A heuristic algorithm for managing inventory in a multi-echelon environment

JOUKNAL OF OPERATIONS MANAGEMENT

Vol. X. No 3, Au@\t 1989

A Heuristic Algorithm for Managiug Inventory in a Mu~ti-~~~e~on environment

ROBERT L. BRECMAN*

LARRY P. RITZMAN**

LEE J. KRA_~EWSKI**

EXECUTIVE SUMMARY

The proper management of finished goods inventory in a multi-echelon environment is an extremely

difficult problem to solve. Optimization approaches for solving this problem are intractable, and

currently available heuristic techniques have serious deficiencies. Pull systems and independent demand

based push systems do not adequately deal with the lumpy demand caused by the dependent relationships

of stocking locations in a multi-echelon environment. Although distribution requirements planning

(DRP) can be modified to handle uncertainties by adding safety stock, the process of deriving demands at

lower echelons implicitly assumes deterministic conditions. In addition, no heuristic method directly

considers the capacity of ~anspo~ation and storage resources, or includes transportation costs. The

inco~(~ration of these additional complexities is left to the discretion of management.

This study introduces a new heuristic algorithm that addresses these additional complexities. The

algorithm is an improvement heuristic that can be implemented as an add-on modufe to a DRP system.

At the core of this heuristic are two search routines (Savings and Chanjie) for improving an initial

solution determined by the DRP explosion process. We demonstrate this heuristic algorithm with two

simple problems to provide some insight concerning its operation.

The heuristic algorithm is then tested against an alternative rolling horizon mixed-integer linear

programming (MILP) procedure that solves each linkage between locations optimally for each planning

horizon. The example scenario used in this research consists of four distribution centers ordering from

two regional warehouses, which in turn order from a central warehouse. Computer simulation is used to

compare the two alternative procedures with rolling horizons and demand uncertainty. The performance

of the procedures is compared for total costs, customer service. and the number of orders placed by

locations within the control of the model. l~ultivariate analysis of variance (MANOVA) techniques are

used to analyze these performance measures.

The experimental results suggest that the heuristic algorithm performs extremely well when compared

to the MILP based procedure. Demand uncertainty is found to have a significant effect on customer

service perfomrance, but safety stock can be added to distribution centers in actual applications to

control this situation. In addition, a qualitative comparison between the MILP approach and the heuristic

algorithm in this study suggests that the introduction of demand uncertainty has the effect of reducing the experimental differences between the two techniques. This result suggests that the heuristic algorithm

presented in this research works best (relative to the MILP approach) in the actual environments for

which it is intended.

This study is of considerabfe value to managers concerned with the management of finished goods in a multi-echelon environment. It represents an initial step toward the development of a heuristic algortthm

Manuscript received February 2, 1989; accepted May IO, 1989, after one revision

*Texas A&M University, College Station, Texas 77843

*“The Ohio State Umversity, Columbus, Ohio 43210

186 Vol. 8, No. 3

that incorporates additional real-world complexities and is tractable for realistically large problems. The

findings from this study provide encouragement that similarly designed heuristics can be implemented

within multi-echelon inventory control systems in the future.

INTRODUCTION

Finished goods inventory control systems often deal with distribution channels that involve more than just one or two stocking locations. A common problem is to determine how best to position and move inventory through a multi-echelon distribution network. A simple example of such a network is shown in Figure 1. Here the demand patterns for regional warehouses are a function of the inventory control methods used at the distribution centers. Similarly, the demand pattern for the plant or central warehouse is a function of the inventory control methods

used at both the distribution centers and the regional warehouses. This research investigates the situation in which the linkages between distribution centers and regional warehouses, as well as between regional warehouses and the central warehouse, are predetermined. When a single firm controls the placement and movement of inventory through a similar multi-echelon distribution system, there is an opportunity to implement a single inventory control system to manage the total distribution channel. The inherent complexity of controlling the inventory for numerous products in a multi-echelon environment makes this an extremely difficult problem.

Firms that operate with multi-echelon distribution systems tend to offer standard goods with relatively large demand rates characterizing the maturity stage of the product life cycles. This environment is often marked by a high degree of price competition, which increases the pressure on operations managers to coordinate the movement of inventory through the distribution channels in their respective firms at the lowest possible cost. As a result, development of new or modified inventory control systems that improve the management of

FIGURE 1 AN EXAMPLE MULTI-ECHELON NETWORK

Customer Demand Customer Demand

Journal of Operations Management 187

multi-echelon distribution systems has historically received considerable attention from

operations researchers and practitioners. One such developmental effort evolved from material requirements planning (MRP). In the

mid-1970s as MRP was gaining acceptance as a viable method for managing multi-stage production systems, it became apparent that this methodology might be applicable to multi-echelon distribution systems. Whybark (1975) noted that multi-echelon distribution systems can be thought of as multistage production systems with the direction of flow reversed. As Whybark described it, the demand in multi-echelon distribution systems “implodes” (rather than explodes, as in multi-stage production systems) from the customers through the distribution centers back to the plant. Although the direction of flows in these two environments are opposite, the dependent relationships are maintained in either direction.

In 1979, Stenger and Cavinato formalized the ideas of Whybark into what is now known as distribution requirements planning (DRP). Stenger and Cavinato noted that, if distribution center (or retail store, as in the case of the state liquor control agency example) demand forecasts are accurate, sizable savings can be achieved by reducing the need for safety stocks at each location in the distribution network. Recent implementations of DRP by Martin (1983) and others further suggest that this method should be given more research attention.

DRP does have some serious deficiencies, however. Although it can be modified to handle uncertainties by adding safety stock, the process of deriving demands at lower echelons implicitly assumes deterministic conditions. In addition, DRP does not directly consider the capacity of transportation and storage resources, or include transportation costs. The

incorporation of these additional complexities remains as another problem for operations managers to address.

