Scope versus Focus: Issues of Flexibility, Capacity, andNumber of Production Facilities

Saifallah BenjaafarDepartment of Mechanical Engineering

University of MinnesotaMinneapolis, Minnesota 55455, USA

Diwakar GuptaMichael G. DeGroote School of Business

McMaster University, HamiltonOntario L8S 4M4, Canada

February,1997

Abstract: How should a multi-product manufacturing firm design production facilities? How

many facilities should it have? How many and which products should be assigned to each facility?

What batch sizes/scheduling rules are appropriate for facilities making more than one product?

These are questions that have become more relevant now as advances in manufacturing technologies

offer an increasing array of equipment choices. In this article, we introduce models that can help

operations managers answer the above questions. For a specific product mix, these models lead to

explicit expressions for the number of facilities, the number products assigned to each facility and

their corresponding capacities. We evaluate the effect of different operating parameters and

scheduling policies on the optimality of different configurations. In particular, we show that the

choice of the scheduling and batch sizing policies can have a significant effect on the nature of the

optimal mix of flexible and dedicated facilities as well the size of these facilities.

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1 Introduction

Increased availability of flexible manufacturing technology gives multi-product manufacturing

firms more choices in how to design production facilities, how to assign products to facilities, and

how to share capacity among products [8] [19]. While manufacturing firms still retain the option to

design dedicated facilities [20], they are increasingly in a position to share production resources

among more than one product type [9]. For example, a firm may choose to pool individual

facilities, which may otherwise be dedicated, so that they are available to satisfy demand from more

than one product [12] [13]. Alternatively, a firm may choose to consolidate capacity by building

fewer facilities in which many different product types can be manufactured [5] [6]. While pooling

mitigates the negative impact of demand variability and helps to balance workloads among facilities

[2] [8] [13], fewer flexible facilities with relatively large capacities offer economies of scale, e.g.,

higher production rates and lower overhead costs, without sacrificing the competitive advantage of

greater product scope [11] [15].

Thus, there exist three broad lines of strategies from which a multi-product firm might choose to

design its own set of custom-fit facilities: (1) product focus, (2) facility pooling, or (3) capacity

consolidation. Naturally, the firm can easily pursue a combination of all three strategies, with some

facilities being dedicated while the remaining being either pooled or consolidated. A focused

strategy is justified when demand for different products is stable and/or when setup times/costs are

high. Facility pooling is desirable when different facilities can be made equally accessible and setup

times/costs are relatively low. The option to consolidate capacity is appropriate when benefits from

the higher efficiency of larger capacities exceed setup time/cost penalties.

In this article, we limit our attention to examining the first and third strategies, namely, product

focus and capacity consolidation. Facility pooling has already been examined in a separate article by

one of the authors of this study [2]. Our goal is to enhance understanding of how strategy choice

affects operational performance, as measured by manufacturing lead time and work-in-process

(WIP) inventory. In particular, we are interested in characterizing conditions under which capacity

consolidation is superior to product focus (and vice-versa) and in determining optimal levels of

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consolidation. To this end, we consider an arbitrary set of products and identify the optimal

configuration of dedicated and flexible facilities to which these products should be allocated. Since

greater capacity consolidation is always accompanied by increased product variety, determining

optimal production capacities is also equivalent to finding optimal product variety.

In order to carry out a fair comparison between different product and capacity allocation

scenarios, we limit our attention to cases in which a facility's capacity is chosen commensurate with

its expected load and product variety. That is, we assume that flexible facilities producing multiple

products can be allocated capacity proportional to their load. Thus, choosing flexible facilities will

result in fewer but faster production units. However, we also account for the fact that flexible

facilities incur more frequent setups due to changeover between products. The frequency of these

setups can, in part, be mitigated by either increasing batch sizes or using a cyclic group scheduling

policy.

Fundamentally, the tradeoff is between fewer, faster, and more flexible facilities or multiple

slower, and dedicated ones. The flexible facilities, since they are producing multiple products,

incur setups when switching from one product to another. The frequency of these setups increases

with the number of products being produced. Thus, while the higher processing speed associated

with a flexible facility could reduce the time a product spends in production, the increase in setups

could have a counteracting effect. The result may or may not be an overal reduction in production

lead time. Therefore, our objective is to characterize this tradeoff and to determine, for a known set

of products, the optimal number of facilities and the optimal number of products to assign to each

facility. It is also our objective to gain managerial insights into the effect of various system

parameters and control policies on the optimal configuration of facilities and products.

The choice between either relatively few flexible facilities with large capacities or many more

dedicated facilities with smaller capacities is frequently encountered by managers in a variety of

industries. For example, in the metal cutting industry, a single CNC machining center capable of

producing a variety of manufacturing features like milling, drilling, and turning can be used to

replace several of the traditional dedicated machines [18]. In the electronics manufacturing industry,

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a fast programmable automated insertion (AI) machine capable of handling components and boards

with various types and sizes can be used to replace either multiple dedicated machines or manual

assembly lines [16] [22]. In the automotive industry, fast multi-purpose sheet metal presses can be

used to replace dedicated press lines [13].

The choice between large and flexible or small and dedicated facilities also arises at the plant

level. For manufacturing firms that produce multiple products in a network of plants, a decision

must be made regarding the number and size of these plants, as well as the set of products assigned

to each plant [12] [13]. Increased globalization of manufacturing has resulted in a shift in many

industries toward building fewer but more flexible plants serving multiple markets with multiple

products [12] [15]. This is evident, for example, in the automotive industry [9], the semiconductor

and electronic manufacturing industry [22], and the textile and paper industries [25].

