is only product specific stock is held and demand can never be rerouted to stock of different product. 4

6 Definition 5.2. It refers thatt the demand for a particular product type which always rerouted to the stock of the flexble product and no product specific stock is held. We determine the optimal inventory control parameters (and corresponding expected total costs) for a one way substitution system with a given set of cost parameters and different demand correlation. The result can be compared using special technique (no pooling and full pooling strategies). The demand for both product type is assumed to follow a discretized bi-variate normal demand distribution, based on a joint continuous normal distribution f(x, y) [ N 2 (µ, ] Σ) with mean vector µ = [20, 20] 1 ρ and covariance matrix Σ = q. The demand correlation is varied ρ 1 (ρ = 0.9, 0, or 0.9) to study influence on the optimal inventory control parameters and the expected total cost. Note that the flexible product (i.e., product 2) has 10% product cost premium over product 1. We apply holding rate of 25% of unit purchasing cost for both product types. The unit penalty cost for not satisfying demand is equal to $2 for both product. The adjustment cost is $0.2/unit of rerouted demand. Product 1 Product 2 E[d i ] σ(d i ) 3 3 Correlation (ρ) 0.9; 0; 0.9 c i h i p i 2 2 a Conclusion In this project we studied on the effect of the one-way substitution strategy on the optimal design of a two item inventory system using an (R, s, S) restocking policy with negligible lead times. We present a DTMC model to evaluate expected total cost for any arbitrary set of inventory control parameters. Numerical result shows that the one way substitution strategy can be outperformed for both no pooling and full pooling strategies (i.e., optimal inventory control parameters for three 5

7 strategies). Furthermore, it seems interesting to observe 1 Demands correlation decrases results in rerouting more demand to the flexible product. 2 This study can also be extended primarily on including positive restocking leadtimes in the analysis. References [1] S. Chopra and P. Meindl, Supply Chain Management: Strategy, Planning, and Operation. Third Edition (Pearson Prentice Hall, New Jersey, 2007). [2] M. S. Hiller, Using Commodity as Backup Safety Stock. European Journal of Operation Research, 136 (2002), [3] Z. Lian, L. Liu, and M. F. Neuts, A Discrete-Time Model for Common Lifetime Inventory Systems. Mathematics of Operation Research, 30, 3 (2005), [4] U. S. Rao, J. M. Swaminathan, and J. Zhang, A Multi-Product Inventory Planning with Downward Substitution, Stochastic Demand, and Setup Costs. IIE Transactions, 36 (2004), [5] S. M. Ross, Introduction to Probability Models. Tenth Edition (Academic Press, Amsterdam; Boston, 2010). [6] W. L. Winston, Operations Research: Applications and Algorithms. Fourth Edition (Thomson Brooks, Australia; Belmont, CA, 2004). 6