There are two alternatives for incorporating capacity restrictions and transportation costs into multi-echelon inventory control models. The first alternative is to model and solve the problem as a mixed integer linear program (MILP). Unfortunately, the large dimensionality of the multi-echelon inventory control environment results in an intractable problem. Consequently, researchers have concentrated on either solving smaller problems (for instance, only two locations) or developing techniques (such as relaxation or decomposition) for reducing the problem size. A second alternative, which is pursued here, is to develop heuristic procedures. This latter approach is attractive because it provides a method for recognizing important practical complexities and results in a more efficient solution procedure.

We present the heuristic algorithm for managing inventory in a multi-echelon environment in the following section. In the remainder of the paper we demonstrate the heuristic algorithm and discuss the results of an experiment designed to test this new procedure. Finally, in the last section we summarize our findings and discuss directions for future research.

THE ALGORITHM

The heuristic algorithm developed in this research can be included as a module within a DRP system. The heuristic attempts to minimize the sum of inventory holding and transportation costs in a multi-echelon environment. Transportation costs are a function of the type of transportation option used, where transportation options are defined as alternate methods for transporting goods on predefined linkages between locations in specific periods. We assume there to be two general transportation options: an option with a fixed cost (FTO) and an option with an entirely variable transportation cost (VTO). FTOs represent fixed contract carrier rates and VTOs represent common carrier rates. Only FTOs are capacitated; it is assumed that

188 Vol. 8, No. 3

common carrier capacity is always available. In this research we also assume that all FTOs have

the same fixed costs and capacities. This last assumption is made without loss of generality, as the heuristic algorithm can be adjusted to handle a menu of fixed costs and multiple capacities. A general outline of this heuristic algorithm for a three-echelon distribution network (distribution centers, regional warehouses, and a central warehouse) follows.

Determination of a Feasible Solution

The first step in the heuristic algorithm is to calculate the net requirements for each item at each of the distribution centers using a standard DRP explosion process. Shipment of these requirements (subject to availability and on a lot-for-lot basis) to each of the distribution centers from the regional warehouses by VTOs represents the lowest cycle inventory holding cost solution for these linkages (possibly at the kxpense of higher transportation costs). This initial solution is improved by the Savings and Change routines, which are the heart of the heuristic algorithm.

The Savings Routine

The Savings routine searches for lower cost solutions for each distribution center-regional warehouse linkage (or the linkages between the two echelons closest to the customer) one period at a time, starting with the last period in the planning horizon. For the period under consideration the savings per unit volume associated with shipping each type of item required for current and future periods on the next lowest cost FTO available in this period is calculated as follows:

where:

SAVINGSi, = [vij, - tit, hi] / ~1 ieF, teT (1)

F = set of items not previously assigned to an FTO.

T = set of periods from the current period to the last period in the fixed planning horizon.

vij = variable costs associated with shipping one unit of item i by transport option j for

the linkage under consideration.

hi = cost of holding one unit of inventory of item i for one period.

Yi = volume required per unit of item i. ‘I J = VT0 that item i is currently being shipped on.

t,t, = the number of periods earlier item i will be required by the system if item i required in period t is instead shipped on the FTO under consideration in the

current period.

Items are ranked in order of SAVINGSi, (highest to lowest) per unit volume. This ranking allows for the comparison of different-sized items. Then these items are tentatively loaded on to the next lowest cost available FTO until all items with a positive savings have been loaded or the capacity of the ITO has been reached.

Then the total savings associated with the FTO under consideration in this period is calculated as follows:

T.SAVINGS = C Qit (SAVINGS)i, (2) iEN. teT


where: N = set of items required in period t loaded on the FTO under consideration in the current period.

Qit = units of item i required in period t loaded on the FTO under consideration

in the current period.

If the total savings exceed the fixed transportation cost, this procedure initializes (includes in the solution) the FTO under consideration with the appropriate items. It then checks the totul

savings for the next lowest cost FTO available in this period. This process repeats until an FTO is found uneconomical, all FTOs available in this period have been assigned, or all positive

savings have been expended.

The Change Routine

To further improve the performance of the heuristic algorithm, a CIzange routine may be initiated after each use of the Savings routine. If the last FTO initialized by the Savings routine in the current period is below capacity or a variable cost transportation option (VTO) is used in the current period, the Change routine is initialized. The Change routine involves the calculation of the “savings” (CHANGE) associated with eliminating or replacing the last FTO previously initialized for a future period with one in the current period. If there are several FTOs initialized in future periods, then the Change routine can be investigated for each potential combination of FTOs that can be elii~inated in future periods. The Cruise routine is needed because it may be more economical to replace an FTO initialized in a later period with an earlier FTO. CHANGE is determined as follows:

CHANGE = IZ{Vi, (8,, _ @,I) + IC fjt _ (f,iYjl)] - tit,hi Sit} je_l (3) ieF’ teT

where, in addition to the previously defined variables:

1 = the current period.

t = a future period where an FTO is being considered for elimination.

F’ = set of items currently shipped in periods 1 through the end of the planning horizon

J = set of FTOs in periods tl and t currently in the solution

fjl = fixed cost of FTO under consideration in the current period.

fjt = fixed cost of FTO being considered for elimination in period t.

yjl = I. if the FTO under consideration in the current period is not currently in the solution

= 0, otherwise.

8iji = units of item i shipped by VT0 j when the FTO under consideration in the current period is initialized.

8ij, = units of item i shipped by VT0 j when the FTO is being considered for elimination in period t is initialized.

6j, = units of item i previousty shipped in period t, now shipped by the FTO under consideration in the current period. Shipment of these items early causes the system to incur additional inventory holding costs.

Each term in the churzge calculation captures the change for a particular type of cost resulting from the initialization of an FTO in the current period and the elimination of an FTO from a

190 Vol. 8, No. 3

future period (or several FTOs from future periods). The Change routine can be performed for

each possible rearrangement of FTOs. For example, a situation with two future FTOs results in

three change calculations (elimination of one of the future FTOs, elimination of the other future FTO, and elimination of both of the future FTOs). The first term in the change calculation (vij

(8,, - Fiji)) equals the change in variable transportation costs. The 8ij, variable defines the items shipped on VTOs before the change is initiated and the 8iji variable defines the items shipped by VTOs after the change is initiated. The second term (&r fjt - (fj,Y,l)) equals the change in fixed costs. Notice that if the FTO under consideration in the current period is already in the solution (Yj, = 0), then the fixed cost savings from eliminating future FTOs are maximized. The final term (tit3 hi Sit) equals the additional inventory holding costs that are incurred (if any) by shipping items earlier as a result of the Change routine. If the maximum change is greater than $0, then the appropriate items from the maximum change are either switched to the FTO in the current period or a VT0 in a future period (items from FTOs that are eliminated may now be shipped by VTOs), and the FTOs from the future periods are

eliminated.

The Complete Procedure

After the Savings and Change routines have been completed for each of the distribution center-regional warehouse linkages, the procedure determines the requirements for the regional warehouses by offsetting the resultant shipments by the appropriate lead times. Then the process repeats for each of the regional warehouse-central warehouse linkages. Also, throughout the heuristic algorithm the capacities of the stocking locations and the transport options, as well as item availability at the next lowest echelon, must be considered.

A general flow diagram of the heuristic algorithm is shown in Figure 2. In the flow diagram, “item” refers to a single unit of an item. Although SAVINGSi, need only be calculated for each type of item, the assignment of items to FTOs is done according to individual units to simplify the programing of the procedure. This diagram also shows how the various routines embedded in the heuristic interact in a logical manner. Nonetheless, this flow diagram does not include many of the detailed nuances of the overall procedure. The following discussion of this diagram

provides some of these details. The initial determination of net requirements for the highest echelon linkages, the first step in

the heuristic algorithm, may result in backorders. When determining these backorders the procedure analyzes inventory balances at all locations since backorders may be caused by unavailable inventory at a lower echelon. The heuristic procedure makes a fair shares allocation (Brown (1977)) of available inventory and satisfies backorders as soon as possible in the future. By adjusting for limited inventory in this manner, the heuristic helps make the initial solution a feasible one.

The Savings and Change routines also include additional complexities not shown in Figure 2. Both routines only consider shipping items earlier than required if the items are available at a lower echelon or the planned lead time is sufficient to allow the lower echelon to order and receive the item. In the latter case, additional inventory costs are incurred only if the central warehouse has to place an order earlier. In essence, both the Savings and Change routines include a small search procedure that monitors inventory levels at lower echelons to check for this condition. In addition, both the Savings and Change routines are limited by capacity considerations and utilize a “switching” maneuver. Switching refers to the reallocation of items on FTOs so that the items with the lowest inventory holding costs are shipped earliest.


FIGURE 2

FLOW DIAGRAM OF HEURISTIC ALGORITHM

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For example, the savings associated with shipping an item early are equal to the tr~spo~ation cost reduction minus the cost of holding the item shipped early (assuming the central warehouse has to place an order early). The item shipped early can be either the item under consideration (ranked in order of SAVINGS,,) or another item (with a lower inventory holding cost or

available inventory at a lower echelon) previously scheduled to be shipped on an FTO in the future period. In the latter case, the allocation of items on FTOs in the future period is switched to allow the lower inventory holding costs.

The features of the heuristic algorithm as presented in this research study are outlined in Table 1. However, for some situations not investigated in this study, it may be desirable to modify the heuristic algorithm. For instance, capacities could be constrained by throughput rather than ending inventories. When location capacity or availability restrictions are reached, it might be beneficial to consider a prioritization scheme that considers customer service as well as cost. If products are large and have irregular package sizes, an optimal procedure for loading FI’Os may be a necessary addition. The heuristic can be modified to address other scenarios. A simple demonstration of the heuristic algorithm fotlows in the next section.

TABLE 1 FEATURES OF THE HEURISTIC ALGORITHM

Feature Description

I.

2. 3.

4. 5.

6.

7. 8.

9.

IO.

II.

12.

Backordering

Explosion process

Initial solution

I. Determines the initial orders that cannot feasibly be met and satisfies

these as soon as possible.

2. Calculates the net requirements for the distribution centers.

3. Calculates an initial feasible solution, which entails shipping all

SAVfNGS routine 4.

CHANGE routine 5.

Stocking location capacity 6.

Transport capacity 7.

Inventory holding costs 8.

Supply feasibility 9.

Lower echelon demands

Lower echelon solutions

10.

11.

Switching maneuver 12.

items on VTOs. Loads FTOs with items from VTOs to improve the initial solution.

Further improves the solution by rearranging the initialization of FTOs

in a predetermined manner. Accounts for distribution center and/or warehouse capacity limits

before shipping items early via the SAVIIVGS or CHANGE routines.

Considers the finite capacity of FTOs when loading.

Searches lower echelons to determine if the shipment of an item

early will cause the system to incur ~~~~jfiu~fi~ inventory holding

costs. Checks lower echelons to determine if it is feasible to ship items,

given the necessary lead times.

Calculates the demand for the next lowest echelon after solving the

echelon above.

Repeats the process to determine a solution for the next lowest

echelon.

Switches items with others on FI’Os so that lower additional inventory

holding costs are incurred.

A Single-Item, Single-FTO Demonstration of the Heuristic

Consider the simple network problem depicted in Figure 3. In this trivial problem (limited only to provide a clearer presentation), there is only one transportation linkage (between the distribution center and the regional warehouse) with a constant lead time of one period and a single item. In addition, there are only two transpo~ation options: (1) an FTO with capacity of 100 units, a fixed cost of $100, and no variable charges; and (2) an uncapacitated VT0 with a


$3 per unit charge. The cost to hold an item in inventory is $1 per period. In this example, the initial feasible solution determined by the heuristic is $345. This solution can be improved by

the Savings and Change routines, which follow. The Suvin~s routine improves the initial solution by determining that it is more economical to

ship the 70 units in period 3 by an FTO rather than a VTO. This trade-off of the fixed transportation costs of an FTO for the variable transportation costs of a VT0 is graphically depicted in Figure 4. Notice that the T.SAVINGS of $210 are greater than the fixed cost of the FTO ($100) for that period, thus the FTO is initialized. Similarly, the Savings routine results in another FTO being initialized for period 2. However, it is not economical to initialize an FTO in period 1 since the T.SAVINGS of $30 in that period are less than the fixed cost of the FTO (SlOO). These changes are reflected in the “improved solution” with total costs of $230 in Figure 3.

FIGURE 3 SINGLE-ITEM, SINGLE-FTO DEMONSTRATION

Distribution Center

Distribution

I

Center

Initial Solution: Additional IN. Holding Costs = $0 Transwnatirm COSIS = Wt35t7OM3) = $345 TOTAL COSTS = $345,

Dislributian Cenkr

Impmved S&ion: Ad~uion~ lnv. Holding Cos,su = $0 T~n~~ti~ Costs = (10)(3) + 100 + 1(x = $230 TOT~~O~=$230

Distribution Center

Best Solution: Addi[i~al bu. Holding Costs = (SS)(l) = $55 T~s~~tion Costs = (10)(3) c 1cO + (5)(3) = $145 TOTAL COSTS = $200

194 Vol. 8, No. 3

FIGURE 4 ANALYSIS OF T.SAVINGS FOR SINGLE-ITEM,

SINGLE-FTO DEMONSTRATION

Total Transportation

cost

f,= $100

Variable costs of VT0 currently used-

L Period 3

INGS = 70(3) = $210 > f,

Transportation

.

Variable costs of VT0 currently used


cost

f, = $100

Variable costs of VT0 currently used -

L Period 1


The Change routine then calculates the benefit from eliminating the FTO in period 3. Since the $100 savings in fixed costs less the additional $15 variable transportation costs (5 units shipped in period 3, times $3 per unit shipped) and additional inventory holding costs of $55 (55 units are required earlier by the regional warehouse) are greater than $0; the change is initiated and the total costs are reduced to $200, as shown in Figure 3. This solution also

happens to be the best solution for this example.

A Multiple-Item, Multiple-FTO Demonstration of the Heuristic

The previous analysis of the Savings and Charr~e routines can be extended to include multiple items and multiple FTOs in each period. In this case, FTOs of equal capacity are introduced in order (lowest to highest) of the fixed cost for each FTO (if the capacities of the FTOs differ, then the introduction is in order of the ratio of fixed cost to capacity for each FTO). In this situation, transportation costs would be variable (since the lowest cost VT0 would be used) until a break-even volume is reached. This break-even volume is the point where it becomes economical to initialize the respective FTO and incur a fixed cost instead of variable charges (T.SAVINGS > fixed cost). Again, assuming FTOs have only fixed costs, transportation costs do not increase until the capacity of this FTO is reached (if FTC.& also result in variable costs, then costs woufd increase, but at a slower rate). Then this process repeats until all the FTOs are considered or all units have been allocated (assigned to an FTO) at this location in the particular period under consideration.

Consider a scenario in which the volumes (reflecting the sum of possibly different sized items) required for each period are as denoted in Figure 5. Atso shown in Figure 5 are the break-even volumes (Ut* for the lowest cost FTO and U?* for the second lowest cost PTO), the FTO capacities, the preliminary nggregare (summed over all products) variable transportation costs associated with each period (TC,, TC?, and TC3), and the fixed cost for each FTO (fl and fz). Savings are realized whenever items are shifted from a higher variable cost option to*an FTO. Starting from the last period (period 3), total savings are equal to only the aggregate variable ~anspo~ation cost reduction (TC3), since there are no additional inventory holding costs when ordering for this single period (TCX = Xi Vii’). Because total savings are less than the fixed cost associated with the second FTO for this period (fz), the second PI’0 is not initialized. Likewise, the calculations for rota1 savings for periods 2 and 1 (assuming sufficient

FTO capacity) are as follows:

Period 2: T.SAVINGS = TC7 + TC2 - HC3

This result comes from equation 2 in the following manner:

Referring to equation 2 for period 3:

2 Qi3 (SAVINGS)i3 = C Qi3 (vij* - tit, h,) ieN ieN

= C Qi3 vi;’ - Qi? tits hi ieN

= TC3 - HC3

Similarly, referring to equation 2 for period 2:

C Qil (SAVINGS)iz = TCZ ieN

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FIGURE 5 MULTIPLE-ITEM, MULTIPLE-FTO DEMONSTRATION


Cost

Period 3 Regional Ware. - DC Linkage

u2 Capacity

Volume

A~gateVolumsof IXmandinPeriod3


cost


A-gateVol"mcof DemmdhPeriod2


Cost



Period 1: T.SAVINGS = TC, + TC2 + TCX - HC? - 2(HC3)

Again, this result comes from equation 2 in the following manner:

C Qi3 (SAVINGS)i3 = TC3 _ 2(HC3) ieN

~ Qi2 (SAVINGS)i2 = TC:! _ HC? ieN

C Q,, (SAVINGS),, = TC, kN

HC2 and HC3 are the aggregate additional inventory holding costs incurred per period when ordering early for periods 2 and 3, respectively. When total savings for a particular period exceed the fixed transportation cost, an FTO is initialized.

For example, consider a situation in which the fixed cost for the second FTO shown in Figure 5 is $100 and other costs are as follows:

TC, = $30

TC2 = $40 HC? = $15

TC3 = $80 HC3 = $10

Assuming that FTO capacities are not binding, the .~~~~ings analysis for this example would be as follows:

Period 3: T.SAVINGS = $80 Since $80 < $100, do not initialize the second FTO in period 3.

Period 2: T.SAVINGS = 40 + 80 - 10 = $1 IO Since $1 10 > $100. initialize the first FTO in period 2.

Period 1: T.SAVINGS = $30 Since $30 < $100. do not initialize the second FTO in period 1. Notice that items from periods 2 and 3 are no longer considered in the calculation of suvings for period 1 since these are all tentatively scheduled to be shipped on an FTO in a later period. Hence, a variable transportation cost of $30 will be incurred in period 1.

Again, the Change routine is needed because it may be more economical to replace an FTO initialized in a later period with an earlier FTO. For this example, the Change routine would

analyze whether the FTO should be switched from period 2 to period 1. Using the data and conditions noted previously, the calculations for the Changr routine would be as follows:

CHANGE = TC,-HC2-HC3 = 30. 15- 10 = $5

Referring to equation 3:

2vi.i (Gij, _ 8iii) = TC, ieF’

Y,, = 1, f,? - (f,,Y,,) = 0

C(-ti,,hiG,,) = - HC? _ HC3 ieF’

Since $5 > $0, the FTO in period 2 is eliminated and an FTO in period I is initialized. The logic here is that it is less expensive to incur the additional inventory holding costs by shipping items for periods 2 and 3 earlier and eliminating the high variable transportation costs in period 1. Now the tentative solution from the Suvin~s routine has been modified so that two FTOs are

198 Vol. 8, No. 3

initialized in period 1, no FIOs are initialized in period 2, one FTO is initialized in period 3, and no items are shipped by a VTO.

It should be noted that additional inventory holding costs (HCZ and HC3 above) are only incurred when there is not sufficient inventory available at a lower echelon. For example, in the

previous single-item example depicted in Figure 3, the best solution (which was determined by using the Savings and Change routines) was to ship 65 units one period early (ship 100 units instead of 35 units in period 2). However, because of an inventory balance at the central warehouse, additional inventory costs were only incurred for the 55 units that must be ordered early at the central warehouse. Hence, inventory balances at lower echelons must be considered

when analyzing both the Savings and Change routines. These two simple examples provide an introduction as to how several segments of the

heuristic algorithm operate. However, neither of these examples fully tests the heuristic in a multi-item, multi-echelon environment for which it is intended. The following experiment

provides a more thorough examination of the heuristic algorithm.

EXPERIMENTAL DESIGN

The heuristic algorithm is tested in an environment with both demand uncertainty and a rolling horizon to closely approximate the conditions under which firms operate. Of course, the inclusion of these complexities sacrifices our ability to measure the absolute performance of the heuristic, since the optimal solution cannot be determined. Instead, the heuristic is compared to

a rolling horizon MILP procedure that optimizes for each planning horizon individually, but does not necessarily yield the globally optimal solution. This comparison provides insight as to how well this heuristic approach to the multi-echelon inventory control problem compares to alternate MILP based approaches. Unfortunately, this type of comparison requires that the test problems be rather small, as the MILP based approach is intractable for real world size problems.

The model used in this research consists of four distribution centers ordering from two regional warehouses, with the two regional warehouses in turn ordering from a central warehouse. Each set of two distribution centers orders from a predetermined regional

warehouse as previously shown in Figure 1. This distribution network is similar to the one used by Eastman Kodak Company described by Rosenbaum (198 1) with the additional complexity of multiple regional warehouses. Lead times from the central warehouse to the regional

warehouses, from the regional warehouses to the distribution centers, and from the sources of supply to the central warehouse are constant and equal to one period. In addition, there are two transportation options (an FTO with a fixed cost and a VT0 with a single variable cost).

The Simulation Model

As with any simulation study, this research must address the unanswered question of how to avoid bias from the initialization of the model. Using the terminology of Kleijnen (1974), this simulation represents a nonterminating system. In other words, the multi-echelon environment modeled here does not have a critical event that terminates the system. Managing inventory in a multi-echelon environment is a continuous process. Thus, the performance measures from the simulation model of this system need to represent steady-state behavior.

Rather than arbitrarily discard observations that appear to be part of a transient phase, this research follows the approach of Bratley, Fox, and Schrage (1983). They argue that for inventory systems similar to the multi-echelon inventory control problem of this research,


steady-state is achieved when a new order arrives at a location. What Bratley et al. describe as steady-state conditions are achieved in this research after new orders for every item have arrived at each of the locations. When this happens, the initialization bias caused by the initial inventory levels is no longer present. By beginning the simulation for the environmental analysis with zero inventory at all locations, it will take exactly three periods (the cumulative lead time) for every location to receive an order for each item and for the inventory levels at

each of the locations to be solely a function of the inventory control system. Consequently, the simulation model was run three periods before the data collection process began. Subsequent graphs of decision variables in this experiment confirmed that steady-state conditions were achieved.

The simulation model implements all the procedures necessary to solve the multi-echelon inventory control problem under conditions of a rolling horizon and uncertain demand. The model first randomly generates forecasted demands for each item over the planning horizon. The solution procedures solve the problem defined by these forecasted demands and implement the resultant policies for the first period. The actual demand for the first period is sampled from a normal distribution with a mean equal to the forecasted demand and standard deviation as defined by the UNCERTAIN factor. Then the performance measures are calculated for the actual demand, the model is rolled forward one period, and the process is repeated. The rolling horizon results in one new forecasted demand for each item being introduced each period. A graphical representation of this procedure is provided in Figure 6. Notice that the planning horizon remains constant throughout each experimental run with the solution representing decisions made in the fourth through seventh periods.

Even though this experiment includes customer demand uncertainty, no safety stocks are maintained throughout the multi-echelon model. In actual applications, uncertain customer demand would involve the maintenance of safety stocks at distribution centers to provide the appropriate service levels. However, for the purposes of this experiment safety stock would only serve to mask the performance of the alternative inventory control systems. If it becomes necessary to backorder a particular item at a particular location, a fair shares allocation procedure (Brown (1977)) is implemented for the available inventory. Unsatisfied demand is then filled in the following period, or as soon as possible.

The simulation model was verified in several ways. The primary component of the simulation model, the heuristic algorithm, was manually verified for small test problems. A stress test (Bratley et al. (1983)) of the model at extremely high and low parameter and factor settings was conducted. In every case the model yielded explainable results. The implementation of the rolling horizons was manually checked over several practice runs and determined to be operating correctly. In addition, the random sampling procedures yielded results that were consistent with their distributions. Finally, the various different modules of the overall model were found to be properly linked and operating as intended.

The proper approach for validating a simulation model is open to question (see Bratley et al. (1983) and Van Horn (1971)). While it would be inappropriate to claim that the simulation model in this research is an identical match to actual multi-echelon inventory control problems that exist in practice, this model is intended as a scaled-down version of actual problems. Actual problems have many more products, transportation linkages, transportation options, as well as non-stationary demand and uncertainty. However, it is believed that this model captures sufficient realism to adequately test the procedures. At the early investigative stage represented by this research, the advantages of better experimental control that come from the limited size and complexity of the test environments far outweigh the loss in generality that results. Of

200 Vol. 8, No. 3

FIGURE 6 DEMONSTRATION OF IMPLEMENTATION OF PLANNING HORIZONS

Period 11213141516171819IlOlll

Experimental Run

Key:

Length of each planning horizon = 5 periods.

-Initial period in planning horizon where demand uncertainty is implemented.


course, an obvious extension to this research is to test the heuristic algorithm in an unquestionably valid actual environment.

In summary, this simulation model does not include all the complexities of actual multi-echelon inventory control problems. However, the careful selection of parameters and experimental factors from industrial data and pilot studies results in a model that closely approximates the complexities involved in solving multi-echelon inventory control problems. Of course, the results from this experiment are subject to the assumptions and limitations outlined throughout this research.

Experimental Factors

This experiment includes both a SYSTEM factor and an UNCERTAIN factor which define the solution method and level of demand uncertainty, respectively. Given two levels for each of these factors, a full factorial experiment results in four experimental cells. Three replications per cell and four rolling horizon iterations per run yield an experiment with 12 runs and 48 individual problems to solve.

The environment for this experiment, as defined by cost, capacity, and coefficient of variation in demand parameters, is set so as to approximate an actual environment and provide the most difficult test for the heuristic algorithm. A pilot study was conducted to determine which environments were potentially the most difficult for the heuristic algorithm to solve. This pilot study involved running the heuristic algorithm for various combinations of factor settings and checking the difficulty the heuristic incurred in solving the resultant problems. Difficulty was measured by the number of times (relative to the problem size) that the heuristic had to utilize the Change routine. Since the Change routine is a correction procedure that eliminates FTOs previously initialized by the Savings routine, the number of Change routine calls suggests difficulty encountered by the heuristic.

The two levels for the SYSTEM factor are the heuristic algorithm and the MILP based approach. The two levels for the UNCERTAIN factor are set to cover a range of potential demand uncertainty. In previous multi-echelon research, Allen (1983) arbitrarily used three levels of demand uncertainty defined by standard deviations ranging from 25% to 100% of forecasted values, while Hummel (1984) anchored his study to actual data. Both of these approaches have merit. Since actual demand data can be market and product specific, selecting experimental factors that replicate interesting environments may provide valuable insights as to

the expected performance of a new methodology. On the other hand, using specific data suggests that there is at least one (or rather, wus one) actual environment where the new methodology performs as experimentally determined. The purpose of the UNCERTAIN factor in this research is essentially to determine if the performance of the heuristic is affected by the

amount of demand uncertainty represented by this factor. Given the limitations of a fixed factor experimental design, the effect of the UNCERTAIN factor can best be determined if the levels for this factor are set rather far apart. However, setting this factor at totally unrealistic extremes may yield results that are neither specific to an actual problem nor interesting in terms of determining expected performance. Hence, the experimental levels of the UNCERTAIN factor for this research are set at standard deviations of 10% and 50% of forecasted demand. These levels cover a wide range and also closely approximate the range of the actual demand uncertainty used by Hummel (1984).

202 Vol. 8, No. 3

Measurement of Performance

This experiment includes three measures of system performance, the first of which is the ratio of the sum of transportation and inventory holding costs (total costs) to a weighted transportation cost for the number of units transported in the experimental run. Dividing total costs by the weighted transportation costs (cost of shipping everything by VTOs) results in a cost performance measure that is “normalized” relative to the number and type of units actually transported in each run. This adjustment is necessary because the heuristic and MILP approaches may result in different shipment schedules. If costs were not normalized in this manner, it would be possible for either of these techniques to improve cost performance relative to the other by not shipping (backordering) items with high transportation cost. Hence, the normalized cost performance measure used in this research provides a better measure as to the relative performance of the alternative inventory control methods. This performance measure

can be formally defined as follows:

Cost performance measure =

c [fL?k Y?kt f v,I k Xilkt + hi Iiktl

ikt

c [V,lk xijkd

i,jeJ(k),kt

where, in addition to the previously defined variables:

J(k) = set of feasible transport options to location k.

units of item i delivered by transport j to location k in period t.

units of item i in inventory at location k at the end of period t.

variable costs associated with shipping one unit of item i by transport option j to location k.

Xijkt =

likt =

Vijk =

fjk =

Yjkt =

thus:

fixed costs associated with using transport option j to location k.

fixed charge variable for using transport j to location k in period t.

c [f2k Y2kt] = fixed transportation costs (j = 2 = ITO). ikt

C [viik x1 I kt] = variable transportation costs (j = 1 = VTO). ikt

x [hi Iikt] = inventory holding costs. ikt

2 tV,lk Xijkt] = cost of shipping exclusively with VTOS.

i,jeJ(k),kt

The lower bound for this performance measure is realized when Y2kt is allowed to be a fractional value. In this situation, all items are shipped by the theoretically lowest cost method and the only inventory holding costs will be due to the uncertainty in demand. The upper bound is realized when the fixed transportation cost term is zero. In this situation, all items incur the highest possible transportation costs. No savings are achieved by incurring additional inventory holding costs in order to ship items on FTOs.

