a numerical analysis of capacitated postponement

8/2/2019 a Numerical Analysis of Capacitated Postponement

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A NUMERICAL ANALYSIS OF CAPACITATEDPOSTPONEMENT*

GREGORY A. GRAMAN AND MICHAEL J. MAGAZINE

Department of Management Science and Information Systems, Wright StateUniversity, Dayton, Ohio 45435, USA

Department of Quantitative Analysis and Operations Management, University ofCincinnati, Cincinnati, Ohio 45221-0130, USA

Customer satisfaction can be achieved by providing rapid delivery of a wide variety of products.

High levels of product variety require correspondingly high levels of inventory of each item to

quickly respond to customer demand. Delayed product differentiation has been identified as a strategy

to reduce final product inventories while providing the required customer service levels. However, it

is done so at the cost of devoting large production capacities to the differentiation stage. We study

the impact of this postponement capacity on the ability to achieve the benefits of delayed product

differentiation. We examine a single-period capacitated inventory model and consider a manufac-

turing system that produces a single item that is finished into multiple products. After assembly, some

amount of the common generic item is completed as non-postponed products, whereas some of the

common item is kept as in-process inventory, thereby postponing the commitment to a specific

product. The non-postponed finished-goods inventory is used first to meet demand. Demand in excess

of this inventory is met, if possible, through the completion of the common items. Our results indicate

that a relatively small amount of postponement capacity is needed to achieve all of the benefits of

completely delaying product differentiation for all customer demand. This important result will

permit many firms to adopt this delaying strategy who previously thought it to be either technolog-

ically impossible or prohibitively expensive to do so.

(DELAYED PRODUCT DIFFERENTIATION; POSTPONEMENT; FLEXIBLE MANUFACTUR-ING; MASS CUSTOMIZATION)

1. Introduction

Companies today have taken the strategic approach of offering a wide variety of custom-

ized products to increase customer satisfaction and improve market share. As firms move

from mass production to mass customization, they are faced with the challenge of staying

cost efficient while providing increased customer satisfaction. Many strategies such as

just-in-time and delayed product differentiation have been identified to reduce the final

product inventories typically needed to meet required customer service levels. While we

recognize that delaying product differentiation may achieve this goal, it is done so at the cost

of devoting large production capacities to this differentiation stage. It is our goal to study the

impact of this postponement capacity on the ability to achieve the benefits of delayed productdifferentiation. Our results indicate that a very small postponement capacity is needed to

* Received March 2000; revisions received August 2000, February 2001, and June 2001; accepted June 2001.

PRODUCTION AND OPERATIONS MANAGEMENT

Vol. 11, No. 3, Fall 2002

Printed in U.S.A.

340

1059-1478/02/1103/340$1.25Copyright 2002, Production and Operations Management Society


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achieve all of the benefits of completely delaying product differentiation for all customer

demand. This important result will permit many firms who previously thought it either

technologically impossible or prohibitively expensive to do so to adopt this delaying strategy.

This research also responds to the make-to-stock (MTS) versus make-to-order (MTO)

dilemma that many firms face today. As firms investigate production processes they attempt

to locate points where the process can switch from push to pull. That is, the points whereproducts can be produced to a generic form and then differentiated only after demand is

realized. We assume such points of differentiation exist but investigate an alternative hybrid

strategy. We suggest that some inventory be held as final product and some in generic form.

This study was motivated by work with a local manufacturer of non-durable household

items. In particular, one product line was differentiated by only the number of units of the

item in the package shipped to retailers. We noted large inventory of some packaged products

and none of others, while current demand required some of both. In Graman and Magazine

(1998), we investigated the impact of waiting until demand was realized before units were

packaged. We showed under various conditions, that inventory reductions were possible

without deteriorating service rates. In discussion with the firm, however, they indicated thattheir packaging capacity was not adequate to wait until all of the demand is realized and still

meet service-window or lead-time requirements. This led to our hybrid strategy of packaging

some products and leaving others in unpackaged bulk storage, which we refer to as partial

postponement. Other examples where a combined strategy has proven successful is in

computer assembly (Swaminathan and Tayur 1998), the blending of gasoline at the pump,

and mixing paint colors at the retail level (Staudt, Taylor, and Bowersox 1976).

Several authors have examined the hybrid approach we advocate in this study. Cook

(1980) showed that partial postponement of safety stocks at the second echelon resulted in

higher customer service levels than either total postponement or total non-postponement.

Howard (1994) identified a company that does postponement and also builds speculativeinventories to a reduced forecasting window. The use of a hybrid system was considered in

Chakravarty (1989), where a rationing policy is shown to have an impact on the required

capacity size. Fine and Freund (1990) developed a model that focuses on the economic

tradeoffs between the acquisition costs of flexible capacity and a firms ability to respond

flexibly to future uncertain demand. Whitney (1993) described how the requirements of a JIT

mixed-model environment can be addressed by exploiting the design process and the

assembly process of a family of products. The capability for variety is built into the product

rather than depending on the manufacturing system to be flexible or reconfigurable. Eynan

and Rosenblatt (1997) consider a single-period model that minimizes total component cost

subject to an aggregate service level. Even though a common component can replace uniquecomponents in each product, the unique components are used first. Hillier (1998) considered

the possibility of using both cheaper unique parts and a more expensive single common part

that can be used as a replacement for any of the unique parts in the event the unique parts

stock out. A thorough treatment of quantitative models used in managing product variety can

be found in Tayur, Ganeshan, and Magazine (1999).

