To appear in IIE Transactions

Scheduling Policies, Batch Sizes, and Manufacturing LeadTimes

Saifallah Benjaafar† and Mehdi SheikhzadehDepartment of Mechanical Engineering

University of MinnesotaMinneapolis, Minnesota 55455, USA

Abstract: In this paper, we examine the impact of scheduling policies on batch sizing

decisions in a multi-item production system. We also investigate the joint effect of

scheduling policies and batch sizing decisions on production lead times. In particular, we

compare the performance of a first come-first-served (FCFS) policy with a group

scheduling (GS) policy and study the effect of both on the optimal batch size. We show

that a group scheduling policy can lead to significant performance gains, as measured by

reduced lead times and higher production rates, relative to the FCFS policy and characterize

conditions under which these gains are realized. We also study the impact of the GS policy

on other system operating parameters. In particular, we find that using a group scheduling

policy eliminates the need for batching, preserves system capacity despite the presence of

setups, and accommodates higher product mix variety. These results are shown to be very

different from those obtained for the FCFS policy and to have important implications for

the management and control of multi-item production facilities.

† The author's research is supported by the National Science Foundation under grant No. DMII-9309631.

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1 Introduction

In this paper, we examine the impact of scheduling policies on batch sizing

decisions in a multi-item production system. We also investigate the joint effect of

scheduling policies and batch sizing decisions on production lead times. In particular, we

compare the performance of a first come-first-served (FCFS) policy with a group

scheduling (GS) policy and study the effect of both on the optimal batch size. We show

that a group scheduling policy can lead to significant performance gains, as measured by

reduced lead times and higher production rates, relative to the FCFS policy and characterize

conditions under which these gains are realized. We also study the impact of the GS policy

on other system operating parameters. In particular, we find that using a group scheduling

policy eliminates the need for batching, preserves system capacity despite the presence of

setups, and accommodates higher product mix variety. These results are shown to be very

different from those obtained for the FCFS policy and to have important implications for

the management and control of multi-item production facilities.

Despite its importance, the relationship between batch sizes and dynamic

performance measures of production systems, such as flow time, work-in-process

inventory, and production rates, has received little rigorous treatment in the literature.

Although a number of analytical models have been proposed for the general performance

evaluation of manufacturing systems [2] [3] [10], none of these models deals explicitly

with the relationships between batch sizes and performance. In the queueing literature, a

significant body of work exists on queues with bulk arrivals and bulk service times [8].

However, exact results exist only for simple models. None of these models account for

setup times between batches and/or the possibility of alternating priority scheduling.

In the production and manufacturing literature, Karmarkar [4] was the first to

examine, using a queuing model of a single machine, the impact of batch sizes and setup

times on levels of WIP and lead times. This model is extended in [5] and [6] to the multi-

item/multi-machine case where a procedure for obtaining optimal batch sizes is also

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described. Using a variation on the single machine model in [4], Kekre [7] studies the

effect of increasing product mix variety on the size of optimal batches. His findings

indicate that increased variety tends to increase the optimal batch size. A queueing model

similar to the one described in [4] is independently proposed by Zipkin [12] and used to

model the aggregate behavior of a batch production facility. Benjaafar [1] evaluates the

relationship between batching and machine sharing decisions in multi-product

manufacturing systems. He shows that in the presence of setup times greater machine

sharing between multiple part types generally induces larger batch sizes.

All of these studies assume a FCFS policy in scheduling arriving batches. That is,

no attempt is made to economize on setups by giving priority to batches of part types for

which the production facility is already setup. Similarly, no attempt is made to

systematically group batches based on their setup requirement and to produce them in an

alternating priority. In fact, Kekre [7] argued that using a setup avoiding, or a look-ahead,

policy leads to little additional benefits, as measured by average waiting time, relative to the

FCFS policy. Using an approximation model, he concluded that "efforts to reduce queue

times by processing all parts available from the queue for which the cell is already setup

and saving setup time results in little gain" [7, p. 330].