Although there exists a large body of research dealing with the relative economic benefits of

different capacity and product allocation strategies (e.g., see [8] for a comprehensive review), very

little of this research is concerned with the impact that these strategies have on operational

performance. A number of analytical models have been proposed for the performance evaluation of

manufacturing systems [7]. However, few of these models are adequate for the analysis of multi-

product systems with setup times. Moreover, barring some exceptions like [14], a majority of these

models do not deal explicitly with issues of batch sizing and group scheduling.

Literature that comes closest in spirit to this study includes articles by Vander Veen and Jordan

[26], Buzacott [6], Karmarkar and Kekre [14], and Li and Qiu [16]. Vander Veen and Jordan [26]

present a deterministic model for determining optimal investment decisions, where capital, labor,

setup, and inventory costs are minimized. Their model is capable of simultaneously allocating

products and capacity to machines. However, it is an aggregate model and does not account for

congestion or queueing delays. Buzacott [6] uses a series of queueing models to compare the

performance of different production system configurations, such as specialization, parallel and serial

processes, and economies of scale. The economies of scale scenario corresponds to some of the

flexible configurations discussed in this paper. Buzacott's models assume that setups are always

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negligible. Thus, our study can be viewed as an extension of his work toward modeling setup

times. However, introducing setup times, which are nearly always non-zero for multi-product

facilities, changes the nature of the problem in a fundamental way and our approach is entirely

different.

Li and Qiu [16] present a multi-period model for determining the optimal capacity to purchase

and the time and type of products to produce so that setup, inventory, and back-ordering costs are

minimized. Their model assumes deterministic unit processing times, and like Vander Veen and

Jordan's study, it too is an aggregate model which does not account for queueuing delays.

Karmarkar and Kekre [14] compare the performance of flexible and dedicated facilities in terms of

investment and operational costs. They do not seek to characterize the optimal choice of flexible and

dedicated facilities for any given production system. Their models are based on M/M/1 queueing

systems with a first-come first-served scheduling priority. They also assume a fixed batch size

policy in which a setup is required before the start of each new batch, irrespective of whether or not

the previous batch was of the same type.

In this article, we address several of the above limitations. In particular, we introduce a new set

of models for examining the performance of multi-product multi-facility systems in the presence of

setup times. The models capture explicitly product variety, setup times, and batch sizes. In

addition to the first-come first-served policy, we also consider a group scheduling policy under

which product types are processed in a cyclical fashion. The models presented in this article are

meant to serve as tools for system designers and production managers in gaining insights into the

effect of product variety and process flexibility on performance under varying operating conditions

and control policies.

The plan of the article is as follows. In section 2, we introduce notation, describe our model in

detail and discuss modeling assumptions. Detailed analysis pertaining to the group scheduling

policy is presented in two sections. Section 3 contains preliminaries and insights from special cases

which are used in building and solving the more general formulation in section 4. Section 4

provides a complete mapping of the solution space in which we characterize the optimal mix of

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flexible and dedicated facilities under all possible parameter values. Section 5 contains a parallel

treatment of the FCFS fixed batch size policy. Implications from our results and an outline of future

research directions are discussed in section 6.

2 Notation and Model Description

Given n products, we are interested in designing a set of m facilities to which these products can

be assigned such that the expected production lead time, or flow time, is minimized. Two extreme

product allocation scenarios can be readily identified: (1) assign each product to a dedicated facility,

and (2) assign all products to a single flexible facility. Between these two extreme scenarios, there

exist feasible allocations where products are partitioned among a set of partially flexible facilities,

with each facility handling only a subset of the n products.

The objective of this article is to characterize the optimal number of facilities, m*, and the

optimal number of products, ki*, assigned to each facility i. In order to isolate the effect of product

variety and to allow for a fair comparison between different product allocations scenarios, we

assume that demands for the n product are independent and identically distributed with a common

mean λ. Total average demand per unit time is denoted by Λ =nλ . To maintain the same overall

capacity across different scenarios, we assume that the capacity of each facility is determined by the

number of products to which it is assigned. Thus, a facility producing k products has k times the

capacity of a facility dedicated to a single product. If we assume, as we do in this article, that each

product can be assigned to only one facility, the average utilization of each facility will be the same

regardless of its size and the number of products which are assigned to it. This makes it possible to

make meaningful comparisons between different product and capacity assignment configurations.

The random variable Xk = S/k is used to denote the processing time per item at a facility with k

products, where S is the item processing time in a dedicated facility. This scaling ensures that

capacity is allocated in accordance to demand load and that the processing time coefficient of

variation remains constant for all facilities regardless of size (see Buzacott [6] for a more detailed

discussion of time scaling). Processing times are arbitrarily distributed with a mean E(Xk) = E(S)/k

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and a second moment E(Xk2) = E(S2)/k2. Since the allocation of a product to a facility is always

accompanied by an average demand load λ, the average utilization of each facility is ρ = λE(S),

which is independent of the number of products k.

Each facility incurs a setup time R whenever it switches from one product type to another.

Setup times have arbitrary distributions with mean E(R) and second moment E(R2). In order to

minimize the frequency of setups, we assume that either a cyclic group scheduling (GS) policy or a

first-come first-served (FCFS) fixed batch size policy is in place. Under the GS policy, also known

as the cyclic exhaustive polling policy in the queueing literature [24], once a facility is setup for a

product, it continues processing units from that product until all current demand is exhausted. It is

then setup for the next product. The facility continues switching from one product type to the next

in a strict cyclic fashion. Owing to its versatility and many desirable attributes, the cyclic exhaustive

policy is used to model a large variety of production, computer, and telecommunication systems

[19] [24]. In particular, it is easy to implement, it minimizes the amount of unfinished work in the

production system [17], and ρ < 1 is both a necessary and sufficient condition for queue finiteness

[1]. The duration of setups does not, therefore, affect system stability, which is not the case for the

FCFS fixed batch size policy.