This paper differs from prior work in that our model is not a cost model, but rather an

inventory-service-level tradeoff model to study the effects of increasing levels of postpone-

ment capacity. We develop a single-period, multiple-product, capacitated postponement

model in recognition that total postponement may not be viable because of capacity

constraints. The model serves as a basis to identify the fundamental relationships betweenpostponement and the inventory-service-level tradeoff and to assess the impact of different

fill rates on inventory level and postponement capacity.

In the next section, we develop the model we will use in comparing total inventory for

fixed service levels and varying postponement capacity. A numerical study and conclusions

follow.

341A NUMERICAL ANALYSIS OF CAPACITATED POSTPONEMENT


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2. Model Development

We consider an existing manufacturing system that produces a single item that is finished

into multiple products. In this study, finishing may mean many things, such as assembly,

painting, packaging, or delivery. The decision as to which final products to offer is set by

management policy and is exogenous to the model. A decision is made about the inventory

levels at the beginning of the period. A single period with a fixed length of time was selectedto gain a clear understanding of the basic tradeoffs. In addition, many products with short life

cycles are amenable to single-period analysis.

2.1. Service Level Measure

Achieving customer satisfaction by meeting all demand requires high inventory levels.

When all demand is not met customer dis-satisfaction occurs. Stockout cost is one measure

of this level of dissatisfaction. However, it is often difficult to determine an exact value for

stockout costs. Common substitutes for stockout cost are service level measures (Nahmias

1997, p. 289). Although there are a number of different ways to quantify service levels

(Fogarty, Blackstone, and Hoffman 1991), it generally refers to either the probability of notstocking out or the proportion of demand satisfied directly from the shelf, or fill rate. (Zeng

and Hayya 1999, a comparative study of these two measures). It seems appropriate to use fill

rate as the service measure in this study because it is generally what most managers mean by

service (Nahmias 1997, p. 290). Other aspects and extensions of service levels are discussed

in Silver and Peterson (1985).

For purposes of this study, filling orders directly from the shelf means that customers

expect delivery within some specific amount of time after the order is placed. Meeting the

target service level implies that orders are filled within a service window offixed length that

is often encountered in practice (Sox, Thomas, and McClain 1997).

2.2. Development of the Capacitated Postponement Model

We examine a single-period, m-product inventory model. Figure 1 shows the process flow

diagram for the partial-postponement scenario. After assembly, some of the common generic

item is completed as non-postponed products and sent to the warehouse to be stored as

finished-goods inventory. In addition, some of the common item is kept as in-process

inventory, thereby postponing the commitment to a specific product. When demand is

realized, some combination of non-postponed and postponed inventories may be used to

satisfy the demand.

FIGURE 1. Process Flow Diagram for the Partial Postponement Scenario.

342 GREGORY A. GRAMAN AND MICHAEL J. MAGAZINE


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The finished-goods inventory of the product is used first to meet demand. Demand in

excess of this inventory is met, if possible, through completion of the common items. It is the

capacity of this completion operation that we will refer to as postponement capacity. The

level of capacity is set to the number of items that could be completed while still meeting

acceptable lead times. Our interest is in setting the non-postponed product inventory levels

given a customer service level and to see how this level is affected by the level ofpostponement capacity. When realized demand exceeds available inventory, an allocation of

postponement capacity is made with a goal of equalizing product fill ratesa policy that may

lead to non-intuitive stocking policies. It is assumed that the inventory of common items will

always be at postponement capacity as this will always result in smaller total inventories for

fixed service levels (Graman and Magazine 1998).

To satisfy a given service-level objective, , it is necessary to obtain an expression for the

fraction of demand that stocks out during the period. Let E(SO) represent the expected

number of stockouts, and be the expected demand during the period. Then,

ESO

1

(1)

If the order-up-to inventory level is chosen so that (1) is satisfied, the service-level objective

will be achieved. In the numerical part of this study, we measure the sensitivity of inventory

levels and postponement capacity to different levels of fill rate ranging from 0.800 to

0.999. When the sensitivity of other model parameters is investigated, the target fill rate

is set at 0.975, reflecting the high fill rate environment of the company that motivated

this research.

Consider the following partial-postponement scenario: let xj equal the realized demand for

product j and SOj equal the number of stockouts for product j. Sj is the non-postponed

inventory level for product j, and C is the amount of postponement capacity in terms of the

common generic item. Let Pj be the amount of postponement capacity used to meet the

demand for product j such that

j

Pj C (2)

Figure 2 is a graphical representation of the inventory levels and possible regions for a

two-product partial-postponement scenario where realizations of demand and stockouts

occur. This diagram is similar to the one used by Tagaras (1989) in solving a two-location

transshipment problem and is similar in structure and interpretation to the diagram used in

Fine and Freund (1990) for describing how to optimally allocate scarce capacity in high-

demand states based on an expected revenue function. The graph is partitioned into k regions,

k 1, . . . , 9. Regions {1} through {6} are de fined by boundaries representing the levels

of S1, S2, and C. Regions {7} through {9} are separated by boundaries representing

equalization offill rates. A detailed explanation of the derivation of the algebraic expressions

of the boundaries can be found in Appendix A. When demand for both products occurs in a

particular region, conclusions about the values of SOj and Pj can be made.