The other two performance measures, customer service as represented by the fill rate for


customer orders and the total number of internal orders placed, provide additional information concerning the, solution procedures. The customer service measure shows whether better cost performance results in poorer customer service. The number of orders placed provides a measure as to the pressure the system exerts on available resources. All three performance

measures are then analyzed using multivariate analysis of variance (MANOVA) techniques.

ANALYSIS OF RESULTS

The experimental results from the environmental analysis are provided in Table 2. As previously described, the cost performance measure for each run is determined by dividing the total actual costs by the total weighted costs. In addition, the cell means for this two-by-two full factorial experiment with fixed factors are shown in Figure 7. Notice that the heuristic algorithm performed quite well in this experiment. Before conducting the experiment it was hoped that the heuristic’s performance would be close to the MILP based approach. However, these results show the heuristic to either meet or exceed the MILP approach on all three performance measures. Although it is not clear at this point whether these differences are significant, these results tend to support the use of the heuristic.

Statistically significant differences in this MANOVA experiment are analyzed with the Wilks’ criterion and Pillai’s trace approaches. Wilks’ criterion is historically the most widely used and Pillai’s trace was shown to be the most robust of the methods available. However, for this experiment it is unimportant as to which significance test is used, because all the commonly available techniques yield identical results for two factor experiments (further details concem- ing MANOVA are available in Bray and Maxwell (1985) and Kirk (1982)). As expected, the differences in the UNCERTAIN factor were found to be statistically significant. Follow-up univariate F tests (summarized in Table 2) show the customer service performance measure to be the primary cause of this significance. However, the factor that we are interested in the most, SYSTEM, is not even remotely significant. Based on the results of this experiment, no significant difference can be determined between the MILP approach and the heuristic algorithm when applied in an environment with demand uncertainty and a rolling horizon.

As expected, as demand becomes more uncertain, customer service suffers for both the MILP approach and the heuristic algorithm. The average level of customer service falls from 93.1% to 66.0% for the MILP approach and from 90.8% to 63.1% for the heuristic algorithm. Somewhat unexpected is the result that additional uncertainty in demand does not significantly affect cost performance. The average cost performance for the MILP decreases slightly from 0.665 to 0.652 for the MILP approach and 0.657 to 0.653 for the heuristic algorithm.? In this case, both methods continue to ship products efficiently, albeit to the incorrect locations.

An alternate analysis of the results would be to remove the number of orders performance measure from the multivariate analysis, since neither method attempts to minimize this variable. The number of orders is only included to provide a measure of the demands placed on the ordering resources of the organization. It is apparent from observation and the analysis of variance results that this measure was not significantly different for the two systems. Furthermore, conducting the multivariate analysis without this variable results in the same conclusions. This alternate analysis yields the following MANOVA results:

Prob. > F

204

SYSTEM Factor 0.7369 UNCERTAIN Factor 0.0003

Vol. 8, No. 3

TABLE 2 EXPERIMENTAL RESULTS

Run SYSTEM UNCERTAIN

Actual Weighted Performance Measures

costs costs cost Service’ Orders’

1

2

3

4

5

6

I

8

9

10

I1

12

MILP

MILP

MILP

MILP

MILP

MILP

Heuristic

Heuristic

Heuristic

Heuristic

Heuristic

0.1 5651.38 8902.08 0.635 0.909 96

0.1 5775.51 8853.30 0.652 0.934 97

0.1 5641.31 7965.54 0.708 0.951 103

0.5 5696.98 8738.34 0.652 0.556 96

0.5 5575.59 8805.60 0.633 0.711 95

0.5 6164.97 9174.66 0.672 0.713 94

0. I 5626.30 8705.70 0.646 0.890 97

0.1 5383.70 8317.02 0.647 0.923 96

0.1 5543.12 8176.38 0.678 0.909 99

0.5 5629.87 8526.42 0.660 0.566 93

0.5 5226.41 8546.70 0.612 0.659 91

0.5 6387.57 9292.80 0.687 0.667 100

‘Customer service measured as the fill rate proportion.

‘Number of internal orders placed

MANOVA and ANOVA Results

MANOVA

SYSTEM Factor

Wilks’ Criterion

Pillai’s Trace

UNCERTAIN Factor

Wilks’ Criterion

Pillai’s Trace

ANOVA

cost

SYSTEM

UNCERTAIN

SYSTEM*UNCERTAIN

Service

SYSTEM

UNCERTAIN

SYSTEM*UNCERTAIN

Orders

SYSTEM

UNCERTAIN

SYSTEM*UNCERTAIN

Prob. 1 F

0.8688

0.8688

0.8406

0.6406

0.8109

0.4229

0.0001

0.9332

0.6602

0.1211

0.7911


FIGURE 7 CELL MEANS FROM EXPERIMENT

MILP Approach Heuristic Approach

Avg. Cost Perform. = 66.5% Avg. Service Level. = 93.1%

Avg. # of Orders = 99

Avg. Cost Perform. = 65.7% Avg. Service Level = 90.8%






Thus, the discussion of results for the environmental analysis need not be tempered in any manner by the inclusion of the number of orders as a performance measure.

The significant UNCERTAIN factor in this research is caused by no safety stock being available to buffer against uncertain demand at the distribution centers and the inability of the system investigated to redistribute inventory among locations to balance the effects from lower than anticipated demands, Safety stocks were eliminated from the experiment so as not to mask the effects of the UNCERTAIN factor. However, in an actual environment it is often preferable to control the effect of uncertain demand (UNCERTAIN) on the level of customer service, In an actual implementation, safety stock can be added as a function of demand uncertainty to yield an expected level of customer service. If demand unce~ainty is so high as to cause severe inventory imbalances, a redistribution policy can also be initiated as an extension to this research. The important point to note is that the degradation of the methods caused by the UNCERTAIN factor can be controlled by adding safety stock and incurring additional inventory holding costs in an actual environment.