Under the FCFS fixed batch size policy, product orders are released and processed in fixed

batches of size Q. Batches are scheduled in the order of their arrival to the system. A setup is

incurred whenever two successive batches are not of the same product. The frequency of setups

can be reduced by increasing the batch size. However, this also increases the batch processing

time. An optimal batch size, Q*, can be identified such that overall average flow time is minimized.

Regardless of the policy we choose, we can see that increasing the number of products assigned

to a facility reduces its processing time. However, increased product variety also means an increase

in the frequency of setups, as the facility spends more time switching between products. The

increased processing efficiency of a larger/more flexible facility can, thus, be limited by the higher

frequency in setups, as well as the length and variability of the setup times. This means that

increased product variety may or may not result in an overall reduction in expected flow time. As

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we show later on, the optimal amount of product variety at a facility is highly sensitive to the

processing and setup time distributions, system utilization, and the scheduling/batch sizing policies.

We use E(Fk) to denote the expected flow time in a single facility with k products and

E(Fk1, k2, …, km) to refer to expected flow time in a system of m facilities, where ki is the number

of products assigned to facility i (i = 1, 2, …, m). The value of E(Fk1, k2, …, km) can be calculated

as

E(Fk1, k2, …, km) = kik1 + k2 + … + km

E(Fki)∑i = 1

m

. (1)

The problem of determining the optimal number of facilities, m*, and the optimal number of

products assigned to each facility, ki*, can now be formulated as the following non-linear integer

optimization model.

Minimize E(Fk1, k2, …, km) (2)

Subject to:

ki∑i = 1

m

= n, (3)

ki ≥ 1 for i =1, 2, …, m, and (4)

ki: integer. (5)

Constraint 3 ensures that the number of products assigned to the m facilities equals n, the total

number of products, while constraint (4) requires that each selected facility be assigned at least one

product. In the next three sections, we simplify the formulations in 2-5 for both the group

scheduling and the fixed batch size FCFS policies. We should note that minimizing the expected

flow time in the objective function is, by virtue of Little's law, equivalent to minimizing expected

WIP.

3 The Group Scheduling Policy: Preliminary Results

We model each production facility with k products as a single server queueing system with k

customers. We assume that unit orders for products arrive dynamically to the production facility

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according to independent Poisson processes. When a facility produces only one product, the

expected flow time of items at that facility is given by the Pollaczek-Khintchine formula [7]. When

the facility produces two or more products, the expected flow time of each product can be obtained

by the pseudo-conservation law for cyclic-exhaustive polling models [19]. Putting it together, we

obtain

E(Fk) = {kλE(Xk

2)2(1 - ρ)

+ E(Xk) for k = 1

kλE(Xk2)

2(1 - ρ) + E(Xk) +

E(R2)2E(R)

+ E(R)(k - 1)

2(1 - ρ) for k ≥ 2.

(6)

The above equation simplifies as follows

E(Fk) = {A

2(1 - ρ) for k = 1

Ak -1 + B + E(R)k2(1 - ρ)

for k ≥ 2,

(7)

where

A = λE(S2) + 2(1 - ρ)E(S), and (8)

B = (1 - ρ)E(R2) - E(R)2

E(R). (9)

The expression of flow time for facilities with more than one product can also be decomposed as

E(Fk) = C1 + C2, where C1 = λE(S2) + 2(1 - ρ)E(S)

2(1 - ρ)k and C2 =

E(R)2(1 - ρ)

k + (1 - ρ)E(R2) - E(R)2

2E(R)(1 - ρ).

It is easy to verify that C1 is decreasing and convex in k, while C2 is linearly increasing in k. The

rate at which C1 decreases in k is determined by the processing time distribution, while the rate at

which C2 increases in k is determined by the setup time distribution. The reduction in C1 is due to

the increased processing efficiency realized with a larger facility. The increase in C2 reflects, on the

other hand, the greater setup time effort associated with a more flexible facility. The counteracting

effects of C1 and C2 conveniently capture the tradeoff that arises from increased product variety.

Whether increased product variety, on balance, improves flow time performance depends on the

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processing/setup time distributions, the facility overall utilization, and the actual number of products

allocated to the facility.

In order to gain some initial insights into the effect of the number of products on performance,

let us first consider a single flexible facility that is assigned k products. We are interested in

identifying the optimal number of products, and the corresponding capacity, that minimizes

expected flow time in such a facility. For the moment, we do not place any constraints on this

selection other than requiring that the flexible facility be assigned at least two products.

Lemma 1: The expected flow time, E(Fk), is a strictly convex function of k.

Proof: From 7, the differences ∆1 = E(Fk) - E(Fk + 1) and ∆2 = E(Fk + 1) - E(Fk + 2) can be

expressed as ∆1 = {A/k(k + 1) - E(R)}/2(1 - ρ) and ∆2 = {A/(k + 1)(k + 2) - E(R)}/2(1 - ρ). Since

A > 0, it is easy to see that ∆1 > ∆2, which completes the proof.

Using the differences ∆1 and ∆2, we can show that E(Fk) is decreasing in k when k(k + 1)

≤ A/E(R), and is increasing in k when k(k + 1) ≥ A/E(R). If we momentarily ignore the integrality

of k and note that E(Fk) is continuously convex in k, the value of k which minimizes expected flow

time, kf*, is given by the first order condition of optimality as follows:

kf* = max {2, A/E(R)} . (11)

Since the value of kf* is not necessarily integer, the optimal number of products is found by

choosing either the integer ceiling or integer floor of kf*, whichever minimizes E(Fk). Note that the

value of kf*, when kf* > 2, is increasing in A and decreasing in E(R). This means that an increase

in either the first or the second moment of processing time increases the optimal number of

products, while an increase in average setup time will reduce the optimal number of products.