In this paper, we show that this conclusion is incorrect and that indeed a setup

avoiding, or group scheduling, policy can lead to significant gains. More importantly, we

show that such a policy has very important and surprising implications for batch sizing

decisions that are very different from those of the FCFS policy. The policy has also

several counter-intuitive implications for planning and managing multi-product

manufacturing facilities regarding system capacity, part mix variety, and setup

requirements.

2 The First Come-First Served Policy

Processing parts in batches is preferable to the processing of parts in lots of size

one when setup times are significant. By batching parts that have similar manufacturing

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requirements, the frequency of setups is reduced, which in turn reduces part lead times and

provides greater production capacity. Excessive batching can, however, result in

performance deterioration. Increasing batch sizes increases the batch processing times

which, in turn, increase the batch lead time (or flow time) through the manufacturing

facility.

The relationship between batch sizes and lead times, under a FCFS batch

scheduling policy, can be illustrated, as suggested by Kekre [7] and Benjaafar [1], by

modeling a machine as a multi-class single server queue. Parts arrive to the machine in

batches of size Q where they wait for service. When the machine finishes its current batch,

it is set up for the next batch, if different from the current one, and starts processing parts

from that batch. The product mix consists of K part types each with an average demand

Di. The demand is assumed to be evenly distributed among part types so that Di = D/K

where D is the total average production demand. This assumption is made in order to be

able to examine the impact of part variety on batch sizing decisions and on lead time. The

processing time is denoted by X and is assumed to be generally distributed with a mean of

1/µ and variance of σ2. This assumption allows for different part types to have different

processing times. A batch incurs a setup time of τ when the previous batch on the machine

is of a different type. Otherwise, no setup is required. The average arrival rate of batches

to the machine is given by λ = D/Q, with batch inter-arrival times assumed to be

exponentially distributed (the robustness of this assumption is discussed in [4]). Assuming

an unlimited buffer size and first-come first-served sequencing of batches at the machine,

average part flow time can be calculated, based on that of a M/G/1 queue, as

FFCFS = E(S) + λE(S2)

2(1 - λE(S)), (1)

where E(S), and E(S2) are respectively the first and second moment of batch processing

and setup time with E(S) = Q/µ + τ(K - 1)/K and E(S2) = E(S)2 + Qσ2 + τ2(K - 1)/K2.

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In order to ensure system stability, there is a minimum feasible batch size that is

given by

Qmin(FCFS) = Dτ(K - 1)K(1 - ρ)

, (2)

where ρ = D/µ and represents actual machine utilization (i.e. excluding time spent on

setups). Equivalently, for a fixed batch size, there is a maximum feasible setup time, τmax,

τmax(FCFS) = Q K(1 - ρ)D(K - 1)

, (3)

and a maximum feasible level of part variety, Kmax,

Kmax(FCFS) = DτDτ - Q(1 - ρ)

. (4)

The value of Qmin is an increasing function of setup time τ and part variety K . The value

of τmax is, on the other hand, a decreasing function of K and an increasing function Q.

The value of Kmax is an increasing function of Q and a decreasing function of τ. These

bounds mean that, in order to ensure production feasibility, part mix variety and setup

requirements must be limited while a minimum batch size must always be maintained.

Since a fraction of system capacity is spent on setups, the maximum feasible

production rate, or maximum throughput, is given by

THmax(FCFS) = Q

Qµ

+ τ(K - 1)

K

(5)

which is an increasing function of Q and a decreasing function of K and τ. In the limit

case, as Q → ∞, THmax → µ, with µ being the machine's maximum throughput in the

absence of any setups.

It is easy to see that increasing Q increases the batch processing time but decreases

the frequency of setups and thus increases system capacity. Depending on the length of

setup times and the level of machine utilization, larger batches may initially result in a net

reduction of part flow times. However, as Q is further increased, the advantages of

batching are eventually eroded and flow time starts to increase with increases in Q. Noting

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that the expression of average flow time is a convex function of Q, the batch size that

minimizes flow time can be obtained by the first order condition of optimality as

Q*(FCFS) = ae + a2e2 + ad(be - cd)

ad, (6)

where a, b, c, d, and e are all positive parameters with a = K(2µ - D), b = 2µτ(K - 1)(µ -

D) + KDµ2σ2, c = Dµ2τ2(K - 1)(K - 2)/K, d = 2µK(µ - D), and e = 2Dτµ2(K - 1). The

value of Q*(FCFS) can be shown to be monotonically increasing with part variety, K , and

setup time, τ, and is, thus, generally greater than one.

