Optimal Consolidation of Single Echelon Inventories

Optimal Consolidation of Single Echelon Inventories

Rosa H. Birjandi

Air Force Institute of Technologies

Department of Operational science, Dayton OH

Feb 2004

ABSTRACT

Effective management of inventory is vital in firm’s survival in today’sbusiness environment where high level of customer service at reduced cost is expected. The effect of inventory consolidation across locations has received a considerable attention in the recent logistics literature. The impact of physical consolidation and risk pooling has been discussed.Previous studies have focused on the allocation decisions assuming that the consolidation locations are pre-determined. In this paper,we extend these analyses to include the selection of consolidation locations. We analyze the impact of ordering cost, inbound shipping and handling cost, outbound transportation costs, lead time variation and demand correlation on the overall cost reduction. We provide an optimization model for the inventory consolidation in the presence of variable lead-times, anddemand correlations. Aschanges in demand quantity might impact the delivery lead times,our modelalso captures the correlation between demand and lead-time at each centralized location. Computational results will be presented to verify the sensitivity of the decisions to changes in cost parameters, and the impact of lead time variation and demand correlation between the potential consolidation locations.

The author would like to thank Professor Philip Evers at the University of Maryland, College Park for his insightful comments on earlier editions of this paper.

1. Introduction

It is well known that consolidation of stock keeping locationsmay reduce the safety stockinventory.The impact of consolidation and the extent that it is influenced by the demand characteristics at the centralized locations is investigated.Studies in the literature have assumed that the consolidation locations are pr-determined. Smykay (1973) and Maister (1976) derived square root laws. Zinn, Levy, and Bowersox (1989) showed that the Smykay square root law represents a special case of their portfolio effect model. They considered only the case where inventory locations are reduced to one. Evers and Beier (1993) developed a model of the square root law associated with safety stocks. Evers (1995) extended this model to include cycle stocks. Mahmoud (1992) considered the impact that safety stock centralization has on various cost factors and developed a method for determining the optimal consolidation scheme based on the portfolio effect model. Tallon (1993) addressed the issue of variable lead times. These square root laws are applicable to situations where certain assumptions hold. For the complete list of these assumptions and reference to their first introduction seeEvers(1995).

In this paper wepropose a cost optimization model for the selection of the centralizedlocations and the physical allocation of inventory to maximize the cost savings.Since demands are often correlated across geographical regions, and changes in demand might impact the delivery lead time, capturing the dependence relationships is necessary in modeling the reality. In this paper, we address the lead-time variability, correlated demands, and non-zero correlation between demand and lead-time at each location.The formulation we obtain is a mixed integer non-linear programming problem. Using this model we analyze the impact of changes in ordering cost, shipping and handling cost and outbound transportation costs. As the result of our experiments show these cost parameters impact the selection of the consolidating locations,the allocation of systems demand, and therefore the overall cost reduction. We also address the importance of factors such as lead time variation and demand correlation and their impact on the savings in safety stocks due to consolidation.

The remainder of this paper is organized as follows. In section II, we describe the methods used to determine the required inventory levels at the stocking locations. We alsodescribe the demand allocation and the formulas used for the computation of the safety stock requirements and cycle stock related cost afterconsolidation. The proposed optimization models are formulated in Section III. In Section IVwe present our computational results. Our conclusions are provided in section V.

II.Background

Consider a single echelon distribution system consisting of n stocking locations. In what follows wemake use of the following notations:

i : Index for decentralized (before consolidation) locations i{1,2,…,n}

Di :Mean aggregate daily demand at decentralized location i during the planning period

Li : Mean lead time at decentralized location i in days

:std. for aggregate daily demand at decentralized location i

: std. for lead time at decentralized location i

K : Safety stock factor: for given fill rate α, K is standard Normal deviate such that

[i]

h : Per unit holding cost during the period

: Coefficient for correlation between demand at locations i and l

ti : Per unit cost of transportation for satisfying demand Di (from the stocking location i to

customers)

Before Consolidation:

We assume that the plannersuse the following formula to determine the level of safety stockrequired at any location.