Although kf* is the optimal number of products that should be assigned to a flexible facility, it

does not mean that a flexible facility with kf* products is necessarily superior to a dedicated one. In

fact, as described in the following lemma, there exist instances where a dedicated facility is more

desirable.

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Lemma 2:

• A flexible facility with kf* products is superior to a dedicated facility if and only if kf* > 2 and

B/A + 2 E(R)/A < 1, or kf* = 2 and A > 2(B + 2E(R)).

• A flexible facility with kf* products and a dedicated facility are equally desirable if and only if kf*

> 2 and B/A + 2 E(R)/A = 1, or kf* = 2 and A = 2(B + 2E(R)).

• A dedicated facility is superior to a flexible facility with kf* products if and only if kf* > 2 and

B/A + 2 E(R)/A > 1, or kf* = 2 and A < 2(B + 2E(R)).

Proof: The proof follows from examining the difference E(Fkf*) - E(F1) when kf* > 2, and E(F2) -

E(F1) when kf* = 2.

Lemma 2 highlights the fact that the relative desirability of flexible and dedicated facilities is

highly sensitive to the processing/setup time distributions, facility utilization, and the number of

products assigned to the flexible facility. A comparison between flexible and dedicated facilities is

shown in Figures 1, 2 and 3 for various values of setup time, facility utilization, and number of

products. From Figure 1, we can see that initial increases in product variety generally improve

performance. However, this improvement is eventually eroded due to the inevitable increase in the

frequency of setups. In fact, for large values of k, the expected flow time grows almost linearly in

k. A result that follows from the fact that for large k, E(Fk) ≈ E(R2)/2E(R) + (k - 1)E(R)/2(1 - ρ).

Figure 2 illustrates the fact that average flow time in a flexible facility is linearly increasing in

average setup time. Thus, flexible facilities become less desirable as setup times get larger. Figure 3

shows that while for lightly loaded systems, a dedicated facility results in lower flow times (since

waiting times are minimal), a flexible facility is preferable for moderately loaded systems. For very

highly loaded systems, a dedicated facility could become once again more desirable, since a highly

loaded system cannot afford to spend much time on setups.

Additional insights into the relative desirability of flexible and dedicated facilities are given by

the following results.

Lemma 3: If setup time is negligible, i.e., R = 0, then the expected flow time is monotonically

decreasing in k, with E(Fk) = (k - 1)E(Fk - 1)/k = E(F1)/k .

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Lemma 3 states that increased flexibility and product variety is always desirable in the absence of

setup times. In fact, expected flow time in a facility with k product is k times smaller than that in a

dedicated facility.

Next, we consider a system with two flexible facilities and an even number, n = 2k, of

products. We show that for flexible facilities producing at least two products, a balanced allocation

of these products among the two facilities is always superior to any unbalanced allocation.

Lemma 4: Let E(Fk1, k2) denote the expected flow time in a production system with two flexible

facilities, where k1, k2 ≥ 2 and k1 + k2 = 2k, then k1* = k2* =k.

Proof: The proof follows from noting that the difference E(Fk - r, k + r) - E(Fk, k) , for k - r ≥ 2, is

given by

E(Fk + r, k - r) - E(Fk, k) = r2

k 2[

E(R)2(1 - ρ)

], (12)

which is clearly positive.

Lemma 4 can also be seen as a corollary to lemma 1, since it follows from the fact that expected

flow time for flexible facilities is convex in k. In fact, lemma 4 can be generalized to an arbitrary

number of facilities and an arbitrary number of products.

Corollary 1: An optimal product allocation vector k* = {k1*, k2*, …, km*} to a set of m flexible

facilities satisfies the property:

max(ki* - kj*) ≤ 1

for all ki*, kj*∈ k* and ki*, kj* ≥ 2.

Thus, if we have n products to be assigned to m flexible facilities, each facility will receive either the

integer ceiling or the integer floor of n/m products. This result, as we show in the next section, will

prove to be useful in formulating and solving the general product allocation problem.

4 The Group Scheduling Policy: Optimal Solution

Given n products to assign to a set of facilities, we are interested in determining the optimal

number of facilities, m*, and the optimal number of products allocated to each facility, ki*. This is

also equivalent to determining the optimal number of dedicated facilities, md*, and the optimal

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number of flexible ones, mf* = m* - md*. The optimal number of products in each dedicated

facility is simply one, and the optimal number of products assigned to each flexible facility is (n -

md*)/mf* (more precisely, the corresponding ceiling or integer floor). This result follows from the

fact that a balanced allocation among flexible facilities is always optimal.

Let us first consider the special case of n = 2. Then, we either have m* = 2 (two dedicated

facilities) or m* = 1 (a single flexible facility with two products). The optimal configuration, can be

obtained by comparing E(F1) and E(F2). Specifically, we have the following result.

Lemma 5: For n = 2, m* = 2 if A < 2B + 4E(R), and m* = 1, otherwise.

In the remainder of this section, we shall assume that n ≥ 3. If we let md and mf denote,

respectively, the number of dedicated and flexible facilities, the expected flow time in a system of m

facilities can be written as:

E(Fk1, k2, …, km) = (md/n)E(F1) + (1 - md/n)E(Fkf

) (13)

where m = mf + md and kf = (n - md)/mf. We note that there are only three possible optimal facility

configurations: (i) all facilities are flexible, (ii) a combination of flexible and dedicated facilities, and

(iii) all facilities are dedicated. If the optimal configuration consists of only flexible facilities, we

know, from corollary 1, that every facility will be allocated the same number of products. Using

the results of lemma 2, the optimal number of facilities and the optimal number of products per

facility are given by the following corollary.