The results from this study suggest that when demand uncertainty is introduced, this heuristic algorithm performs quite well (and sometimes even performs better) than the MILP procedure that is optimal for static and deterministic problems, These results are consistent with previous

206 Vol. 6, No. 3

single-level and multi-level MRP lot-sizing research by Blackburn and Millen (1982) and

Wemmerlov and Whybark (1984) which found heuristic techniques to match (or even out-perform) optimum seeking procedures when rolling horizons and demand uncertainty are introduced. This study yields the result that the heuristic algorithm works best (relative to the MILP approach) in the actual environments for which it is intended.

SUMMARY OF FINDINGS AND SUGGESTIONS FOR FURTHER RESEARCH

This study represents the initial step in the development and testing of a heuristic algorithm for solving the multi-echelon inventory control problem. The heuristic developed in this research balances transportation costs against the cost of holding inventory in a multi-echelon environment. By solving the network in a hierarchical manner, the heuristic has the good result of being efficient, but also the potentially bad result of yielding a suboptimal solution. Although the heuristic algorithm performed very well in this study, future implementations may be improved by limiting potential suboptimalities in a more systematic manner within the search procedure. The Change routine and the linkage between echelons are two areas where one

might look first for improving the heuristic algorithm. Probably the most direct application for this heuristic algorithm is as an add-on module for

DRP applications. The determination of the initial solution in the heuristic procedure is essentially the DRP explosion process. The subsequent routines in the heuristic search for improvements to the initial solution given by the DRP explosion. In this manner, transportation and inventory holding cost considerations can be added to DRP in a systematic manner. This will relieve operations managers from the burden of addressing these issues separately.

Another much needed extension to this research is in the area of experimental testing. This study shows that this heuristic algorithm performs quite well in rather constrained test environments. The question remains as to whether these results can be extended to larger, more realistic environments. Even though larger environments preclude the use of optimal seeking procedures for benchmarks, a study of a larger environment could provide information concerning the performance of this heuristic relative to any alternative heuristic procedures. In addition, there is also a need for a more specific test of the heuristic algorithm. A specific test would be one in which the heuristic is implemented for an actual firm. The settings for this experiment were rather general in nature. This generality was necessary to eliminate potential testing biases and allow for general conclusions regarding the performance of the heuristic algorithm. Now that the heuristic algorithm has been shown to perform well under general conditions, it would be interesting to test this procedure in an actual distribution setting.

With regard to the appropriate environment for the heuristic algorithm, it should be noted that this method was designed to balance transportation costs against inventory holding costs. Since the procedure generates savings from reduced transportation costs, transportation costs need to be a major cost item (which is usually the case). However, an environment where inventory holding costs are also relatively large may provide the best match with this heuristic algorithm. Otherwise, a simple heuristic that loads all transportation options to capacity may be adequate. In addition, this heuristic is specifically designed to operate in environments beset by lumpy demand, fixed cost transportation options which create nonlinear costs, and large problem sizes. Since only heuristics can be used in these environments, a direction for future research would be to test this heuristic against others that can be used in practice.

Journal of Operations Management

ENDNOTE

tThe complex interactions associated with this multivariate analysis are not accounted for in these comparisons

REFERENCES

1. Allen, W.B. A Comparative Simulation of Central Inventory Control Policies for Positioning Safety Stock in a

Multi-Echelon Distribution System. Ph.D. dissertation, Indiana University, 1983.

2. Blackbum, J.D., and R.A. Millen. “The Impact of a Rolling Schedule in a Multi-Level MRP System.” Journal qf

Operations Management, vol. 2. no. 2, 1982, 125.135.

3. Bratley, I?, B.L. Fox, and L.E. Schrage. A Guide to Simulation. New York: Springer-Verlag. 1983.

4. Bray, J.H., and S.E. Maxwell. Multivuriare Analysis of Variance. Beverly Hills, CA: Sage Publications, 1985.

5. Brown, R.G. Materials Management Systems: A Modular Libra?. New York: John Wiley & Sons, 1977.

6. Hummel, J.W. An Evaluation of Inventory Management Alternatives in a Multi-Echelon, Multi-Location Logistics

System. Ph.D. dissertation, The Pennsylvania State University, 1984.

7. Kirk, R.E. Experimental Design. Belmont, CA: Brooks/Cole Publishing Company, 1982.

8. Kleijnen, J.P.C. Statistical Techniques in Simulation. New York: Marcel Dekker, Inc., 1974.

9. Martin, A.J. DRP: Distribution Resource Planning-Distribution Management’s Most Powerjiul Tool. Englewood Cliffs, NJ: Prentice-Hall, 1983.

10. Rosenbaum, B.A. “Inventory Placement in a Two-Echelon Inventory System: An Application.” In TIMS Studies

in the Management Sciences, vol. 16, L. Schwarz (ed.) New York: North-Holland, 1981, 195.207.

I I. Stenger, A.J., and J.L. Cavinato. “Adapting MRP to the Outbound Side-Distribution Requirements Planning.”

Production und Inventov Management, Fourth Quarter, 1979, I- 13.

12. Van Horn, R.L. “Validation of Simulation Results.” Management Science, vol. 17. no. 5, 1971. 247.258.

13. Wemmerlov, U., and D.C. Whybark. “Lot-Sizing under Uncertainty in a Rolling Schedule Environment.”

internationul Journal of Production Research. vol. 22, no. 3, 1984, 467-484.

14. Whybark, D.C. “MRP: A Profitable Concept For Distribution.” Proceedings of the Fifth Annual Transportation

and Logistics Educators Conference, 1975. 82-93.

208 Vol. 8, No. 3

Documents

A heuristic algorithm for managing inventory in a multi-echelon environment