(1)

Theplanner would like to maintain a level of service defined byrequiring a pre-specified probability of no stock out during the replenishment lead time (L, ) which translate to thesafety factor (K). (D, ) characterize thedistribution of demand for the location during the period. The last term under the square root,represents the covariance of demand and lead time at the location with correlation coefficient τ. For more detail see Tersine (1994).

In our cost optimization model in section III among other costs, we need to capture the total cost related to the required cycle stock at each location. We will compute the total cost of holding, ordering, shipping and handling the cycle stock during the planning period. This cost depends on thenumber of order cycles for each locationi. The cost of ordering,shipping, handling, and holdingthe cycle stock, as a function of the number of replenishment ordersNi,would be:

, (2)

where is the mean demand during the planning period. Taking the first derivative of the cost function (1) and solvingfor Ni as in (3);

(3)

The total cost related to ordering, holding, and shipping and handling the required cycle stock Cjfor location i can be obtained by substituting the optimal Ni from (3) into (2) as follows:

(4)

After Consolidation:

When the number of stocking locations are reduced,mn locationsare selected to hold the systems inventory and satisfy the total demand. In what follows we make use of theseadditional notations:

j : Index for centralized (after consolidation) locations

Lj : Mean lead time at centralized location j

: std. for lead time at centralized location j

: Coefficient for correlation between demand and lead time at location j

Wi,j : Proportion of demand for location i(before consolidation) assigned to

centralized location j(due to consolidation).

ei,j : The extra per unit cost of transportation due to satisfying location i demand by location j

Oj :Fixed order cost.

Fj : Fixed cost of operation at location j if holding the item inventory.

fj : Fixed shipping and handling cost from supplier to the locationj.

vj : Variable shipping and handling cost from supplier to the location j.

The demand for the n stocking locations need to be partially or fully directed (allocated) to them centralized locations. For any location j selected to hold consolidated inventory, the effective demand (centralized demand) will be the sum of the demands from locations 1,…,n allocated to j. For the purpose of this definition only, we denote the mean effective demand at centralized location j, by Djand let denote the standard deviation of effective demand at centralized location j after consolidation.Then the centralized demand seen by locationj, will have the following statistics:

,

and

,

where Wi,jis the proportion of demand for location i demand allocated to location j.The allocation derives the order size, and the level of safety and cycle stocks. Incorporating the above representations in (1)and using we obtain the required safety stock SSjat each centralized location jas a function of the allocation Wi,j as follows:

Birjandi and Golovashkin (1998) showed that the terms under the big square root are convex and treated this as the square root of a convex function. Equation (5)indicates that, the planned safety stock inventory at each centralized location j after consolidation depends on proportion of mean demand during the period directed from each decentralized location (Wi,j), the safety stock factor (K),mean and standard deviation of lead time (L σLi),mean and standard deviation of demand (Di σD i ), demand correlation coefficients(ρi,j)and the coefficients of correlationbetween demand and lead time (τj).

Also incorporating the above allocation in (4), derives the total cost related to the required cycle stock Cjat each centralized location j and the average cycle stock held at the location during the replenishment cycle CSj

(6)

(7)

As indicated in (7) the fixed order cost (Oj), the fixed shipping and handling cost (fj ), and the holding cost (h), all impact the planned inventory level at location j.

III.The optimization Models:

In this section we first provide an optimization model (model 1) that deals with the allocation of safety stock capturing the supply lead-time, demand correlations, and the correlation between demand and lead times. Assuming that the selection decision is made a priory,mlocationsare selected to hold the inventory. We experiment with model 1 and analyzethe impact of lead-time variations,demand correlations, and the correlation between demand and lead times on the savings in safety stocks. We then present the proposed cost minimization model (model 2)that deals with the selection decisions and the allocationdecisions simultaneously to maximize the overall cost savings due to consolidation while a specified level of service is maintained.