Corollary 2: When only flexible facilities are possible and there are n products to assign, then the

optimal number of facilities is given by m* = n/kf*, where kf* is the optimal number of products

allocated to each facility and is given by:

• kf* = 2, if and only if A/E(R) ≤ 2,

• kf* = A/E(R) if and only if 2 ≤ A/E(R) ≤ n, and

• kf* = n, if and only if A/E(R) ≥ n.

A configuration consisting of all flexible facilities is superior to one consisting of all dedicated

facilities under the conditions described in the following lemma.

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Lemma 6: All flexible facilities are preferred over all dedicated facilities if and only if one of the

following conditions holds:

• A/E(R) ≤ 2 and A ≥ 2[B + 2E(R)],

• 2 ≤ A/E(R) ≤ n and A ≥ B + 2 AE(R) , or

• AE(R) ≥ n and A ≥ n[B + nE(R)]/(n - 1).

Consider now the case when we have both flexible and dedicated facilities. At first, let us

ignore integrality requirements in order to make exposition simpler. Impact of requirements that m ,

md and kf must be integers will be dealt with later. Due to the balanced allocation property, each of

the flexible facility will have the same number (kf) of products assigned to it. Hence, the decision

concerning the number of dedicated and flexible facilities will depend only on the relative

magnitudes of E(F1) and E(Fkf). Specifically, we will choose all dedicated facilities when E(F1) <

E(Fkf), and all flexible facilities when the opposite is true. When E(F1) = E(Fkf

), we will be

indifferent between choosing either all flexible, or all dedicated or a combination of the two.

Conditions for any one of the possible configurations to be optimal are summarized in proposition

1.

Proposition 1: The optimal number of dedicated and flexible facilities is given by the following.

1. When A/E(R) ≥ n,

(a) m* = 1 and md* = 0 if and only if A ≥ n[B + nE(R)]/(n - 1),

(b) m* = md* = n, if and only if A < n[B + nE(R)]/(n - 1),

2. When 2 ≤ A/E(R) ≤ n,

(a) m* = n A/E(R) and md* = 0 if and only if A ≥ B+ 2 E(R)A,

(b) any feasible m* and md* such that n - md* = (m* - md*) A/E(R) if and only if A =

B+ 2 E(R)A,

(c) m* = md* = n if and only if A < B+ 2 E(R)A,

3. When A/E(R) ≤ 2,

(a) m* = n/2 and md* = 0 if and only if A ≥ 2[B + 2E(R)] ,

(b) any feasible m* and md* such that md* = 2m* - n if and only if A = 2[B + 2E(R)], and

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(c) m* = md* = n if and only if A < [B + 2E(R)] .

The above proposition provides a complete mapping of the solution space. From the values of m*

and md*, the optimal number of products allocated to each facility can be directly obtained for any

possible combination of parameters.

Although the above discussion ignores the integrality constraint on m*, mf*, and kf*, the

optimal solution we obtain in proposition 2 is always integer under conditions 1(a and b) and 2(c)

and 3(c). Under conditions 2(b) and 3(b), all facilities can be made dedicated when kf* is not an

integer. If kf* is an integer, an arbitrary number of flexible facilities of size kf* can be selected,

with the remaining n - mf*kf* products assigned to dedicated facilities. However, under conditions

2(a) and 3(a), it might not always be possible to allocate n products consistent with proposition 2

because n might not be an integer multiple of kf*. In this case, if kf* is not an integer, we set it

equal to either its integer floor, arg(kf*), or integer ceiling, arg(kf*) + 1, whichever results in the

smaller expected flow time and compare the resulting E(Fkf*) with E(F1). If E(Fkf*) is larger, all

facilities are made dedicated. Otherwise, we consider combinations of flexible and dedicated

facilities. Fortunately, owing to the convexity of E(Fkf), only three configurations need to be

evaluated:

1. mf* = arg(n/kf*), and md* = n - mf*kf*. In other words, after choosing as many flexible

facilities of size kf* as possible, the remaining products are assigned to dedicated facilities.

2. mf* = arg(n/kf*), and the remaining products are assigned arbitrarily one each to existing

facilities. Thus, n - mf*kf* facilities will have kf* + 1 products while all others will have kf*

products.

3. mf* = arg(n/kf*) + 1, and the products are redistributed among these facilities so that the number

of products assigned to any two facilities differ at most by 1.

The expected flow time for each of the above configurations is evaluated and the configuration with

the minimum flow time is selected.

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5 The Fixed Batch Size FCFS Policy

We now consider the situation in which parts are released and processed in batches of fixed size

Q. Batches are scheduled in order of their arrival to the system. The batch arrival process for each

product is assumed to be Poisson distributed with an average rate λb = λ/Q. A setup is incurred

whenever two successive batches are not of the same product. The batch processing time (setup

time + processing time of all the Q units in the batch) in a facility with k products is given by:

Xk = {QS/k with probability 1/k, and

QS/k + R with probability 1 - 1/k, (20)

from which we get:

E(Xk) = QE(S) + (k - 1)E(R)

k, (21)

and

E(Xk2) =

Q2E(S 2) + (k - 1)(2QE(S)E(R) + kE(R2))

k 2. (22)

The expected flow time can now be obtained from the Pollaczek-Khintchine formula [7], which,

after simplification, yields

E(Fk) = λ[Q2E(S 2) + (k - 1){2QE(S)E(R) + kE(R2)}]