When demand for both products occurs in region {1}, it is met by S1

and S2

. When the

demands occur in regions {2}, {3}, or {6}, they are met by S1, S2, and C, respectively. If

the demands occur in region {4}, the demand for product 1 is met from S1, while total

demand for product 2 is not met because it exceeds the combination of S2 and C. Theconverse is true for region {5}. In regions {7}, {8}, and {9} total demand for both products

is not met. Total demands in region {7} cannot be met, but capacity can be allocated to

equalize the fill rates. Total demands in regions {8} and {9} also cannot be met, and the fill

rates cannot be equalized because the demand for one product exceeds the other by more

than C.



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The expected number of stockouts for product j can be stated by the expression

ESOj k

ESOkj (3)

where E(SO {k}j) is the expected number of stockouts for product j when demand occurs inregion k. A general expression for E(SO {k}j) in each region k where shortages can occur can

be written as

ESOkj xj Sj Pjk fx1 , x2dx 1 dx 2 (4)where (xj Sj Pj)k is the amount of the shortage of product j in region k and f(x1, x2)

is the joint p.d.f. of the random variables of demand X1 and X2 for products 1 and 2,

respectively. The limits of integration are given in the algebraic expressions for the bound-

aries of the region of interest and can be found in Appendix A.The left-hand side of (3) is given by (1). The next step is to use this equation to solve for

S1 and S2, given the capacity, fill rates, and the demand distributions. We will do this for a

variety of scenarios, including different variabilities of demand, various correlations of

demand, and the number of products being postponed. The development of a model for more

than two products is similar but requires many more regions and does not provide additional

insight to the model development process.

2.3. Solution Methodology

There is, unfortunately, no simple, closed-form expression for the order-up-to levels in the

two-product, capacitated postponement model described in the previous section. An addi-tional difficulty lies in the interaction between stocking levels of the non-postponed products

and the postponed items. To illustrate this interaction, consider the equation for the expected

number of stockouts in region {7} for a two-product model (ref. (A13) and (A14)). Some

demand for each product is not met by the non-postponed inventory, and all of the

postponement capacity is allocated to equalize the fill rates, i.e., P1 P2 C. The value

FIGURE 2. Graphical Depiction of Inventory Levels and Regions for a Partial Postponement Scenario Where

Realizations of Demand and Stockouts Can Occur.



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of S1 not only depends on x1 and P1, but also on S2, C, and x2, which are contained in the

limits of integration. The converse is true for S2

.

This complexity and the number of regions in the m-product model led us to develop a

stochastic root-finding algorithm that uses a Monte Carlo integration method to determine the

appropriate values for the Sjs. The algorithm provides the capability to solve the i.i.d.

m-product model, as well as two-product models involving different demand distributionsand correlated demands by iteratively selecting different values S1 and S2 until the right-hand

side of (3) is equal to the left-hand side. An explanation of the algorithm is presented in

Appendix B.

3. Numerical Results

Because we are not able to develop closed-form results, our goal is to gain insights through

computational experiments. Rather than doing a full experimental design and statistically

attempt to verify hypotheses, our goal of insight development leads us to observations

findings that will be present in most cases although not necessarily true in every instance.

Our approach to the problem is best illustrated using a baseline case example. Consider asingle-period, two-product system with demands that are i.i.d. N(j , j) with j 1,000, j 200, j 1, 2, and 0.975. The postponement capacity is set to several

different levels. For any level of postponement capacity, the solution to the capacitated

postponement model is obtained using the approach described in the previous section to find

the values of S1 and S2 required to produce E(SO1) and E(SO2), respectively. Because of

the symmetry of the baseline case, the values of S1 and S2 will be the same. The parameters

and computational results of the baseline case for several levels of postponement capacity are

shown in Table 1. The values in the column headed C as a % ofE(De ma nd) can go above

100% because the amount of capacity necessary to have total postponement may exceed the

total item expected demand. If C is set so high (note C 2,200) that the service level isexceeded through postponement alone, then S1 and S2 will become negative in the model as

we attempt to minimize inventories at the target service level. Clearly, negative inventory

levels are not possible in practice, and this value of C should never be used. The total

inventory required is C S1 S2.

The benefit-capacity curve in Figure 3 is a graphical representation of the data in the

columns headed Percent Reduction and C a s a % o f E(De ma nd) in Table 1. A

benefits-capacity curve shows the relationship between the benefits of postponement, denoted

R, and the postponement capacity, denoted C. The benefits of postponement are expressed

as a percent reduction in finished-goods inventory from zero capacity (non-postponement),

and the postponement capacity is expressed as a percent of total expected demand. We findit convenient to express the benefit as a relative measure rather than in absolute quantity of

items so that comparisons can be made across different product cases.

The solutions to the extreme cases of total non-postponement (capacity 0, inventory

2,311) and total postponement (capacity inventory 2,161) were obtained in Graman and

Magazine (1998). The maximum reduction in item inventory level because of postponement,

denoted Rmax

, is 6.48%. However, we find that only a small amount of capacity is needed to

realize these benefits. Incremental savings are obtained by adding some capacity, but

diminish rapidly beyond capacity of approximately 30% of expected demand. Virtually no

gains in inventory savings would be realized beyond this point. The amount of capacity

needed to realize most of the benefits is small relative to total postponement! This isattributed to a reduction in the variance of demand caused by the risk pooling effect that

occurs when demand for several items is filled from an inventory of common items.

OBSERVATION 1. Only a small amount of postponement capacity is needed to achieve

almost all of the benefits of postponement.

Determination of the point beyond which benefits caused by postponement become



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marginal depends in part on the cost of postponement. Because costs are not part of this

analysis, we arbitrarily define the point of marginal benefit as the amount of capacity required

to realize 90% of the maximum benefit, denoted C90%

. Ninety percent of Rmax

for the

baseline case is equal to 5.83%. The two levels of percent reduction in total finished-goods

inventory that bracket 5.83% are 5.58% and 6.08% with associated capacities of 300 and 400

items, respectively (see Table 1). A linear interpolation between the capacities associated

TABLE 1

Summary of Results for the Baseline Case of Two Products with Independent, Identical-Distributed Normal

Demands, N(1,000,200), N(1,000,200)

Computational Results

S1

E(SO1) S2 E(SO2) C S1 S2

Percent

ReductionC

C as a % of

E(Demand)

0 0.0% 1,155.67 25.00 1,155.37 25.00 2,311.04 0.00%

100 5.0% 1,072.81 25.00 1,072.93 25.00 2,245.75 2.83%

200 10.0% 1,002.23 25.00 1,002.80 25.00 2,205.03 4.59%

300 15.0% 940.65 25.00 941.45 25.00 2,182.10 5.58%

400 20.0% 884.84 25.00 885.59 25.00 2,170.43 6.08%

500 25.0% 832.14 25.01 832.85 24.99 2,165.00 6.32%

600 30.0% 781.00 25.02 781.66 24.98 2,162.66 6.42%

700 35.0% 730.55 25.02 731.17 24.98 2,161.72 6.46%

800 40.0% 680.41 25.01 680.99 24.99 2,161.40 6.48%

900 45.0% 630.38 25.01 630.92 24.99 2,161.30 6.48%

1,000 50.0% 580.39 25.01 580.88 24.99 2,161.27 6.48%

1,100 55.0% 530.40 25.01 530.85 24.99 2,161.26 6.48%

1,200 60.0% 480.43 25.01 480.83 24.99 2,161.26 6.48%

1,300 65.0% 430.45 25.01 430.81 24.99 2,161.26 6.48%

1,400 70.0% 380.47 25.01 380.79 24.99 2,161.26 6.48%

1,500 75.0% 330.49 25.01 330.77 24.99 2,161.26 6.48%

1,600 80.0% 280.51 25.01 280.75 24.99 2,161.26 6.48%

1,700 85.0% 230.53 25.01 230.73 24.99 2,161.26 6.48%

1,800 90.0% 180.55 25.01 180.71 24.99 2,161.26 6.48%

1,900 95.0% 130.57 25.01 130.68 24.99 2,161.26 6.48%

2,000 100.0% 80.60 25.01 80.66 24.99 2,161.26 6.48%

2,100 105.0% 30.62 25.01 30.64 24.99 2,161.26 6.48%

2,200 110.0% 19.36 25.01 19.38 24.99 2,161.26 6.48%

FIGURE 3. Percent Reduction in Item Inventory Versus Capacity as a Percent of Total Item Expected Demand for

Two Independent, Identical, Normal-Distributed Products, N(1,000, 200), N(1,000, 200).



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with the bracketed percents gives the capacity of 350.25 items for 5.83%. Division of this

amount by the total expected demand (1,000 1,000) gives C90% 17.51%.

We note that a more exact value of C90% (17.05%) can be found by using a smallerincrement of capacity (1 versus 100 units). The difference between these two values of C

90%

corresponds to a decrease of approximately 10 units of capacity. We note that interpolation

provides conservative estimates ofC90%. The difference between the values is small and does

not justify the additional computational effort needed to produce this more exact value of

C90%.

3.1. Fill Rate

Because fill rate is a measure of customer satisfaction, we would expect high levels of

customer service to require correspondingly high inventory levels. The effect of fill rate on

the realization of the benefits of postponement at various levels of capacity is examined inthis section. Six different levels offill rate were used in the baseline case. Table 2 and Figure

4 summarize the results of this experiment. Each level of fill rate is shown along with its

respective value of Rmax and C90%. We observe that Rmax and C90% increase in an almost-

linear fashion for low fill rates (0.800 0.950) and increase exponentially as fill rate become

very high (0.950). From these observations we reach the following conclusion:

OBSERVATION 2. The benefits of postponement increase at an increasing rate as the fill rate

increases (the capacity must also increase accordingly to realize any specified benefit).

This may influence our choice of the level of customer satisfaction we provide. High levels

of satisfaction are possible with a relatively small amount of postponement capacity up to

TABLE 2

Sets of Two Products with Independent

and Identically Distributed Demands

with Selected Fill

Fill Rate Rmax

C90%

0.800 1.80% 16.67%

0.900 3.64% 16.72%

0.950 5.21% 17.09%

0.975 6.48% 17.51%

0.990 7.88% 17.99%

0.999 10.50% 20.17%

FIGURE 4. Maximum Benefit From Postponement, Rmax, and the Capacity Required to Realize 90% of the

Maximum Benefit from Postponement, C90%

, as a Function of Fill Rate.



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a point. As we know, filling all customer demands cannot be guaranteed. This presents some

measure of making that cutoff.

3.2. Variability of DemandThe effect of demand variability on the realization of the benefits of postponement at

various levels of capacity is examined. Five different levels of the coefficient of variation

were selected with js equal to 1,000 among the two-product pairs and different js.

Table 3 and Figure 5 summarize the results of this experiment. Each i.i.d. two-product

pair along with its corresponding coefficient of variation are shown along with their

respective values of Rmax

and C90%

. We observe that Rmax

and C90%

increase in an

almost-linear fashion as the coefficient of variation increases. From these observations

we reach the following observation:

OBSERVATION 3. The benefits of postponement increase as the coefficient of variation

increases (the capacity must also increase to realize any specified benefit).

The effect of the variability of demand on inventory levels was also analyzed by holding

the coefficient of variation constant in 12 sets of two independent, identical products with

normal, gamma, and uniformly distributed demands. Each pair differs from every other pair

in their respective values of the mean and SD of demand as well as different demand

probability distributions. We observe in Figure 6 that the benefits of postponement increase

at a decreasing rate as the capacity increases. We also note that different levels of demand

with the same coefficient of variation generate nearly identical benefit-capacity curves that

are independent of the demand distributions used in this study. The curves for the gamma-

distributed pairs are slightly higher than those for the normal and uniform distributions.

TABLE 3

Sets of Two Products with Independent

and Identically Distributed Demands

with Selected Coefficients of Variation

Product Sets Rmax

C90%

c.v. 0.00 0.00% 0.00%

c.v. 0.10 2.69% 9.03%

c.v. 0.20 6.48% 17.51%

c.v. 0.30 9.87% 26.02%

c.v. 0.40 12.60% 34.75%

FIGURE 5. Percent Reduction in Item Inventory Versus Capacity for Sets of Two Independent, Identical, Normally

Distributed Products with Selected Coefficients of Variation.



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OBSERVATION 4. The amount of capacity required to realize the benefits of postponement

is determined overwhelmingly by the coefficient of variation, not by the mean or the SD of

demand alone, nor the demand probability distributions considered in this experiment.

3.3. Correlation of Demands

We performed experiments to isolate the effect of correlated demands on the realization of

the benefits of postponement at various levels of postponement capacity. Nine different levels

of the correlation coefficient are used in the baseline case. We observe in Figure 7, that for

normally distribute demands, Rmax

increases at an increasing rate as the demands become

more negatively correlated and C90%

increases at a decreasing rate. These observations lead

us to our next finding:

OBSERVATION 5. The benefits of postponement increase as the demands become more

negatively correlated (the capacity required to meet a given benefit must also increase to

realize those benefits).

Similar results were obtained for the component commonality problem in Eynan (1996)

where the author reports that larger savings result when demands correlation is negative

compared with the independent demand case.

3.4. Number of Products Being Postponed

We also examine the effect of the number of products being postponed on the realizationof the benefits of postponement at different levels of postponement capacity. The total

FIGURE 6. Percent Reduction in Item Inventory Versus Capacity for Sets of Two Independent, Identical Products

with Normal, Gamma, and Uniformly Distributed Demands and Identical Coefficients of Variation Equal to 0.20.

FIGURE 7. Percent Reduction in Item Inventory Versus Capacity for Sets of Two Identical and Normally

Distributed Products, N(1,000,200), with Correlated Demands and Seven Different Correlation Coefficients (c.c.).



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expected demand is set at 2,000 items regardless of the number of products being postponed.

The number of products ranges from two to five. We observe in Figure 8, for normally

distributed demands, that Rmax increases at a decreasing rate as the number of i.i.d. products

being postponed increases and that C90%

is near constant regardless of the number of

products being postponed. These observations lead us to our next insight:

OBSERVATION 6. The benefits of postponement increase as the number of i.i.d. products

being postponed increases; the amount of capacity (as a percent of total expected demand)

required to obtain this effect stays constant.

3.5. Non-Identically Distributed Demands

To this point, all products being compared were identically distributed. We next want to

compare independent but not identically distributed demand for final products. If only the

mean demands are different, E(SO1) E(SO2) and S1 will not be equal to S2. On the other

hand, if only the SDs are different, E(SO1

) E(SO2

) but S1

may still not be equal to S2

. The

assumption of setting S1 equal to S2 for the identically distributed case in the stochastic

root-finding algorithm was relaxed.

Investigation of non-identically distributed demand will proceed using three approaches:

constant coefficient of variation, constant expected demand, and constant SD of demand.

3.5.1. APPROACH 1. CONSTANT COEFFICIENT OF VARIATION. The baseline case is modified with

the mean demands ranging from 100 to 1,900 such that 1 2 2,000. The SDs are chosenso that each product has the same coefficient of variation. Results are obtained for each of the

two-product pairs in Table 4. We observe that Rmax and C90% increase at a decreasing rate

FIGURE 8. Percent Reduction in Item Inventory Versus Capacity for Sets of m-products, Independent and

Identically Distributed, N(1,000,200).

TABLE 4

Sets of Two Products with Independent and

Non-Identically Distributed Demands and Identical

Coefficients of Variation Equal to 0.20

Product Sets Rmax C90%

N(100,20), N(1,900,380) 1.03% 4.50%

N(200,40), N(1,800,360) 2.00% 4.86%

N(400,80), N(1,600,320) 3.72% 9.32%

N(600,120), N(1,400,280) 5.10% 12.92%

N(800,160), N(1,200,240) 6.04% 15.24%

N(1,000,200), N(1,000,200) 6.48% 17.51%



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as 1

gets closer to 2

. We conclude the following:

OBSERVATION 7. The benefits of postponement increase as the distribution of demand of

one product approaches that of the other; capacity must also increase to realize these

benefits.

It is not surprising that if 2

1and

2

1, then larger amounts of non-postponed

inventory must be held for product 2 to attempt to equalize fill rates.

3.5.2. APPROACH 2. IDENTICAL EXPECTED DEMANDS. The results of four sets of two-product

pairs with 1 2 1,000 and different js that sum to 400 were obtained for each entry

in Table 5. The ratio of the js of each two-product pair is shown along with the values of

Rmax and C90%. We observe that Rmax and C90% increase at a near-constant rate as the SD of

one product approaches the other. Our conclusion is stated as:

OBSERVATION 8. When expected demands are identical, the benefits of postponement

increase as the SDs of the demands become equal; an increase in capacity is required to

realize those benefits.

An interesting variation of the baseline case is one product with deterministic demand and

the other with probabilistic demand. Consider a two-product model with independent and

non-identically distributed normal demands with the same means (1 2 1,000),

different SDs of demand (1 0,

2 400), and 0.975. Figure 9 shows inventory levels

for two products at increasing levels of postponement capacity. In the total non-postpone-

ment scenario (capacity 0), S1 and S2 are found to be equal to 975 and 1,458, respectively,

to achieve the required service level for each product.

If a policy of postponement is adopted, we would intuitively assume that the non-

postponement inventory of product 1 stays at 975 because its demand is known with

certainty. The demand for product 2 (in excess of its non-postponed inventory) would be met

TABLE 5

Sets of Two Products with Independent and Non-Identically Distributed

Demands, Identical Expected Demands, and the Sum of the Standard

Deviations of Demand Equal to 400

Product Sets Ratio of SD Rmax

C90%

N(1,000,0), N(1,000,400) 0.000 2.92% 8.42%

N(1,000,100), N(1,000,300) 0.333 4.59% 11.28%

N(1,000,150), N(1,000,250) 0.600 5.77% 14.20%

N(1,000,200), N(1,000,200) 1.000 6.48% 17.51%

FIGURE 9. Inventory Levels for Two Products with Independent and Non-Identically Distributed Demands,

N(1,000,0), N(1,000,400), and Identical Expected Demands.



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using the postponement capacity. However, there will be realizations of demand where the

demand for product 2 is greater than S2, resulting in low fill rates, say 85%, and if the fill rate

for product 1 remains at 97.5%, we will not have equal fill rates.

In our postponement environment the non-postponement inventory levels are set to

equalize the fill rates. The results in Figure 9 show that a target fill rate can be achieved when

S1

(as well as S2

) decrease as C increases! A target fill rate can be obtained with less total

item (postponed and non-postponed) inventory than would be required in the intuitive

non-postponement scenario above. This is one of the consequences of the equal-fill-rate

allocation rule we have chosen to adopt.

3.5.3. APPROACH 3. CONSTANT SDS OF DEMANDS. The four sets of two-product pairs in Table

6 have the same SDs of demand and different expected product demands that sum to 2,000.

The values of Rmax and C90% are shown for each two-product set. We observe that Rmax is

near-constant and C90%

increases slightly at a near-constant rate as the demand of one

product approaches the other. We state this finding as follows:

OBSERVATION 9. When the SDs of demand are constant, the benefits of postponement do not

change as the expected demands become equal; however, a slight increase in capacity is

required to realize those benefits.

We make the following conclusion from our analysis of non-identically distributed

demands:

OBSERVATION 10. The benefits of postponement caused by changes in the coefficient of

variation are attributed to changes in the SD of demand rather than changes in the expected

demand. If the two demands are differently distributed, the benefits of postponement will

increase as the demand distributions approach each other.

3.6. Summary of Numerical Results

The capacitated-postponement model demonstrates that non-postponement requires higher

inventory levels than those required for any level of postponement. Stated another way, some

benefit, in terms of reduced inventory, can be realized from any amount of postponement.

Increasing reductions in inventory occur as the fill rate increases, the coefficient of vairation

increases, the demands are more negatively correlated, the number of products being

postponed increases, and the demand distributions of the products approach each other.

The baseline case shows a result that is representative of all the cases studied, i.e., a major

portion of the possible benefits of postponement are realized at a capacity substantially lessthan the capacity required for total postponement. Total postponement produces benefits that

are only marginally better than can be realized at substantially lower capacity levels. This

suggests that there may be a level of partial postponement that is at least as bene ficial as total

postponement especially if there is a cost to delaying product differentiation and converting

the generic product to a final product.

TABLE 6

Sets of Two Products with Independent and Non-Identically Distributed

Demands, Identical SDs of Demand, and the Sum of the Expected

Demands Equal to 2,000

Product Sets 1/

2Rmax

C90%

N(400,200), N(1,600,200) 0.250 6.61% 14.49%

N(600,200), N(1,400,200) 0.429 6.51% 15.30%

N(800,200), N(1,200,200) 0.667 6.42% 16.35%

N(1,000,200), N(1,000,200) 1.000 6.48% 17.51%



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4. Conclusions

A single-period inventory-service-level model was constructed to compare results of

different postponement capacity levels. The benefit of postponement is defined as the percent

reduction in total non-postponed inventory attributed to postponement while the customer

service level is held constant. A comparison of the inventory levels required for eah scenario

shows that increasing postponement requires less inventory.The main result of this study is that a substantially large portion of the benefits of

postponement can be realized at capacities far below the capacity required for total post-

ponement. Higher capacities provide virtually no additional benefits. Although the benefits of

postponement increase as the levels of customer satisfaction increase and the model param-

eters change in the preferred direction, the capacity must also increase to realize those

benefits. This would have a significant impact on a firms decision to adopt a postponement

strategy and the cost of that adoption. One may envision many environments using post-

ponement with the realization that very little needs to be postponed to gain all of the benefits.

A similar notion is described in Jordan and Graves (1995), where the authors showed that

only a little flexibility is needed to achieve almost all the benefits of total flexibility.In addition, these results suggest that the debate firms have in using a make-to-stock versus

a make-to-order strategy may have a simpler answer. Use the hybrid approach and expect no

deterioration in service capability, but expect benefits in inventory reduction even with a

small make-to-order capacity. A strategy relying completely on delayed product differenti-

ation may, in some circumstances, not be as cost effective as the hybrid strategy described

in this paper. Also, such a hybrid approach may be superior in some instances to a complete

make-to-order or complete make-to-stock inventory strategy.

While this study focused on the inventory-service-level tradeoff, further research can be

performed on a production cost basis. Given the cost of the postponed item, the cost of the

finished product, and the cost to finish the postponed item into the finished good, the total costat each level of postponement capacity can be determined. The best choice would be the one

with the lowest total cost. The impact of the various parameters examined in this study on the

optimal combination of postponed and finished inventory could also be investigated. Exam-

ples of such a cost structure can be found in Eynan and Rosenblatt (1995) and Hillier (1998).

It is worth noting that the choice of the value of these costs could produce different results

than those presented in this study.

There are issues not addressed in this study that may have a strong influence on the

decision to adopt a postponement strategy. The stage of the product in its life cycle, product

characteristics, delivery frequency and delivery time, economies of scale, product/process

design, organizational readiness, and barriers between functional areas within the organiza-tion are topics that should be given consideration during the decision-making process

(Graman 2001).

Appendix A

Derivation of Expressions for the Expected Number of Stockouts for the Various Regions in Figure 2

Ifx1 S

1and x

2 S

2, then the demands occurred in region {1} and were met using the non-postponed inventory

of each product. Ifx1 S1 and S2 x2 S2 C, then the demands occurred in region {2}. Demand for product

1 is met from its non-postponed inventory and demand for product 2 is met from its non-postponed inventory plus

some amount of the postponement capacity. Conversely, ifx2 S2 and S1 x1 S1 C, then demands occurred

in region {3}. Demand for product 2 is met from its non-postponed inventory, and demand for product 1 is met from

its non-postponed inventory plus some amount of the postponement capacity. Expressions for the expected number

of stockouts of both products in regions {1}, {2}, and {3} are not necessary because all demands are met.

If x1 S1 and x2 S2 C, then the demands occurred in region {4}. Again, the demand for product 1 is met

from non-postponed inventory. The demand for product 2 cannot be met using both its non-postponed inventory and

all of the postponement capacity. The following expression describes the expected number of stockouts for product

2 in region {4}



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ESO42 x10

S1 x2S2C

x2 S2 C f2x2 f1x1dx 2 d y1 (A1)

An expression for the expected number of stockouts of product 1 in region {4} is not necessary. Because of

symmetry, demands occurring in region {5} are met in the same way as those in region {4} except the references

to products 1 and 2 are reversed.

If S1 x1 S1 S2 C x2 and S2 x2 S1 S2 C x1, then demands occurred in region {6}.

Demand for each product are met using all non-postponed inventory and some amount of capacity allocated to each

product such that P1 P2 C. Therefore, expressions for the expected number of stockouts of either product in

region {6} are not necessary.

If x1 S1 S2 C x2 and x2 S1 S2 C x1, then the demands occurred in region {7}, {8}, or

{9}. The demands will not be satisfied and stockouts for both products will occur. All of the non-postponed

inventory for each product will be used and all of the postponement capacity will be allocated in some amount, Pj ,

to each product. The goal of equalizing fill rates requires that

1 2 (A2)

where, in these regions,

1S1 P1

x1, (A3)

2S2 P2

x2, (A4)

and

P1 P2 C. (A5)

Combining (A3), (A4), and (A5) yields the following expressions for P1 and P2:

P1C x1 S1 x2 S2 x1

x1 x2(A6)

P2C x2 S1 x2 S2 x1

x1 x2(A7)

Realizations of demand can occur where it is not possible to equalize the fill rates as the demand for one product

may be large relative to the other product. Allocation of all the capacity to the large demand product does not raise

its fill rate to that of the product with the lower demand. If all of the capacity is allocated to product 2 and none to

product 1, where x2 x

1, then (A3) and (A4) become

1 S1

x1(A8)

and

2S2 C

x2(A9)

Equating (A8) and (A9) and solving for x1

yields

x1S1

S2 C x2 (A10)

If S1 x

1 (S

1/( S

2 C)) x2 and x2 S2 C, the demands occurred in region {8}. Demand is not met

for either product. The fill rate of product 1 is greater than the fill rate for product 2, which was allocated all of the

capacity. The following expression describes the expected number of stockouts for product 1 in region {8}

ESO81 x2S2C

x1S1

S1/S2Cx2

x1 S1 f1x1 f2x2dx 1 dx 2 (A11)



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The expected number of stockouts for product 2 in region {8} is

ESO82 x1S1

x2S2C/S1 x1

x2 S2 C f2x2 f1x1dx 2 dx 1 (A12)

The equations for region {9} are the converse of those for region {8}.

Regions {6}, {8}, and {9} are bound by region {7}. The total demands in region {7} for both products are notmet, and all capacity is allocated to equalize the fill rates. The following expression describes the expected number

of stockouts for product 1 in region {7}

ESO71 x2S2

S2Cx1S1S2Cx2

S1C/S2 x2

x1 S1 P1 f1x1 f2x2dx 1 dx 2

x2S2C

x1S1/S2Cx2

S1C/S2 x2

x1 S1 P1 f1x1 f2x2dx 1 dx 2 (A13)

The expected number of stockouts for product 2 in region {7} is

ESO72 x1S1

S1C

x2S1S2Cx1

S2C/S1 x1

x2 S2 P2 f2x2 f1x1dx 2 dx 1

x1S1C

x2S2/S1Cx1

S2C/S1 x1

x2 S2 P2 f2x2 f1x1dx 2 dx 1 (A14)

The limits of integration in (A13) and (A14) reflect the fact that region {7} is further partitioned at x2 S2 C

for product 1 and at x1 S

1 C for product 2 to facilitate the development of the expressions.

Appendix B

Stochastic Root-Finding AlgorithmWe observe in Figure A1 that the expected number of stockouts, E(SOj), is a decreasing function of the inventory

level, Sj , for the demand probability distributions used in this study. Therefore, a bisection method will allow us to

find the Sj that gives the target E(SOj). We refer to the procedure as a stochastic root- finding algorithm because it

is composed of a bisection routine to find Sj and a Monte Carlo integration method to compute E(SOj) for given

values ofSj. Monte Carlo integration (simulation) is a scheme employing random numbers to solve certain stochastic

problems where the passage of time plays no substantive role (Law and Kelton 1985, p. 113). Monte Carlo

integration evaluates a function at a random sample of points and estimates its integral based on that random sample.

The algorithm proceeds as follows:

1. Bisection Method

a. Specify the demand distributions, fill rate, and postponement capacity.

FIGURE A1. Expected Number of Stockouts Versus Item Inventory for Three Demand Distributions with Identical

Expected Demands and SDs of Demand.



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b. Find upper and lower bounds on each Sj that produce values for E(SOj) that are above and below the target

E(SOj)s.

c. Use a bisection method to converge on the value of Sj that meets E(SOj).

2. Monte Carlo Integration Method

a. Generate random variates for each product demand.

b. Allocate the postponement capacity using a policy of equalizing fill rates.

c. Update the sample statistics on E(SOj).

d. Calculate the confidence interval half-width.

e. If the calculated half-width is greater that the half-width criterion, go to step 2a. Otherwise, report E(SOj).

3. If the calculated values of the E(SOj)s are within some of the targeted values of the E(SOj)s, then go to step

4. Otherwise, update the upper and lower bounds on each Sj and go to step 1c.

4. Report the Sjs.

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Gregory A. Graman is an Assistant Professor of Operations Management in the Raj Soin Collegeof Business at Wright State University, where he teaches in the areas of operations management andquantitative analysis. He has a Ph.D. in Operations Management, M.Sc. in Quantitative Analysis,M.B.A. in Operations Management and B.S. in Metallurgical (Material Science) Engineering from the



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University of Cincinnati. He has several years of experience in process and industrial engineering,supervision, quality management, and systems analysis in the primary metals industry. He is a memberof APICS, holds CPIM certification, and is active in several professional organizations. His currentresearch interests include supply chain management issues, delayed product differentiation, andlogistics and distribution systems.

Michael J. Magazine is Professor of Quantitative Analysis and Operations Management and OhioEminent Scholar. After completing his PhD at the University of Florida in Industrial and Systems

Engineering, he taught at North Carolina State University and at the University of Waterloo. Inaddition, he has had visiting appointments at PUC in Brazil, INRIA in France and Georgia Tech, MIT,and the University of Michigan. He is a Professional Engineer. Professor Magazines research interestsinclude supply chain management, scheduling and other applications of manufacturing systems,especially applied to e-commerce. He has worked on the design and analysis of heuristics andapplications in the production and manufacturing area and has been the holder or co-holder of severalresearch grants in this area. He has published over 50 papers in these areas. Professor Magazine hasserved on the editorial boards of several journals and was the Area Editor of Operations Research forManufacturing, Operations and Scheduling and is currently a Senior Editor for M&SOM. He is theCo-Editor of Quantitative Models in Supply Chain Management, published by Kluwer in 1999. Hehas been on the ORSA, TIMS and CORS Councils and has been VP-International for INFORMS andPresident of the INFORMS Section on Manufacturing and Service Operations Management.


a numerical analysis of capacitated postponement

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