The allocation decision:

Consider the case where there are nlocations i =1,2,..,n and without loss of generality, assume thatlocations j =1,2,..,m with known expansion capacities have been selected to hold the consolidated inventory. In order to maximize the consolidation effect, we use the following nonlinear program:

Model 1

Minimize: (8)

Subject to:

(9)

(10)

(11)

(12)

Where SSjis the aggregate safety stock allocated to location j as defined in (5) and the objective is to minimize the total safety stock after consolidation. The constraint set (9)ensures that the total system demand will be covered after consolidation. Note that when some of the assumptions mentioned above are relaxedand when there are no limitation on capacity expansions, the non linear model of Evers(1995) will be equivalent to this model.

The selection and allocation decisions:

We now develop a cost optimization model to optimally select the locations and optimally allocate the demands among the selected locations. As described in Section II, 5) and 6)capture theoverall ordering, shipping & handling, and holding costfor each centralized location after consolidation. Othercost components we need to consider includethe cost of in bound transportation and out bound transportation. The total out bound transportation cost (13) needs to include theextra cost ei,jassociated with locationjserving the demand originated at locationi after consolidation. The increase in transportation cost is due to the increase inthe average delivery distance from the stocking locationto the demand source. The inbound transportation is also influenced by the number and choice of the locations selected. In our study we capture this by assigning a constant percentage increasefor any additional location to a base value as shown in (14) (e.g. v = 5%).

(13

(14)

The objective ofoptimally selecting a number of locations amongst 1,2,…n to continue as stocking locations and satisfythedemand for all the locations at minimum cost can be defined as in (15). Constraints (16) through (20)describe the desired allocation.

MINIMIZE: (15)

(16)

(17)

(18)

(19)

(20)

However the non linear terms due to multiplication of two variables in the objective function and in the constraint set (15) adds to the complexity. We remove these non-linear terms byredefinition of the boundary condition for Wi,j and modification of the objective functionas follows:

Model 2

(15’)

SUBJECT TO:

(16’)

(17)

(18’)

(20)

If Xj=1, location j is selected for centralization. The constraint set (17) describes the limitation on possible capacity expansion imposed through the input data ηj for the locations if they are selected to hold the inventory. Constraint set (18’) ensures that the proportion of demand from any location i allocated to location j, can be positive only if location jis selected. We removed the constraint set (19) as the increase in outbound transportation cost enforces that the locationsserve their own demand if they are selected. This constraint can be added if the increase in outbound transportation costis not significant and the allocation of demand originated at some selected locationsto other locations is not desirable for other managerial reasons.

IV.Experimental Results

In this section, we outline the result of the computational tests with the two models. We experimented with a program written in GAMS. The nonlinear solvers exploit the convexity of the terms under the big square root and solve the problems to optimality in a fraction of a second. For analysis in the ten cases listed bellow, we solved problems consisting of seven decentralized locations (n = 7) to be considered for consolidation. In experimentation with model 1, four locations are pre-selected (m = 4 locations).No cost is explicitly involved and we will examine the impact of demand and supply characteristics on the safety stock consolidation. In experimenting with model 2, the optimal number (m) will be determined by the optimization and thecentralized locations are selectedto hold the optimally allocated inventory.Many cost trade offs and demand and supply parameters collectively influence the selection and allocation decisions and impact the overall cost saving. It is very difficult to measure and report on the impacts in all the combinations. We will define a base case for each experiment by setting equal levels for almost all the parameters at all locations and create seven cases:three cases for model 1 and four cases for model 2, by variation of one parameter of interest at a time to see the impact. The equal settings for all the locations might diminish the magnitude of improvements due to consolidation. The purpose of these tests is not to show the magnitude of improvement as much as it is the sensitivity of the decisions to change in the parameters. Table 1 and Table 2 will summarize the results for an instance in each case.

Model 1 (The allocation problem):

Base case:In our base case for location i,we set themean daily demandsequal to 200+10 iand the mean lead times equal to 18 + idays. We assume the demand standard deviationsareequal to 10% of mean daily demands. The standard deviations in lead time are also assumed to be equal to 10% of mean lead times.The safety stock factor of 2.0 is required for all the centralized locations. The coefficients for the demand correlation were generated randomly from U(-0.4, 0.3). We used a C program to test thepositive semi-definiteness of our input correlation matrices making sure they aresymmetric positive semi definite and all diagonal elements are equal to 1. The coefficients of correlation between demand and lead time are set equal to 0.1 for all the locations.Finally, we assumed that capacity of each location if selected can be expanded to support twice its original demand (ηj = 3 for all j=1,..,n). The locations selected after solving the base case, are 1, 2, 3, and 4. The optimal allocation is provided bellow.