For a given batch size, it should be noted that the average time spent on setups per

batch is given by

Tsetup(FCFS) = τ(K - 1)/K (7)

which means that only K - 1 out of every K batches incur a setup. In other words, with a

probability of 1/K, two consecutive batches are of the same type. This, in turn, means that

the length of a production run per setup is variable and may be greater than a single released

batch. Thus, in optimizing the batch size under a FCFS scheduling policy, we are not

optimizing the length of each production run but only the size in which parts are released to

the system.

3 The Group Scheduling Policy

In practice, there is often an attempt to minimize the frequency of setups by

dynamically reordering batches so that part types for which the machine is currently setup

are processed first. This means that the machine would process all the batches of a

particular type, that are currently in queue, before switching to another part type. In this

section we examine the effect of using such a batch sequencing rule on system

performance, on the optimal batch size, and on the setting of other system operating

parameters. Specifically, we consider a sequencing rule where once the machine is setup

(setups are assumed to be non-zero) for a particular part type, it continues processing

batches from that type until all batches are exhausted. It is subsequently setup for the next

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part type. The machine is assumed to switch from one part type to the next in a cyclic

order. This sequencing rule is known in the queuing literature as a cyclic and exhaustive

alternating priority policy and is usually studied in the context of polling systems [9]. In

the manufacturing literature, such rules are often referred to as family or group scheduling

(GS) rules.

Proposition 1: Q*(GS) = 1, Qmin(GS) = 1 and τmax(GS)= ∞, Kmax = ∞.

Proof: The expression of average part flow time for the GS scheduling policy is

given by [9]

FGS = E(X) + λE(X 2) + τ(K - λE(X))

2(1 - λE(X )), (8)

where E(X) and E(X2) are respectively the first and second moment of batch processing

time, excluding setup time, with E(X) = Q/µ and E(X2) = E(X)2 + Qσ2. Expression (8)

can also be rewritten as

FGS = ad

Q + fd

, (9)

where a and d are as defined earlier and f = K2[τσµ(µ - D/K) + Dµ2σ2/K]. Since a, d and

f are all positive parameters independent of Q, average part flow time is a linearly

increasing function of Q and, thus, the optimal batch size, Q*, is equal to one. The stability

condition for this policy is simply λE(X) < 1, or equivalently D/µ < 1. Consequently, the

minimum feasible batch size, Qmin, is one, and both the maximum feasible setup time,

τmax, and the maximum feasible product variety, Kmax, are unbounded. ◊

These results lead to the following corollary:

Corollary 1: For τ > 0, THmax(GS) > THmax(FCFS) and THmax(GS) is independent of

τ, where THmax(•) is the maximum feasible throughput rate.

Proof: It follows from the stability conditions for the two policies that the maximum

feasible throughput rates (i.e., system capacity) are, respectively, given by

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THmax(FCFS) = Q

Qµ

+ τ(K - 1)

K

(10)

and

THmax(GS) = µ. (11)

Noting that THmax(FCFS) is a strictly decreasing function of τ, it is easy to show that

THmax(GS) > THmax(FCFS). ◊

Proposition 1 and corollary 1 have important implications for system operation and

management. They go counter to long held beliefs regarding the inevitability of batching in

the presence of setups. In fact, these results not only do they show that batching is not

necessary when a setup avoiding policy is in place but that it is not even optimal.

