An EOQ Model for Multi-Item Inventory with Stochastic Demand

An EOQ Model for Multi-Item Inventory with Stochastic Demand

Kizito Paul Mubiru

Makerere University

Abstract

Traditional approaches towards determining the economic order quantity (EOQ) in inventory management assume deterministic demand of a single item, often at a constant rate. In this paper, an optimization model is developed for determining the EOQ that minimizes inventory costs of multiple items under a periodic review inventory system with stochastic demand. Adopting a Markov decision process approach, the states of a Markov chain represent possible states of demand for items. The decision of whether or not to order additional units is made using dynamic programming. The approach demonstrates the existence of an optimal state- dependent EOQ and produces optimal ordering policies, as well as the corresponding total inventory costs for items.

1. Introduction

   Business enterprises continually face the challenge of optimizing inventory levels and decisions of items in a stochastic demand environment. Two major problems are usually encountered: (i) determining the most desirable period during which to order additional units of the item in question and (ii) determining the economic order quantity given a periodic review inventory system when demand is uncertain. In this paper, an inventory system is considered whose goal is to optimize the economic order quantity and total costs associated with ordering and holding inventory items. At the beginning of each period, a major decision has to be made, namely whether to order additional units of the items in stock or postpone ordering and utilize the available units in inventory. The paper is organized as follows. After reviewing the previous work done, a mathematical model is proposed where initial consideration is given to the process of estimating the model parameters. The model is solved thereafter and applied to a special case study. Some final remarks lastly follow.

2. Literature Reviews

   Tadashi and Takeshi[1] formulated the stochastic EOQ type models with discounting using Gaussian processes in the context of the classical EOQ model. Numerical properties of the order quantities that minimize expected costs for various model parameters were examined. However, the model is restricted to a single item. The stochastic EOQ-type models to establish inventory policies were examined by Berman and Perry [2].Output can be interpreted by a random demand and the input by a deterministic production plus random returns. In this study, the authors examine a single item and the decision of how much to order at least cost is not explicit. Research results also explore the structure of optimal ordering policies for stochastic inventory system with minimum order quantity by Yao & Kateshaki [3]. In this article, ordering quantity is either zero or at least a minimum order size. The impact of such inventory policies on the cost structures of the stocked item is however not explicit. In a similar context, Broekmeullen [4] proposed a replenishment policy for a perishable inventory system based on estimated aging and retrieval behavior. The model takes into account the age of inventories and which requires only very simple calculations. The model has profound insights especially in terms of the randomness of demand. However, the model is restricted to perishable products. According to Roychowdhury [5], an optimal policy for a stochastic inventory model for deteriorating items with time-dependent selling price is feasible. The rate of deterioration of the items is assumed to be constant over time. The demand and lead time both are random. A profit-maximization model is formulated and solved for optimum order quantity. The model provides some intriguing insights to the problem. However, model extensions to handle multiple items must be embedded in the formulation to provide optimal results. The current literature presented examines modeling in the context of a single product. In this paper, an inventory system is considered whose goal is to optimize the EOQ, ordering policy and the total costs associated with multiple items in inventory. At the beginning of each period, a major decision has

   to be made, namely whether to order units of the stocked items or to postpone ordering and utilize the available units in stock.

   The paper is organized as follows. After describing the mathematical model in §3, consideration is given to the process of estimating the model parameters. The model is solved in §4 and applied to a special case study in

   §5.Some final remarks lastly follow in §6.

3. Model Formulation

We consider a designated number of items in inventory whose the demand during each time period over a fixed planning horizon is classified as either favorable (denoted by state F) or unfavorable (denoted by state

U) and the demand of any such period is assumed to depend on the demand of the preceding period. The

aZ(m,n) = [aZ (m,n), aZ (m,n)]T where T denotes matrix transposition.

F U

F U

3.2 Finite period dynamic programming formulation

Recalling that the demand can either be in state F or in state U, the problem of finding an optimal EOQ may be expressed as a finite period dynamic programming model.

Let Cn(i, m) denote the optimal expected total inventory costs of item m accumulated during the periods n,n+1,…,N given that the state of the system at the beginning of period n is i{ F,U }.The recursive equation relating Cn and Cn+1 is

Cn(i, m)= minZ {QZi (m)(CZi (m) + Cn+1(F, m) ,

U

U

transition probabilities over the planning horizon from f f

one demand state to another may be described by means of a Markov chain. Suppose one is interested in determining an optimal course of action, namely to order additional stock units (a decision denoted by Z=1) or not to order additional units (a decision denoted by Z=0) during each time period over the planning horizon, where Z is a binary decision variable. Optimality is defined such that the lowest expected total inventory costs are accumulated at the end of N consecutive time periods spanning the planning horizon under consideration. In this paper, a three item (m =3) and two-period (N=2) planning horizon is considered.

3.1 Assumptions and notation

Varying demand is modeled by means of a Markov chain with state transition matrix QZ(m) where the entry QZij(m) in row i and column j of the transition matrix denotes the probability of a transition in demand

CZi (m) + Cn+1(U, m))}

i {F,U }

n=1,2..N (1)

m = 1, 2, 3

together with the final conditions CN+1(F, m) = CN+1(U, m) = 0

.

This recursive relationship may be justified by noting that the cumulative total inventory costs

CZij(m) + Cn+1(j,m) resulting from reaching state j { F,U } at the start of period n+1 from state i { F,U } at the start of period n occurs with probability QZij(m). The dynamic programming recursive equations become:

Cn(i,m) = minZ{eZi(m) + QZi (m) Cn+1(F,m)

from state i{F , U} to state j{U , F} for item m

f

+ QZ (m) C

(U,m)}

{1,2,3}under a given ordering policy Z{0,1}. The number of customers observed in the system and the number of units demanded during such a transition is

iU

i{F , U}

n+1

i i

i i

F

F

U

U

captured by the customer matrix NZ(m) and demand matrix DZ(m) respectively. Furthermore, denote the number of units in inventory and the total (ordering, holding and shortage) cost durig such a transition by the inventory matrix IZ(m) and the cost matrix CZ(m) respectively. Also, denote the expected future cost , the already accumulated total cost at the end of period n when the demand is in state i{ F, U }for a given ordering policy Z{0,1} by respectively eZ (m) and aZ (m ,n) and let eZ(m) = [eZ (m), eZ (m)]T and

n= 1, 2.N (2)

m = {1, 2, 3}

Z {0,1}

CN(i, m) = minZ{eZi(m)} (3) result where (3) represents the Markov chain stable

state.

3.2.1 Computing QZ(m) , CZ(m) and OZ(m)

The demand transition probability from state i{ F, U

} to state j { F, U },given ordering policy Z { 0,1 }

1 if e1

F

Z =

( ) < e0

m

m

F

(m)

may be taken as the number of customers for item m

0 if e1

F

( ) e0

m

m

F

(m)

observed with demand initially in state i and later with demand changing to state j, divided by the sum of

The associated total inventory costs and EOQ are then

customers over all states. That is,

NZij(m)

C1(F, m) =

e1 ( ) if Z = 1

m

m

F

QZij(m) =

f U

f U

[NZi (m) + NZi (m)]

and

e0 ( ) if Z = 0

m

m

F

m

m

i{F , U} , m = {1, 2, 3}, Z {0,1} (4)

[D1

FF

( ) – I1

m

m

FF

( )] + [D1

m

m

FU

(m)

When demand outweighs on-hand inventory, the

OZ ( ) = – I1

F FU

(m)] if Z = 1

inventory cost matrix CZ(m) may be computed by means of the relation

CZ(m) = [c0(m) + ch(m) +cs(m)][DZ(m) IZ(m)]

where c0 (m) denotes the unit ordering cost, ch (m)

0 if Z = 0

respectively. Similarly, when demand is Unfavorable (ie. in state U), the optimal ordering policy during period 1 is

denotes the unit holding cost and cs(m) denotes the unit shortage cost.

Therefore,

[c0(m)+ch(m)+cs(m)][DZij(m) IZij(m)]

1 if e1

U

Z =

0 if e1

U

( ) < e0

m

m

U

m

m

( ) e0

U

(m)

(m)

CZij(m) =

if DZij(m) > IZij(m)

0 if DZij(m) IZij(m) (5)

In this case, the associated total inventory costs and EOQ are

for all i,j{ F, U } , Z{0,1} and m {1,2,3}

C1(U, m) =

e1 ( ) if Z = 1

m

m

U

A justification for expression (5) is that DZij

(m) –

e0 ( ) if Z = 0

m

m

U

IZij(m) units must be ordered in order to meet the excess demand. Otherwise ordering is cancelled when

and

[D1

UF

( ) – I1

m

m

UF

( )] + [D1

m

m

UU

(m)

demand is less than or equal to the on-hand inventory. The following conditions must however, hold.

OZ ( ) = – I1

m

m

U UU

(m)] if Z = 1

1. Z=1 when c0 (m) > 0 and Z = 0 when c0 (m) = 0

2. cs(m) > 0 when shortages are allowed, and cs

(m) = 0 when shortages are not allowed.

4 Optimization

The optimal EOQ and ordering policy are found in this section for each time period separately.

0 if Z = 0

Using (2),(3) and recalling that aZi(m,2) denotes the already accumulated total inventory costs at the end of period 1 as a result of decisions made during that period, it follows that

aZi(m,2) = eZi(m) + QZi (m) min { e1 (m) , e0 (m)}

+ QZi

f F F

(m) min { e1 (m) , e0 (m)}

f

f

U

U

U U U

1. Optimization during period 1

   = eZi(m) + QZi (m) C1(F, m) + QZi

   (m) C1(U, m)

   When demand is Favorable (ie. In state F), the optimal ordering policy during period 1 is

2. Optimization during period 2

   Using the dynamic programming recursive equation(1), and recalling that aZi(m) denotes the already accumulated total cost of item m at the end of period 1 as a result of decisions made during that period, when demand is favorable (ie. in state F),the optimal ordering policy during period 2 is

1. Implementation

   1. Case Description

In order to demonstrate use of the model in §3-4, a real case application from Shoprite supermarket in Uganda is presented in this section. The category of items

1 if a1

F

Z =

( ) < a0

m

m

F

(m)

examined included 400gms Full cream Milk powder, 250gms Black coffee and 250gms Cadbury Cocoa;

0 if a1

F

( ) a0

m

m

F

(m)

whose demand fluctuate every week. The supermarket wants to avoid excess inventory when demand is

While the associated total inventory costs and EOQ are

Unfavorable (state U) or running out of stock when demand is Favorable (state F) and hence seeks decision

C2(F, m) =

and

a1 ( ) if Z = 1

m

m

F

m

m

a0 ( ) if Z = 0

F

support in terms of an optimal inventory policy, the associated inventory costs and specifically, a recommendation as to the EOQ of Full cream milk powder, Black coffee and Cadbury Cocoa over the next two-week period is required.

[D1

m

m

FF

( ) – I1

m

m

FF

( )] + [D1

m

m

FU

(m)

1. Data collection

   OZ ( ) = – I1

   F FU

   (m)] if Z = 1

   Samples of customers were taken for each item. Past data revealed the following demand pattern and

   respectively.

   0 if Z = 0

   inventory levels of the three items over the first week of the month when demand was Favorable (F) or Unfavorable (U).

   Similarly,when demand is unfavorable(ie. in state U

   ),the optimal ordering policy during period 2 is

   Considering Full cream milk powder (m =1), when additional units were ordered (Z=1),

   1 if a1

   U

   Z =

   ( ) < a0

   m

   m

   U

   (m)

   m m

   m m

   0 if a1 ( ) a0 ( )

   U U

   In this case, the associated total inventory costs and EOQ are

   a1

   U

   C2(U, m) =

   (m) if Z = 1

   and

   a0 ( ) if Z = 0

   m

   m

   U

   [D1

   m

   m

   UF

   ( ) – I1

   m

   m

   UF

   ( )] + [D1

   m

   m

   UU

   (m)

   OZ ( ) = – I1

   U UU

   (m)] if Z = 1

   respectively.

   0 if Z = 0

   When additional units were not ordered (Z=0),

   When additional units were ordered (Z=1) for Black coffee (m=2),

   while these matrices were

   for the case when additional units were not ordered.

   When additional units were ordered (Z=1) for Cadbury cocoa (m=3),

   while these matrices were

   for the case when additional units were not ordered.

   The following unit ordering, holding and shortage costs (in UGX) were captured for each individual product at the supermarket:

   400 gms Full Cream Milk powder (m=1) c0(1) = 4500 , ch(1) = 1200, cs(1) = 300

   250 gms Black Coffee (m=2)

   c0(2) = 4800 , ch(2) = 900, cs(2) = 300

2. Solution procedure for Milk powder, Black Coffee and Cadbury Cocoa

   Using (4) and (5), the state transition matrices and Inventory cost matrices for each respective item in week 1 are

   250 gms Cadbury Cocoa (m=3)

   c0(3) = 5100 , ch(3) = 600, cs(3) = 300

   for the case when additional units are not ordered(Z=0).

   When additional units are ordered (Z = 1), the matrices Q1 (1), C1(1) , Q1 (2) , C1(2) , Q1 (3) and C1(3) yield

   the costs(in million UGX)

   However, when additional units are not ordered (Z= 0), the matrices Q0 (1), C0(1) , Q0 (2) , C0(2) , Q0 (3) and C0(3) yield the costs (in million UGX)

   The results are summarized in Tables 1 and 2 below:

   Table 1:

   Values of Z , eZi(m) and OZi(m,n) for items during week 1(=1)

   Full Cream Milk Powder (m=1)	Z = 1	Z = 0
   eZ (1)	0.265	0.371
   f eZ (1) u	0.087	0.142
   OZ (1,1) f	83	0
   OZ (1,1) u	14	0
   eZ (1) f	0.265	0.371
   Black Coffee (m=2)	Z = 1	Z = 0
   eZ (2) f	0.063	0.005
   eZ (2) u	0.026	0.015
   OZ (2,1) f	0	0
   OZ (2,1) u	0	0
   Cadbury Cocoa (m=3)	Z = 1	Z = 0
   eZ (3) f	0.118	0.024
   eZ (3) u	0.067	0.018
   OZ (3,1) f	0	0
   OZ (3,1) u	0	0

   i

   i

   The cumulative total costs aZ (m,n) are computed using (1) for week 2 and results are summarized in Table 2 below:

   Table 2:

   Values of Z , aZ i (m,n) and OZi(m,n) during week 2(n=2)

   Full Cream Milk Powder (m=1)	Z = 1	Z = 0
   aZ (1,2)	0.265	0.371
   f aZ (1,2) u	0.087	0.142
   OZ (1,2) f	83	0
   OZ (1,2) u	14	0
   Black Coffee (m=2)	Z = 1	Z = 0
   aZ (2,2) f	0.073	0.015
   aZ (2,2) u	0.032	0.022
   OZ (2,2)	0	0
   f OZ (2,2) u	0	0
   Cadbury Cocoa (m=3)	Z = 1	Z = 0
   aZ (3,2)	0.139	0.044
   f aZ (3,2) u	0.090	0.041
   OZ (3,2)	0	0
   f OZ (3,2) u	0	0

3. The Optimal ordering policy and EOQ

   Week1

   Full Cream Milk powder

   Since 0.265 < 0.371, it follows that Z=1 is an optimal ordering policy for week 1 with associated total inventory costs of 0.265 million UGX and an EOQ of (156- 95) + 115-93) = 83 units when demand is favorable. Since 0.087< 0.142, it follows that Z=1 is an optimal ordering policy for week 1 with associated total

   inventory costs of 0.087 million UGX and an EOQ of (107 93) = 14 units if demand is unfavorable.

   Black Coffee

   Since 0.005 < 0.063, it follows that Z=0 is an optimal ordering policy for week 1 with associated total inventory costs of 0.005 million UGX when demand is favorable. Since 0.015 < 0.026, it follows that Z=0 is an optimal ordering policy for week 1 with associated total inventory costs of 0.015 million UGX if demand is unfavorable.

   EOQ = 0 units regardless of the state of demand. Cadbury Cocoa

   Since 0.024 < 0.118, it follows that Z=0 is an optimal ordering policy for week 1 with associated total inventory costs of 0.024 million UGX when demand is favorable. Since 0.018 < 0.067, it follows that Z=0 is an optimal ordering policy for week 1 with associated total inventory costs of 0.018 million UGX if demand is unfavorable.

   Week 2

   Full Cream Milk Powder

   Since 0.449 < 0.568, it follows that Z=1 is an optimal ordering policy for week 2 with associated accumulated inventory costs of 0.449 million UGX and an EOQ of (156- 95) + 115-93) = 83 units when demand is favorable. Since 0.320 < 0.531, it follows that Z=1 is an optimal ordering policy for week 2 with associated accumulated inventory costs of 0.320 million UGX and an EOQ of (107 93) = 14 units if demand is unfavorable.

   Black coffee

   Since 0.015 < 0.073, it follows that Z=0 is an optimal ordering policy for week 2 with associated accumulated inventory costs of 0.015 million UGX when demand is favorable. Since 0.022 < 0.032, it follows that Z=0 is an optimal ordering policy for week 2 with associated total inventory costs of 0.022 million UGX if demand is unfavorable. In this case, EOQ = 0 regardless of the state of demand.

   Cadbury Cocoa

   Since 0.044 < 0.139, it follows that Z=0 is an optimal ordering policy for week 2 with associated accumulated inventory costs of 0.044 million UGX when demand is

   favorable. Since 0.041 < 0.090, it follows that Z=0 is an optimal ordering policy for week 1 with associated accumulated inventory costs of 0.041 million UGX if demand is unfavorable; and EOQ = 0 regardless of the state of demand.

1. Conclusion

An inventory model with stochastic demand was presented in this paper. The model determines an optimal ordering policy, inventory costs and the EOQ of a multi-item inventory problem with stochastic demand. The decision of whether or not to order additional stock units is modeled as a multi-period decision problem using dynamic programming over a finite planning horizon. The working of the model was demonstrated by means of a real case study. It would however be worthwhile to extend the research and examine the behavior of EOQ for items under non stationary demand conditions. In the same spirit, our model raises a number of salient issues to consider: Lead time of items during replenishment and customer response to abrupt changes in price of items. Finally, special interest is thought in further extending our model by considering EOQ determination in the context of Continuous Time Markov Chains (CTMC).

Acknowledgments

The authors wish to thank Makerere-Sida Bilateral Research program for the financial support and the staff of Shoprite supermarket during the data collection exercise.

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