2k[Q - λ{QE(S) + (k - 1)E(R)}] +

QE(S) + (k - 1)E(R)k

. (23)

In order to ensure that the expected flow time is finite, the following stability condition must be

satisfied:

Q - λ{QE(S) + (k - 1)E(R)} > 0. (24)

This condition can be interpreted as stating that, for a fixed number of products k, there is a

minimum feasible batch  size, Qmin, which is given by:

Qmin = λ(k - 1)E(R)

1 - λE(S). (25)

Equivalently, for a fixed batch size Q, there is a maximum feasible number of products, kmax,

which is given by:

17

kmax = λE(R) + Q(1 - λE(S))

λE(R). (26)

This means that either a batch size of at least Qmin must be used or the total number of products

must be kept below kmax. The effect of Q and k on the expected flow time is illustrated in Figures 4

and 5.

Proposition 2: The expected flow time for a single facility with k products, and operating under

the fixed batch size FCFS policy, is strictly convex in k, 1 ≤ k < kmax, if αo ≥ 0, where αo =

λQ2E(S2) + 2Q(1 - λE(S))(QE(S) - E(R)) - 2λE(R)2, and is strictly increasing in k otherwise.

Proof: See Appendix A.

Using proposition 2, the number of products, k*, that minimizes the expected flow time in a

single facility when αo ≥ 0 can be obtained from the first order condition of optimality as:

k * = max{1, -U + V 2 - 4UW2U

}, (27)

where

U = λ[QE(R2)(1 - ρ) + 2λE(R)3],

V = 2λE(R )[λQ2E(S 2) + 2Q(1 - ρ)(QE(S) - E(R)) - 2λE(R)2)], and

W = -[Q(1 - ρ) + λE(R)][λQ2E(S 2) + 2Q(1 - ρ)(QE(S) - E(R) - 2λE(R)2)].

When αo < 0, the optimal number of products is simply one.

Lemma 7: Let E(Fk1, k2) denote the expected flow time in a production system with two flexible

facilities, operating under the FCFS fixed batch size policy, where k1, k2 ≥ 1, k1 + k2 = 2k, then

k1* = k2* =k when αo ≥ 0.

The proof of lemma 7 is similar to that of lemma 3. The result of lemma 7 can be generalized to an

arbitrary number of facilities and products as follows.

Corollary 3: An optimal product allocation vector k* = {k1*, k2*, …, km*} to a set of m facilities

satisfies the property:

max(ki* - kj*) ≤ 1

for all ki*, kj*∈ k* and ki*, kj* ≥ 1 when αo ≥ 0.

18

The implication of corollary 2 is that a balanced product allocation is always optimal when αo ≥

0. Thus, if we have n products to allocate among m facilities, each facility should be assigned n/m

products. For m facilities, each with n/m products, expected flow times is given by

E(Fn/m) = mλ(E(S2) + ( n

m - 1)(2E(S)E(R) + nmE(R2)))

2n(1 - λE(S) - λ( nm - 1)E(R))

+ mE(S) + (n - m)E(R)

n (28)

Noting that the expected flow time at each facility is convex in the number products, the optimal

number of facilities, m*, can be obtained as:

m* = n/k* (29)

where k* is given by 27. This leads to the following proposition.

Proposition 3: Given n products, and assuming a FCFS fixed batch size policy, then m* = n if

αo ≥ 0, and m* = n/k* otherwise.

The proof follows from the fact that the optimal number of products to allocate to any facility is

always one when αo < 0, and k* when αo ≥ 0. It is interesting to note that the optimal

configuration always consists of either (1) all dedicated facilities, (2) a set of flexible facilities with

an equal number of products, or (3) a single flexible facility. However, it is never a mix of flexible

and dedicated facilities.

An alternative method for finding the optimal allocation of products, as well as the optimal

number of facilities, when αo ≥ 0, is via marginal analysis [10]. This can be accomplished by first

choosing m, then obtaining optimal allocation via marginal analysis (see Appendix B). Values of m

ranging from 1 to n, are evaluated. The value of m that minimizes expected flow time is selected.

It must be noted that the performance of each flexible facility can be further improved by

choosing batch sizes in an optimal fashion. Observing that the expected flow time, E(Fk), is convex

in Q, the batch size that minimizes expected flow time can be obtained from the first order condition

of optimality as:

Q* = max(1, -F + F2 - GHH

), (30)

where

19

F = -λ(k - 1)E(R)[λE(S2) + 2 E(S)(1 -λE(S))],

G = [1 -λE(S)][λE(S2) + 2 E(S)(1 -λE(S))], and

H = -λ(k - 1)E(R2)[1 -λE(S)].

When the batch size is chosen optimally, the number of products that can be assigned to a

facility can be shown to be unlimited (i.e., kmax = ∞). In fact, when Q = Q*, the expected flow

time is found to converge to a finite limit.

Proposition 3: Let E(Fk*) denote the expected flow time for a facility with k products operating

under the FCFS policy and with Q = Q*. Then, limk→∞(E(Fk*)) = γ, where γ is constant and finite.

Proof: From the ratio

Q*k

= (k - 1k

)F 'G

+ (k - 1k

)2(F 'G

)2 + (k - 1

k)H '

G,

where

F' = λE(R)[λE(S2) + 2 E(S)(1 -λE(S))] and

H' = λE(R 2)[1 -λE(S)],

we obtain the limit

limk→∞Q*k

= F 'G

+ (F 'G

)2 + H '

G = F' + F '2 + GH'

G = β.

Rewriting the expression of the expected flow time as:

E(Fk*) =

λ((Q*

k)2

E(S2) + k - 1k

(2Q*

kE(S)E(R) + E(R2)))

2(Q*

k(1 - λE(S)) - (k - 1

k)λE(R))

+ Q*

kE(S) + k - 1

kE(R)

The limit of the expected flow time can now be calculated as:

limk→∞(E(Fk*)) =

λ(β 2E(S 2) + 2βE(S)E(R) + E(R2))

2(β(1 - λE(S)) - λE(R)) + βE(S) + E(R),

which is clearly independent of k and finite.

6 Discussion and Conclusions

20

In this article, we present a set of models for evaluating the performance of multi-product

facilities with setup times. We used the models to systematically identify the optimal mix of flexible

and dedicated facilities for manufacturing a given set of products. Because the models provide

explicit expressions for the optimal solution, computational effort is minimal. The models and the

corresponding solutions can serve as useful tools for gaining insight into the effect of product

variety and process flexibility on performance of manufacturing facilities. Several such insights are

summarized below.

The optimal amount of product variety at a facility, and the corresponding process flexibility, is

highly sensitive to the processing/setup time distribution, system utilization, and the

scheduling/batch sizing policies. The effect of product variety on manufacturing performance is

generally not monotonic. In fact, when greater flexibility at a facility is accompanied by increases in

capacity, initial increases in product variety typically improve performance. This improvement is

eventually eroded due to the inevitable increase in the frequency of setups. The amount of setups

tolerated is itself determined by the scheduling rule and the batch sizes. Therefore, it is always

important to identify the optimal number of products, and the corresponding capacity, to allocate to

a particular facility. Similarly, in a multi-facility system, performance is not monotonic in either the

number of facilities or the flexibility of these facilities. Both the optimal number of facilities and the

size of each facility are highly dependent on the operating parameters and the control policies.

For systems with similar operating parameters but with different scheduling/batching policies,

the optimal configuration of flexible and dedicated facilities can be very different. For example,

under the GS policy, the optimal solution could consist of a mix of flexible and dedicated facilities.

In contrast, under the FCFS fixed batch size policy, the optimal solution is either a single flexible

facility or multiple facilities with an equal number of products, but never a mix of dedicated and

flexible ones. The stability conditions for the two policies are also different. Under the GS policy,

system stability is unaffected by setup times. Thus, there is no requirement for a minimum batch

size and no limit on the number of products that can be assigned to a given facility. This is not the

case for the FCFS fixed batch size policy, where there is a requirement for either a minimum batch

21

size or a maximum number of products. This also means that the maximum feasible throughput,

i.e., available capacity, is affected by batch size and product variety. In fact, under the FCFS

policy, the maximum throughput of a single facility with k product and a batch size of Q is given by:

THmax = kQ

QE(S) + (k - 1)E(R). (31)

which is clearly increasing in Q and decreasing in k.

However, despite their differences, neither the GS nor the FCFS policy dominates the other. In

fact, it is easy to find examples to the contrary (see Figure 6). When batch sizes cannot be selected

optimally, the performance of the FCFS policy tends to deteriorate significantly with increases in

either product variety, setup time, or utilization. Since the stability of the GS policy is independent

of product variety and setup time, its performance is more robust with respect to increases in either

of these two factors (see Figure 6). On the other hand, if batches can be selected optimally, then,

by virtue of proposition 3, the FCFS policy will be superior to the GS policy when the number of

products allocated to a facility is very large (i.e., when k→∞). A result that follows from the fact

that, under the GS policy, E(Fk)→∞ as k→∞. Comparisons of the two policies are illustrated in

Figures 7 and 8. We should also note that the performance of the optimal configuration obtained

under the FCFS policy can be further improved if m*, k*, and Q* can be selected simultaneously.

However, the joint convexity of the expected flow time in m*, k*, and Q* is difficult to establish.

Without which, explicit expressions of m*, k*, and Q* are difficult to obtain.

We should note that in comparing the GS and FCFS policies, we have assumed that the arrival

process is always Poisson distributed. Thus, we are implicitly assuming that the randomness in the

arrival process remains the same regardless of batch size. This assumption is appropriate for

situations where changing the batch size does not necessarily change the variability in which orders

arrive and/or are released to the system. In instances, where orders always arrive individually and

then wait until a full batch is formed before being released to the facility, the additional queueing

time must be modeled and the input process to the facility must be changed accordingly. This input

process is not necessarily Poisson. Although these changes will affect the performance of the

22

FCFS policy for specific parameter values, we do not expect it to change its general behavior (see

for example, reference [3]).

When product demands are comparable, and capacity can be freely allocated among flexible

facilities, a set of facilities of equal size and and with identical flexibility is always optimal.

Similarly, when the number of facilities is fixed, but product and capacity allocation is flexible,

allocating products and capacity in a balanced fashion is always preferred. This means that it is

more desirable to have a set of multiple facilities that are partially flexible than a mix of highly

flexible/large facilities and small/dedicated ones.

In many industries, there is often the possibility of producing the same product in more than one

facility. This may be desirable when the demands for different products are not symmetric.. In a

recent paper [4]. we addressed the problem of optimally allocating asymmetric workloads to

multiple facilities with asymmetric capacities. Our findings showed that whenever possible,

demand from a product should be entirely assigned to a single facility. In fact, an effective heuristic

for product allocation is one that maintains balanced utilization among facilities while minimizing the

sharing of products among facilities. Our assumption, in this article, of equal utilizations and

uniquely assigned products are, therefore, in line with these findings. However, this does not mean

that instances do not exist where sharing of products among facilities is preferable. This would

certainly be the case, for example, when different facilities are geographically distant and

transportation costs are high.

In this article, we focused on operational rather than economic benefits, which obviates the need

to quantify benefits in economic terms. Also, we did not include the costs of acquiring and

operating a production facility. Although this would be relatively straightforward to incorporate in

our models, we believe that the underlying cost structures are too industry (or even plant) specific to

permit building of reasonable generic models [23]. It is nevertheless worthwhile to examine, for

specific cases, the impact that technology acquisition and operating costs have on the optimal

configuration. Because upgrading capacity/flexibility is not necessarily linear in cost, its effect on

the optimal solution is not always easy to predict.

23

This article is an initial attempt at answering questions relating to size, flexibility, and number of

production facilities. Extensions of our models that consider additional features present worthwhile

directions for future research. For example, manufacturing firms typically have existing facilities

whose capacities cannot be altered easily. Also, demand levels and processing requirements of

products might not be identical. In addition to size, production facilities might also differ in

reliability, precision, and required operator skills. All these features need to be taken into account

when considering design of new facilities and all offer interesting avenues for future research.

Acknowledgements

This research has been supported in part by the National Science Foundation (grant DMII-

9309631) and the Natural Sciences and Engineering Research Council of Canada (grant

OGP00454904).

Appendix

A) Proof of Proposition 2

24

We first show that if αo ≥ 0, then E(Fk) is convex in k. The derivative of the expected flow

time with respect to k can be written as follows:

∂E(Fk)∂k

= 2α1k 2 + 2α2αok - α3αo

k 2(α3 - α2k)2 (A.1)

where

α1 = λ[E(R2)Q(1 - λE(S)) + 2λE(R)3],

α2 = 2λE(R), and

α3 = 2[Q(1 - λE(S)) + λE(R)],

which leads to

∂2E(Fk)

∂k 2 =

2α1α2k 3 + αo[α32 - 3α2α3k + 3α2

2k 2]

k 3(α3 - α2k)3 (A.2)

Given that

i) k3(α3 - α2k)3 > 0, since 1 ≤ k ≤ kmax = α3/α2,

ii) 2α1α2k3 > 0, since α1, α2 > 0, and

iii) α3 - 3α2α3k + 3α2k2 > 0 since α3 > 0 and α3 - 3α2α3k + 3α2k2 = (α3 - 3α2k/2)2 +

(3α2k/2)2 > 0, then if αo ≥ 0, we have

∂2E(Fk)

∂k 2 > 0. (A.3)

Next, we show that if αo < 0, then E(Fk) is strictly increasing in k. To this effect, we show that the

first derivative is always positive when αo < 0, which is equivalent to showing that

2α1k2 + 2α2αok - α3αo > 0. (A.4)

The right hand side of inequality B.4 is a second order polynomial of k. Therefore, in order to

show that C.4 holds, it is necessary to show that the determinant

∆ = (α2αo)2 + 4(2α1)(α3αo) = 4αo[α22αo + 2α1α3] < 0. (A.5)

Noting that

α22α0+2α1α3 = α2

2[λQ2E(S2)+Q(1 -λE(S)){2λQE(R2)E(S)+E(R2)(Q(1-λE(S))+λE(R))}/λE(R)2] > 0,

25

∆ is negative if and only if αo < 0. The fact that ∆ is negative means that E(Fk) is either strictly

increasing or strictly decreasing in k. Using the fact that, as k→kmax, E(Fk)→∞, it is

straightforward to show that E(Fk) is indeed strictly increasing in k.

B) Marginal Allocation Algorithm

Fox's marginal allocation procedure can be summarized in the following four steps [10]:

1. Start with k(0) where ki = 1 for i = 1, 2, …, m.

2. Set j = 1.

3. Set k(j) = k(j - 1) + ei where ei is the ith unit vector and i is any index for which

E(Fki(j - 1)) - E(Fki(j - 1) + 1)

is maximum.

4. Stop if j = n - m. Otherwise set j = j + 1 and go to step 3.

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26

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28

Table 1 Flow time comparisons between the FCFS and the GS policies(E(S) = E(S2) = 1, E(R) = 2, E(R2) = 1, λ = 0.7)

K Q* E(Fk*|FCFS) E(Fk|GS)1 1 2.17 2.173 19 28.55 7.645 38 34.21 14.027 56 36.63 20.569 75 37.98 27.1611 94 38.83 33.7813 113 39.43 40.4215 132 39.86 47.0620 178 40.57 63.6950 460 41.84 163.63100 930 42.27 330.27500 4686 42.61 1663.58

29

Figure 1 The effect of product variety on expected flow time (GS policy)(E(S) = 1, E(S2) = 6, E(R) = 0.4, E(R2) = 1, λ = 0.9)

Figure 2 The effect of setup time on expected flow time (GS policy)(E(S) = 1, E(S2) = 10, k = 5, λ = 0.9, exponentially distributed setup time)

30

Figure 3 The effect of utilization on expected flow time (GS policy)(E(S) = 0.55, E(S2) = 10, E(R) = 0.2, E(R2) = 1, k = 4)

31

Figure 4 The effect of product variety on expected flow time (FCFS policy)(E(S) = 0.55, E(S2) = 10, E(R) = 0.2, E(R2) = 10, λ = 0.9, Q = 30)

Figure 5 The effect of batch sizes on expected flow time (FCFS policy)(E(S) = 0.55, E(S2) = 10, E(R) = 0.2, E(R2) = 10, λ = 0.9, k = 5)

32

Figure 6 Flow time comparisons between the GS and FCFS policies(E(S) = 0.55, E(S2) = 10, E(R) = 0.2, E(R2) = 10, λ = 0.9, Q = 30)