Furthermore, the GS policy improves system capacity by maintaining a maximum feasible

production rate that is unaffected by increases in setup times. In practice, this capability is

important for systems where sustaining high production volumes is desirable and/or where

parts with highly different setup requirements are simultaneously produced. The GS policy

also preserves system capacity despite increases in part variety. This capability is important

for manufacturing systems with a highly diversified product portfolio or those that compete

based on customized products, among others. Note that with a FCFS policy, system

capacity quickly deteriorates with increases in setup time or part variety. The impact of the

FCFS and GS policies on Q*, Qmin,τmax, and Kmax is graphically depicted in Figures 1,

2, and 3.

The above somewhat counterintuitive results can, in part, be explained by the

following proposition:

Proposition 2: Let Tsetup(GS) be the average time spent on setup per batch, then for

fixed D, µ and Q, Tsetup(GS) is a constant independent of both product variety, K, and

setup time, τ. The value of Tsetup is given by Tsetup(GS) = Q(1/D - 1/µ).

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Figure 1 The effect of setup time on average flow time in the FCFS andGS policies (K = 2, µ = 1, D = 0.6, Q = 11)

Figure 2 The effect of batch size on average flow time in the FCFS and GSpolicies (K = 2, µ = 1, D = 0.8, τ = 1.3)

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Figure 3 The effect of part variety on average flow time in the FCFS andGS policies (Q = 10, µ = 1, D = 0.8, τ = 2.63)

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Proof: Let us first define average cycle time, Tc, as the average time interval between two

successive setups of the machine for the same part type. The value of Tc can be calculated

as

Tc = Kτ + NcQ/µ, (12)

where Nc is the average number of batches produced during a cycle time and is itself given

by

Nc = λTc = DTc/Q. (13)

The average time spent on setup per batch can now be obtained as

Tsetup (GS) = (K/Nc)τ, (14)

where the ratio K/Nc corresponds to the frequency with which setups are performed

during a cycle time. Substituting for the values of Nc and Tc, the above expression can be

rewritten as

Tsetup(GS) = Q(1/D - 1/µ). ◊ (15)

The fact that Tsetup(GS) is independent of both τ and K is itself surprising and

counter-intuitive. It can however be explained by the fact that as either τ or K increases,

the machine would spend more time processing batches after each setup, making these

setups increasingly less frequent. We should note that this is not the case for the FCFS

policy where the average time spent on setups per batch is given by

Tsetup(FCFS) = (K - 1)τ/K (16)

and is indeed increasing in both τ and K. If we divide Tsetup by the batch size Q to obtain

the average time spent on setup per part, we find that in the case of the GS policy the value

of this average to be independent of Q, which partly explains why a batch size of one is

desirable; on the other hand, for the FCFS policy the average time spent on setup per part is

decreasing in Q which similarly justifies the need for larger batch sizes in this case.

As it is the case with the FCFS policy, the length of each production run under the

GS policy is also variable. That is, the number of parts produced after each setup may be

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greater than the size, Q, in which parts are released to the machine. In fact, the average

number of batches produced following each setup is given by (see Appendix)

NQ = DτQ(1 - D

µ), (17)

which means that the corresponding number of parts produced per setup is

NP = Dτ(1 - D

µ). (18)

This number can be viewed as the realized average production batch size. The value of NP

is an increasing function of setup and machine utilization and is generally greater than one.

Thus, although the optimal release batch size is one, parts could be produced in larger

quantities. The GS policy will, in effect, result in a dynamic batch sizing policy that

responds to changes in system loading and setups. It is interesting to note that Np is

independent of Q and K so that regardless of the size of the release batch and the level of

part variety, the same average number of parts will be produced per setup. This is not the

case in the FCFS policy, where the realized batch size is determined by both Q and K and

is independent of system loading conditions (under the FCFS policy, the probability that a

given batch of size Q would incur a setup is (K- 1)/K, independently of machine

utilization). An additional advantage to the GS policy is, thus, relieving production

managers from determining the batch size in which parts should be released and in letting

the actual production batch size be determined dynamically as a function of the system's

loading conditions.

In view of proposition 1 and Corollary 1, it is easy to find instances where the GS

policy performs significantly better than the FCFS policy. For example, this will be the

case when Q is sufficiently small, setup time is long, or product variety is high. In fact,

when Q is below Qmin(FCFS), τ is greater than τmax(FCFS) or K is larger than

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Kmax(FCFS), the FCFS policy leads to an infinitely large flow time (see Figures 1, 2 and

3). More generally, we have the following result:

Proposition 3: FFCFS ≤ FGS if and only if (1) ρ ≤ ρ0 and (2) Q ≥ Q0, where

ρ0 = [(K 2 - K + 2) - (K - 1)(K 3 - K2 + 4)]/2,and

Q0 = D(K - 1)[τ(2(1 - ρ) + K(K - 1)) + KDσ2)]/[ρ2 - ρ(K 2 - K + 2) + (K 2 - K + 2)].

Proof: Let FASY(FCFS) denote the asymptote of average flow time as a function of Q for

the FCFS policy (a lower bound for FFCFS). We first show that FASY(FCFS) ≤ FGS if

and only if ρ ≤ ρ0 Noting that the expression of average flow time for the FCFS policy

can be written as

FFCFS = aQ2+ bQ - c

dQ - e, (17)

its asymptote can be obtained as

FASY(FCFS) = ad

Q + bd + aed2

. (18)

Therefore, FASY(FCFS) ≤ FGS is equivalent to

bd + aed2

≤ fd

(19)

or,

d(b - f) + ae ≤ 0 (20)

which reduces to

ρ2 - ρ(K 2 - K + 2) + (K 2 - 2K + 2) ≥ 0. (21)

The above inequality can be rewritten as

(ρ - ρ0)(ρ - ρ1) ≥ 0,

with

ρ0 = [(K 2 - K + 2) - (K - 1)(K 3 - K2 + 4)]/2

and

ρ1 = [(K 2 - K + 2) + (K - 1)(K 3 - K2 + 4)]/2.

Noting that ρ1 ≥ 1, the above inequality is equivalent to

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ρ ≤ ρ0. (22)

Next, we show that FFCFS ≤ FGS implies ρ ≤ ρ0 and Q ≥ Q0. Using the fact that if FFCFS

≤ FGS then FASY ≤ FGS, it follows that if FFCFS ≤ FGS then ρ ≤ ρ0 . Now, noting that

FFCFS ≤ FGS is equivalent to

aQ2 + bQ - cdQ - e

≤ ad

Q + fd

,

and using the fact that dQ - e ≥ 0 and d(b - f) + ae ≤ 0 (see (20)), then the above ineqality

can be rewritten as

Q ≥ cd - efd(b - f) + ae

, (23)

or

Q ≥ Q0,

where

Q0 = D(K - 1)[τ(2(1 - ρ) + K(K - 1)) + KDσ2)]/[ρ2 - ρ(K 2 - K + 2) + (K 2 - K + 2)].

Hence, we have shown FFCFS ≤ FGS ⇒ ρ ≤ ρ0 and Q ≥ Q0. To show, the reverse, we

note that when ρ ≤ ρ0 and Q0 ≤ Q, we have

aQ2 + bQ - cdQ - e

≤ ad

Q + fd

,

which is equivalent to FFCFS ≤ FGS. ◊

The value of ρ0 is an increasing function of K so that the range of utilizations over

which the GS policy is more desirable increases as K decreases. The value of Q0 is

similarly an increasing function of K which in this case means that as K increases the range

of batch sizes over which the GS policy is more desirable increases. The value of Q0 is

also an increasing function of setup time and processing time variance so that with

increases in either τ or σ2 the range of batch sizes that makes the GS policy superior

increases. We should note that condition (2) could have been equivalently expressed in

terms of either a critical setup time parameter or a critical part variety parameter.

Numerical comparisons of average flow time between the GS and the FCFS

policies are provided in Table 1. The GS policy can be seen to yield significant gains in

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performance over a wide range of operating conditions. It is particularly important to note

that the difference in flow time at the optimal batch size for both the GS and the FCFS

policies can be substantial. As anticipated from proposition 3, the difference in performance

between the two policies tends however to diminish with increases in batch size. Using

simulation, similar behavior was observed for the flow time variance. A small sample from

the simulation is listed in Table 2.

As mentioned in the introduction, Kekre [7] argued that a group scheduling policy

(a look-ahead policy in Kekre's terminology) would result in little reduction in part flow

times. Clearly, the results of this paper show that this is not necessarily true. Kekre bases

his argument on a group scheduling policy similar to the one described here, except that the

switching between part types follows a first come-first served discipline with setups

occurring only if there is at least one part already in the queue. Such a policy is evidently

superior to the strict cyclic GS policy described here and, thus, should result in at least

equal benefits. Using simulation, this result was verified and two policies were, indeed,

found to have similar characteristics and to provide relatively equal performance. A sample

of the simulation results is included in Table 3 - the notation GS/FCFS and GS/CYC is

used to differentiate between the policy described by Kekre and the strict cyclic policy

discussed in this paper. As expected, the dynamic GS/FCFS performs slightly better than

the cyclical GS/CYC policy, especially for larger batch sizes and higher number of part

types. Similar results were also obtained recently by Wemmerlöv and Vakharia [11] in a

simulation study . Because of the state-based nature in which changeover between batch

types are made under a first come-first served discipline, analytical results are, however,

difficult to obtain.

4. Conclusion

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The objective of this paper was to show that a group scheduling policy can lead to

substantial improvements in performance relative to a first-come first-served policy. In

addition to its impact on flow time, such a policy is found to have important and surprising

implications for other system operating parameters. In particular, we found that using a

group scheduling rule eliminates the need for batching, preserves system capacity despite

the presence of setups, and accommodates higher product mix variety. The capability to

produce a large variety of items in small batch sizes without significant setup time penalties

and in short lead times has become in recent years increasingly important to the design and

operation of most manufacturing systems. When setup times cannot be significantly

educed, a group scheduling policy, similar to the one proposed here, certainly offers an

attractive alternative for system designers and managers. Since the proposed group

scheduling is not necessarily optimal, further performance improvements, through

additional refinement of the policy, may be possible. Future research should thus focus on

characterizing the optimal scheduling policy and examining its effect on batch sizing

decisions.

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Table 1 Flow time comparisons between the FCFS and the GS policies

K Q FFCFS FGS1 ∞ 8 . 92 ∞ 11.93 87.4 14.94 41.9 17.95 36.8 20.9

2 6 3 6 . 4 23.97 37.6 26.910 44.1 35.915 57.6 50.920 72.0 65.930 101.4 95.950 161.0 155.91 ∞ 1 5 . 42 ∞ 18.43 ∞ 21.44 586.3 24.45 80.6 27.46 59.3 30.4

4 7 53.7 33.48 5 2 . 3 36.49 52.7 39.410 53.9 42.415 64.7 57.420 78.2 72.430 106.8 102.450 165.9 162.41 ∞ 2 1 . 92 ∞ 24.93 ∞ 27.94 ∞ 30.95 132.8 33.96 74.8 36.9

6 7 62.5 39.98 59.0 42.99 5 7 . 6 45.910 58.1 48.915 67.4 63.920 80.4 78.930 108.7 108.950 167.6 168.9

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Table 2 Flow time variance comparisons between the FCFS and the GSpolicies

K Q VFCFS VGS1 ∞ 65.63 7638.8 193.26 1354.2 517.6

2 10 1944.8 1139.115 3317.8 2237.320 5184.0 3760.130 10282.0 7921.050 25921.0 21054.01 ∞ 148.83 ∞ 324.06 3516.5 660.5

4 10 2905.2 1197.215 4186.1 2460.220 6115.2 4019.630 11406.2 7956.650 27522.8 20793.61 ∞ 228.03 ∞ 400.06 5595.0 812.25

6 10 3375.6 1505.4415 4542.8 2560.420 6464.2 4342.830 11815.7 8354.050 28089.8 21083.0

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Table 3 Flow time comparisons between the GS/FCFS and the GS/CYCpolicies

K Q FGS/FCFS FGS/CYC1 8.6 8 . 92 10.8 11.93 13.7 14.94 16.7 17.95 19.9 20.9

2 6 22.0 23.97 25.2 26.910 33.5 35.915 48.5 50.920 61.4 65.930 89.7 95.950 146.6 155.91 1 4 . 0 1 5 . 42 16.8 18.43 18.9 21.44 21.4 24.45 24.4 27.46 26.8 30.4

4 7 29.3 33.410 40.5 42.415 50.1 57.420 65.8 72.430 96.9 102.450 150.7 162.41 2 0 . 6 2 1 . 92 22.1 24.93 23.6 27.94 25.1 30.95 25.9 33.96 29.5 36.9

6 7 33.0 39.910 41.1 48.915 55.4 63.920 70.8 78.930 94.8 108.950 142.0 168.9

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Appendix

Proposition 4: The average number of parts produced per setup is given by Np = Dτ/(1

- D/µ). Proof: Noting that

Tc = Kτ + NcQ/µ,

and

Nc = λTc = DTc/Q,

we get

Nc = D KτQ(1 - D

µ).

Since Nc is the total average number of batches produced per cycle, the average number of

batches of each part type produced per cycle is

NQ = D τQ(1 - D

µ)

and the corresponding avearge number of parts is

NP = D τ(1 - D

µ). ◊

References

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[1] Benjaafar, S., "Modeling and Analysis of Machine Sharing in AutomatedManufacturing Systems," The European Journal of Operational Research, 91 , 1, 56-73,1996.

[2] Bitran, G. R. and S. Dasu, "A Review of Open Queueing Network Models ofManufacturing Systems," Queueing Systems, 12, 95-132, 1992.

[3] Buzacott, J. A. and J. G. Shanthikumar, Stochastic Modeling of ManufacturingSystems, Prentice Hall, New Jersey, 1993.

[4] Karmarkar, U. S., "Lot Sizes, Lead Times and In-Process Inventories," ManagementScience, 33, 3, 409-418, 1987.

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[6] Karmarkar, U. S., Kekre, S., Kekre and S. Freeman, "Lot Sizing and Lead-timePerformance in a Manufacturing Cell," Interfaces, 15, 2, 1-9, 1985.

[7] Kekre, S., "Performance of a Manufacturing Cell with increased Product Mix," IIETransactions, 19, 3, 329-339, 1987.

[8] Kleinrock, L., Queuing Systems, Vol. I, John Wiley, New York, New York, 1975.

[9] Takagi, H., "Queueing Analysis of Polling Models," ACM Computing Surveys, 20, 1,5-28, 1988.

[10] Viswanadham, N. and Y. Narahari, Performance Modeling of AutomatedManufacturing Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1992.

[11]Wemmerlöv, U. and A. J. Vakharia, "On the Impact of Family SchedulingProcedures," IIE Transactiona, 25, 4, 102-104, 1993.

[12] Zipkin, P. H., "Models for Design and Control of Stochastic, Multi-Item BatchProduction Systems," Operations Research, 34, 1, 91-104, 1986.

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Biographical Sketch

Saifallah Benjaafar is an Assistant Professor in the Department of Mechanical Engineeringat the University of Minnesota. He holds Ph.D. and M.S. degrees from the School ofIndustrial Engineering at Purdue University and a B.S. degree from the Department ofElectrical and Computer Engineering at the University of Texas, Austin. His researchinterests are in design, modeling and control of manufacturing systems. He is the author ofover 30 technical papers in this area. He has served as area editor for the 1994 IERCConference Proceedings and as a guest editor for a special issue of the InternationalJournal of Flexible Manufacturing Systems on manufacturing flexibility. He is also amember of the editorial board of the International Journal of Industrial Engineering.

Mehdi Sheikhzadeh is a doctoral student in the Department of Mechanical Engineering atthe University of Minnesota. He holds a M.S. degree in Industrial Engineering from theUniversity of Minnesota and a B.S. degree from Tehran University. His research interestsare in aplying queueing models to multiple product manufacturing systems.