1 2 3 4 5 6 7

1 1 0.493 0.473 0.644

2 1 0.507 0.527 0.356

3 1

4 1

Case 1:

In this case we examine the impact of increased variation in replenishment lead times by changing the standard deviation for all the locations from 10% of mean lead time to 30% of mean lead time. The optimal overall safety stocklevel increases from6541 for the base case to 19431 units. The safety stock reduction due to consolidation in this case is equal to11% which is 1% more than the base case.The allocation is changed and location 1 serves more demand from 5, 6, and 7 as before 1, 2, 3, 4, also serve their own demands.

5 6 7

1 0.522 0.522 0.569

2 0.478 0.478 0.431

Case 2:

In this case we examine the impact of increased correlations between demand and lead time at each location. For the base case we used 0.1 at all the location and 10% lead time deviation. For this case we used the highcorrelation coefficient 0.9 at all the locations and applied the 30% deviation. Comparing with case 1, the same locations are selected but the allocation is changed and the extra saving due to high lead time deviations is reduced back to provide 10.1% improvement as in case 1.

5 6 7

1 0.478 0.478 0.431

2 0.522 0.522 0.569

Case 3:

In this case we investigate the impact of increased demand correlations. For the base case and the instance of case 3, we randomly generated and checked the following coefficient matrices for positive semi definiteness.

Coefficient matrix for the Base case: Coefficient matrix for case 3:

1 -0.3 -0.4 -0.3 -0.3 -0.3 -0.3 1 0.7 0.7 0.9 0.7 0.7 0.7

-0.3 1 0.1 0.1 0.1 0.1 0.2 0.7 1 0.7 0.7 0.7 0.7 0.8

-0.4 0.1 1 0.1 0.1 0.1 0.1 0.7 0.7 1 0.7 0.7 0.7 0.7

-0.3 0.1 0.1 1 0.1 0.1 0.2 0.9 0.7 0.7 1 0.7 0.7 0.8

-0.3 0.1 0.1 0.1 1 0.1 0.1 0.7 0.7 0.7 0.7 1 0.7 0.7

-0.3 0.1 0.1 0.1 0.1 1 0.1 0.7 0.7 0.7 0.7 0.7 1 0.7

-0.3 0.2 0.1 0.2 0.1 0.1 1 0.7 0.8 0.7 0.8 0.7 0.8 1

As we can see in Table 1, the high correlation between demands is increasing the safety stock requirement at the centralized location and the saving in safety stock has decreased by 18%.

Test Cases / Decentralized
Safety Stock / Centralized
Safety Stock / Percent Reduction
In Safety Stock
Base Case / 7278 / 6541 / 10.1
Case 1 / 21834 / 19431 / 11
Case 2 / 21392 / 19226 / 10.1
Case3 / 7278 / 6672 / 8.32

Table 1: The result of experiment with Model 1

Model 2 (The selection and allocation problem):

Base case: For the cost optimization model, in addition to the parameters described for the Model 1 base case,we set the cost parameters as follows:

For all the locations, theper unit out boundtransportation cost is equally set to $10 with a $2 per unit increase due to satisfying the demand originated to i by location j (ei,j =2 & ej,j= 0).The base variable in bound transportation cost is set equal to $10 and increases by a factor of 0.05(m). We also equally assignedfixed location cost of $2000, fixed per order cost of $50,fixed shipping & handling cost of $30 (f =30), and carrying charge of$50(25% of the inventory value for annual holding cost) for all the locations.

The optimal selection is 1,2, and 7 and the overall cost improvement is 48%.These locations serve their own inventory additional allocationfrom the rest of the locations is shown bellow: