Design of Stockless Production Systems

Abstract

In make‐to‐stock production systems finished goods are produced in anticipation of demand. By contrast, in stockless production systems finished goods are not produced until demand is observed. In this study we investigate the problem of designing a multi‐item manufacturing system, where there is both demand‐ and production‐related uncertainty, so that stockless operation will be optimal for all items. For the problem of interest, we focus on gaining an understanding of the effect of two design variables: (i) manufacturing speed—measured by the average manufacturing rate or, equivalently, the average unit manufacturing time, and (ii) manufacturing consistency—measured by the variation in unit manufacturing times. We establish conditions on these two variables that decision makers can use to design stockless production systems. Managerial implications of the conditions are also discussed.

Keywords

production system design manufacturing operations inventory management stochastic processes

1. Introduction

In make‐to‐stock production systems finished goods are produced in anticipation of demand. By contrast, in stockless production systems finished goods are not produced until demand is observed. In this sudy we examine the problem of designing a multi‐item manufacturing system, where there is both demand‐ and production‐related uncertainty, so that stockless operation will be optimal for all items. Note that we use the term stockless in lieu of make‐to‐order as the latter term is sometimes also used to denote the mode of operation where finished goods are manufactured according to customer specifications, which we do not consider herein.

Our interest in the problem of designing stockless production systems grew out of consulting work we performed for firms in two very different industries. One consulting project was done with a manufacturer of commercial airplanes, the other with an oil and gas company. As shall become apparent to the reader, our findings have applicability to firms in a variety of industries. For the problem under study we focus on gaining an understanding of the effect of two variables that are encountered in the design of production systems. One variable considered is manufacturing speed, measured by the average manufacturing rate or, equivalently, by the average unit manufacturing time. The other variable considered is manufacturing consistency, measured by the variation in unit manufacturing times.

Interest in stockless production systems now spans a number of decades with the earliest work on the subject being that of Popp (1965), which looked at the make‐to‐stock versus stockless operation question for a single item. Since then multi‐item manufacturing settings have been investigated. Some recent work includes that of Federgruen and Katalan (1999), where the impact of adding a stockless item to a make‐to‐stock production system was studied, and that of Arreola‐Risa and DeCroix (1998) and Rajagopalan (2002), which explored which items of a collection ought a firm to produce make‐to‐stock and which items are best produced under stockless operation. The work of Rajagopalan assumed a setup time between items while the earlier work of Arreola‐Risa and DeCroix did not. It should also be mentioned that the problem studied in Arreola‐Risa and DeCroix is different from our work in this study in three fundamental ways. First, they focused on determining the stockless condition for one item within a set of items, while we seek conditions under which stockless operation will be optimal for all items. Second, they studied an existing production system where manufacturing speed and consistency were known and fixed, and the decision variables are the values of the base‐stock levels. In contrast, our production system is being designed, and hence the decision variables are the values of manufacturing parameters so that stockless operation is optimal. Third, we model unit manufacturing time as having a continuous phase‐type or a gamma distribution. Arreola‐Risa and DeCroix used different probability distributions to model unit manufacturing time.

Other recent work has primarily examined scheduling‐related issues and includes that of Zheng and Zipkin (1990), Veatch and Wein (1996), de Vericourt et al. (2000), Veatch and de Vericourt (2003), and Youssef et al. (2004). A significant amount of research has also been conducted on assemble‐to‐order systems (see Song and Zipkin 2003), however that research stream lies outside the focus of this work. Lastly, zero‐inventory situations due to product perishability or duopoly competition can be found in Geismar et al. (2011) and Sun et al. (2008), respectively.

We make three major contributions with this article. First, we establish conditions on the design variables manufacturing speed and manufacturing consistency that decision makers can use to design a production system whose operation is optimally stockless for all items. Second, we identify theoretical properties of our findings that may provide guidance to researchers in their study of similar production systems. Third, we discuss several managerial implications of our results that should be of interest to supply chain managers considering a stockless strategy. Of special interest is the dynamic between risk pooling and stockless production.

The article has five more sections. Section 2 describes the problem setting and the model we employ in this study. Section 3 investigates multi‐item systems, while section 4 looks at the single‐item case. Section 5 illustrates the application of our results in a real‐world project. Section 6 presents our conclusions and future research directions.

2. Problem Setting and Model

We consider a firm that is a manufacturer of several types of finished goods, where the different types of finished goods are called items. Items are manufactured one unit at a time, have setup times and setup costs that are negligible, share the same production facility, and experience manufacturing times that may exhibit variability from unit to unit, which we assume without loss of generality, that is, the special case of deterministic manufacturing times is, not necessarily excluded. The distribution of unit manufacturing times is the same for all items.

Let M denote unit manufacturing time. To maximize the applicability of our results, we first model M as having a continuous phase‐type (PH) distribution. The continuous PH distribution is dense in the field of all positive‐valued distributions, that is, it can approximate any distribution on the nonnegative reals (see e.g., Kao 1997). Let

Z = (Z_{t}, t \geq 0

) be a continuous‐time, time‐homogeneous Markov chain without instantaneous states on the state space {1,2,…,k,k + 1}, where {1,2,…,k} are transient states and k + 1 is an absorbing state. Let S denote the generator of Z restricted to the transient states, so that S is of order k. The k × k matrices S that arise in this way, which are called PH‐generators, are characterized by nonnegative off‐diagonal entries, negative diagonal ones, and nonsingularity. Let

α = (α_{1}, α_{2} . \dots, α_{k})

denote the initial distribution of Z on the transient states and set

α_{k + 1} = 0

for the probability that Z is initialized in absorbing state k + 1 (to avoid the trivial case when unit manufacturing time is zero). Let M be the time at which Z is absorbed in state k + 1. The distribution of M is said to be continuous phase‐type with representation (α,S) and order k, and will be denoted by PH(α,S,k). Letting

f_{X} (\cdot)

denote the probability density function of random variable X, we then have that when M is distributed according to the continuous phase‐type family

f_{M} (t) = α e^{tS} S^{0}, t > 0

where

S^{0} = - S

1 and 1 is the k × 1 unit vector.

To measure manufacturing speed, we will use the average manufacturing rate, or equivalently, the average unit manufacturing time. Letting μ and E(·) respectively denote the average manufacturing rate and the expected value operator, from Equation we have that

E (M) = 1 / μ = - α S^{- 1} 1 .

To measure manufacturing consistency, we will use the squared coefficient of variation, which is defined as the ratio of the variance to the mean squared. Letting V(·) and SCV(·) respectively denote the variance and squared coefficient of variation operators, from Equation we have that since

E (M^{2}) = 2 α S^{- 2} 1

and

V (M) = E (M^{2}) - [E (M {)]}^{2}

, then

SCV (M) = \frac{2 α S^{- 2} 1}{[- α S^{- 1} 1]^{2}} - 1 .

Note that the lower the value of SCV(M), the greater the consistency of unit manufacturing times. Also, because the SCV(·) is a relative measure, it allows for comparison across different manufacturing environments. Although there is no generally accepted standard as to what constitutes a “high” or “low” squared coefficient of variation, a value of < 0.25 is typically viewed as signaling low variability (high manufacturing consistency), while a value > 0.75 is typically viewed as signaling high variability (low manufacturing consistency). See Knott and Sury (1987) for an empirical study that offers further guidance on this subject matter.

It is worth noting that modeling M as having a continuous phase‐type distribution is appealing as it holds with some generality. However, use of continuous phase‐type makes the manipulation of the design variables, that is, manufacturing speed and manufacturing consistency (captured in Equations and , respectively), very challenging. In our consulting work the system designers ultimately decided to model M using the gamma family of distributions. This was done for three reasons:

The gamma distribution does not take negative values and manufacturing times were being modeled.

The gamma distribution can adopt both symmetric and asymmetric shapes.

The gamma distribution allowed for easy manipulation of the design variables.

When M is modeled as distributed according to the gamma family (see e.g., Taylor and Karlin 1998)

f_{M} (t) = \frac{(μ k)^{k}}{Γ (k)} t^{k - 1} e^{- k μ t}, t > 0

where μ > 0 and k > 0 are the parameters of the gamma distribution, and Γ(·) is the gamma function. When k is integer, the gamma distribution becomes an order‐k Erlang distribution. For brevity, Equation will be referred to as Gamma(μ,k).

It follows that for the Gamma(μ,k)

E (M) = \frac{1}{μ} .

Hence the average unit manufacturing time is equal to the reciprocal of the parameter μ, the manufacturing rate or speed, which is one of the design variables of interest. Furthermore, we have that (see e.g., Allen 1990)

SCV (M) = \frac{1}{k},

where the parameter k represents manufacturing consistency, which is the other design variable considered in this article. As k is strictly positive, it follows that the higher the value of k, the greater the consistency of unit manufacturing times. In fact, deterministic unit manufacturing times result when k → ∞.

A comparison of Equations and with Equations and makes it clear that the design variables of interest are much easier to manipulate when M has a Gamma(μ,k) distribution than when M has a PH(α,S,k) distribution. The Gamma(μ,k) distribution offers two more advantages that should also be of interest to real‐world system designers.

First, Theorem 2 in O'Cinneide (1991) states the following: “The coefficient of variation of an order k PH‐distribution is at least

1 / \sqrt{k}

, and the order k Erlang distribution is the only one to attain the bound.” Thus in our research the SCV—our measure of manufacturing consistency—is ≥ 1/k. That is, for PH(α,S,k) distributions, the order k and the SCV are inversely related. So, if for example, one wanted to study cases with very low SCV values, say 0.0001, then the order of the needed PH(α,S,k) distribution would be at least equal to 1/0.0001 = 10,000. Based on Equations and , we know that the production system designer would need to manipulate matrices of at least dimension 10,000 × 10,000. Of course, lower SCV values would lead to matrices of even higher dimensions. Consequently, as the SCV value of the system being designed decreases, the required calculations in Equations and become increasingly burdensome and eventually intractable. In contrast, consideration of very low SCV values with the Gamma(μ,k) distribution is not only feasible but elementary as well. We can achieve any very low SCV value by simply selecting a large enough value of k. In the example above where SCV = 0.0001, we just set k = 1/0.0001 = 10,000.

A second advantage of the Gamma(μ,k) distribution is subtle but very relevant to real‐world system designers. We also know from Theorem 2 in O'Cinneide (1991) that the Erlang distribution minimizes the order k of PH(α,S,k) distributions required to achieve a SCV value, and we know that when k is integer, the gamma distribution becomes an order‐k Erlang distribution. Consequently, if a production system designer wants to model M as a PH(α,S,k) distribution, and at the same time wants to minimize the order k of the PH‐generator S, then the designer would have to model M as having an Erlang distribution, and consequently our gamma distribution results would apply by simply selecting k to be integer.

Having established the manufacturing time distributions that we consider in this study, that is, the PH(α,S,k) and Gamma(μ,k) distributions, we next turn our attention to item demand. The items being manufactured face exogenous, discrete, independent, and identically distributed (IID) demands, whose distribution is item‐specific. Back orders are allowed and incur a cost penalty per unit back ordered per unit time, while produced and unsold items incur a per unit inventory holding cost per unit time. Furthermore, both types of costs are stationary with respect to time. Given that demands are discrete and IID, we model them as being generated by a Poisson process. Letting demand for item i by time t be denoted by

D_{i} (t)

and letting

p_{X} (\cdot)

denote the probability mass function of the random variable X, we have that

p_{D_{i} (t)} (x) = \frac{e^{- λ_{i} t} (λ_{i} t)^{x}}{x!}, x = 0, 1, \dots

and

E (D_{i} (t)) = λ_{i} t,

where the parameter

λ_{i}

represents the average demand per unit time for item i or equivalently the average demand rate for item i.

To complete the description of our manufacturing setting requires that we posit production‐inventory and production‐scheduling policies, that is, when to release production orders, how many units of each item to produce and stock, and in what sequence to process the production orders. Unfortunately, for the system of interest the optimal forms of these policies are unknown. However a base‐stock policy has been shown to be optimal for a similar system under periodic review (see DeCroix and Arreola‐Risa 1998) and under continuous review (Zipkin 2000). Hence we adopt a base‐stock policy as our production‐inventory policy for the manufacturing setting under study. At the same time, given its widespread usage, we select first‐in first‐out (FIFO) as the production‐scheduling policy.

We now formally define the notation of our model. Let:

I_{i}

= steady‐state finished goods inventory level of item i

h_{i}

= inventory holding cost rate of item i

π_{i}

= back ordering cost rate of item i

K = long‐run average inventory holding and back ordering cost per unit time

S_{i}

= base‐stock level of item i

n = number of items.

The long‐run average inventory holding and back ordering cost per unit time is then:

K (S_{1}, \dots, S_{n}) = \sum_{i = 1}^{n} {h_{i} \sum_{x = 1}^{S_{i}} x p_{I_{i}} (x) + π_{i} \sum_{x = 1}^{\infty} x p_{I_{i}} (- x)} .

If we define

S_{i}^{*}, i = 1, \dots, n

, as the minimizers of

K (S_{1}, \dots, S_{n})

, it follows that stockless production is optimal for the manufacturing setting considered when

S_{i}^{*} = 0

for all i. The fundamental design problem under study is to determine those values of the manufacturing speed and manufacturing consistency variables for which this condition is met.

3. Multi‐Item Production Systems

In this section we investigate production systems with multiple items. We divide our study of such systems into two cases: homogeneous items and heterogeneous items. In the homogeneous items case the cost and demand parameters of items are identical, but the items are physically different, such as in flavor or color. In the heterogeneous items case not only are the items physically different, but it is also true that the cost and demand parameters are no longer the same.

In one of the consulting projects alluded to in section 1, items were heterogeneous, while in the other consulting project, which involved multiple plants, in one of the plants items were homogeneous, in several other plants the items were heterogeneous, and in one plant a single item was going to be produced. We consider production systems with heterogeneous items in subsection 3.1, while in subsection 3.2, we turn our attention to production systems with homogeneous items. In section 4, we present results for single‐item production systems.

3.1 Heterogeneous Items

In this subsection we investigate manufacturing systems that produce multiple heterogeneous items. Such systems are widespread and can be found in industries as varied as food, clothing, and electronics.

Before presenting our first major results on production systems with multiple heterogeneous items, we need to establish some additional notation. Let λ denote the average total demand rate for the n items produced, where

λ = \sum_{i = 1}^{n} λ_{i}

, and let

δ_{i}

denote the fraction of the average total demand rate coming from item i, where

δ_{i} = λ_{i} / λ

Theorem
For a production system with multiple heterogenous items, when M has a PH(α,S,k) distribution or a Gamma(μ,k) distribution, stockless operation is optimal, if and only if for each item i
$p_{N_{i}} (0) \geq \frac{π_{i}}{h_{i} + π_{i}},$
where
$N_{i}$
is the steady‐state number of outstanding production orders of item i, and
$p_{N_{i}} (0)$
is the probability that there are no outstanding production orders of item i.

Proof of Theorem
Let
$Ψ_{N_{i}} (\cdot)$
denote the probability generating function of
$N_{i}$
, where (see e.g., Allen 1990)
$Ψ_{N_{i}} (s) \equiv \sum_{x = 0}^{\infty} p_{N_{i}} (x) s^{x}$
and
$p_{N_{i}} (x)$
is the probability that there are x outstanding production orders of item i.

In Online Appendix S1 we derive that when M has a PH(α,S,k) distribution,
$Ψ_{N_{i}} (s) = \frac{(1 - λ / μ) δ_{i} (1 - s)}{1 - [1 - δ_{i} (1 - s)] {[Π (1 - δ_{i} (1 - s))]}^{- 1}}$
12a
where
$Π (s) = \sum_{j = 0}^{\infty} \int_{0}^{\infty} \frac{e^{- λ t} (λ t s)^{j}}{j!} α e^{tS} S^{0} dt .$
12b
Next, it can be easily demonstrated that since
$I_{i} = S_{i} - N_{i}$
and
$Ψ_{N_{i}} (\cdot)$
in Equation only depends on
$λ_{i}$
and λ (and not on the individual values of the
$λ_{j}$
's for j ≠ i), the minimization of
$K (S_{1}, \dots, S_{n})$
in Equation can be achieved by minimizing
$K_{i} (S_{i})$
for each i, where
$K_{i} (S_{i}) = h_{i} \sum_{x = 1}^{S_{i}} x p_{N_{i}} (S_{i} - x) + π_{i} \sum_{x = 1}^{\infty} x p_{N_{i}} (S_{i} + x) .$
Define
$Δ K_{i} (S_{i}) \equiv K_{i} (S_{i} + 1) - K_{i} (S_{i})$
and
$Δ^{2} K_{i} (S_{i}) \equiv Δ K_{i} (S_{i} + 1) - Δ K_{i} (S_{i})$
. From Equation , it is easy to derive
$Δ^{2} K_{i} (S_{i}) = (h_{i} + π_{i}) p_{N_{i}} (S_{i} + 1) \geq 0 .$
Hence it is optimal for item i to be stockless (
$S_{i}^{} = 0$
), if and only if
$K_{i} (S_{i} = 0) \leq K_{i} (S_{i} = 1) .$
Otherwise, it is optimal for item i to be produced make‐to‐stock (
$S_{i}^{} \geq 1$
). Now
$K_{i} (S_{i} = 0) = π_{i} \sum_{x = 1}^{\infty} x p_{N_{i}} (x),$

$K_{i} (S_{i} = 1) = (h_{i} + π_{i}) p_{N_{i}} (0) + π_{i} \sum_{x = 1}^{\infty} x p_{N_{i}} (x) - π_{i},$
so the condition in Equation in combination with Equations and leads to the condition in inequality .

When M is modeled as having a Gamma(μ,k) distribution, in Online Appendix S2 we derive that
$Ψ_{N_{i}} (s) = Ψ_{N} [(1 - δ_{i}) + δ_{i} s]$
17a
where
$Ψ_{N} (s) = \frac{(1 - λ / μ) (1 - s)}{1 - s [1 + λ (1 - s) / μ k]^{k}} .$
17b

Because
$Ψ_{N_{i}} (\cdot)$
in Equations and only depends on
$λ_{i}$
and λ, paralleling the steps for the PH(α,S,k) distribution, but now using Equations and instead of Equations and , one can obtain again the condition in inequality .

Note that the condition in inequality is equivalent to the newsvendor argument: if
$[1 - p_{N_{i}} (0)] π_{i} \leq p_{N_{i}} (0) h_{i}$
, then
$S_{i}^{} = 0$
. Also, observe that although the condition in inequality applies to both the PH(α,S,k) and Gamma(μ,k) distributions, since
$p_{N_{i}} (0) = Ψ_{N_{i}} (0)$
, the value of
$p_{N_{i}} (0)$
does depend on the distribution of M.

We next discuss several research findings and managerial implications that follow from the result in Theorem , and then present two corollaries of the theorem, respectively called Corollaries 1 and 2.

First observe that the right‐hand side (RHS) of the inequality in is a version of the well known critical fractile of the newsvendor model. This fractile plays an important role in many un‐capacitated, one‐period inventory management problems. It is intuitively pleasing to find the same fractile playing a role in our capacitated, multi‐period, production‐inventory system setting. Similar findings have been reported in some of the research already cited (de Vericourt et al. 2000, Veatch and de Vericourt 2003, Veatch and Wein 1996, Zheng and Zipkin 1990).

Further examination of Theorem allows us to deduce an intriguing property of production systems with heterogeneous items: because
$p_{N_{i}} (0) = Ψ_{N_{i}} (0)$
, the presence of λ, as itself and as part of
$δ_{i} = λ_{i} / λ$
in Equations (12a)–(12b) and in Equations (17a)–(17b), suggests that for item i, the number of other items and their average demand rates (
$λ_{j}, j \neq i$
) have no effect on the condition (11) of item i; it is only their sum
$\sum_{j \neq i}^{n} λ_{j}$
in
$λ = λ_{i} + \sum_{j \neq i}^{n} λ_{j}$
that matters. To illustrate, consider Company A, which has a production system with two heterogeneous items, with
$λ_{1} = 30$
and
$λ_{2} = 20$
. Suppose now that due to a product redesign, the demand for item 2 ends up being split between new items 3 and 4, where say
$λ_{3} = 13$
and
$λ_{4} = 7$
. Before the product redesign λ = 30 + 20 = 50, and
$δ_{1} = 30 / 50 = 0.6$
, while after the product redesign λ = 30 + 13 + 7 = 50, and
$δ_{1} = 30 / 50 = 0.6$
. As the value of the left‐hand side (LHS) in expression (11) is the same before and after the redesign because the values of λ and
$δ_{1}$
did not change, the optimality of item 1 being or not being stockless is unaffected by item 1 sharing the production system with one or two items.

Corollary
For a production system with multiple heterogenous items, when M has a PH(α,S,k) distribution, stockless operation is optimal, if and only if for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[α (λ_{i} I - S)^{- 1} S^{0}]}^{- 1}} \geq \frac{π_{i}}{h_{i} + π_{i}} .$

Proof of Corollary
From Equation we have that
$Ψ_{N_{i}} (0) = \frac{δ_{i} (1 - λ / μ)}{1 - [1 - δ_{i}] {[Π (1 - δ_{i})]}^{- 1}},$
and from Equation we have that
$\begin{matrix} Π (1 - δ_{i}) & = \sum_{j = 0}^{\infty} \int_{0}^{\infty} \frac{e^{- λ t} [λ t (1 - δ_{i} {)]}^{j}}{j!} α e^{tS} S^{0} dt \\ = \int_{0}^{\infty} e^{- λ t} \sum_{j = 0}^{\infty} \frac{[(λ - λ_{i}) t]^{j}}{j!} α e^{tS} S^{0} dt \\ = \int_{0}^{\infty} e^{- λ t} e^{(λ - λ_{i}) t} α e^{tS} S^{0} dt \\ = \int_{0}^{\infty} e^{- λ_{i} t} α e^{tS} S^{0} dt \\ = α (λ_{i} I - S)^{- 1} S^{0}, \end{matrix}$
where the interchange of the infinite sum and the integral is justified by the monotone convergence theorem (Royden 1988, p. 87), and I is the k × k identity matrix.

Replacing
$Π (1 - δ_{i})$
in
$Ψ_{N_{i}} (0)$
yields
$Ψ_{N_{i}} (0) = \frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[α (λ_{i} I - S)^{- 1} S^{0}]}^{- 1}}$
which in combination with
$p_{N_{i}} (0) = Ψ_{N_{i}} (0)$
and Theorem conduce to the desired result.

To illustrate the application of Corollary , say for example that after some experimentation the production system designer decides to capture the behavior of M as the mixture of two exponential distributions with parameters
$(γ_{1}, γ_{2})$
and
$(μ_{1}, μ_{2})$
, where
$f_{M} (t) = γ_{1} μ_{1} e^{- μ_{1} t} + γ_{2} μ_{2} e^{- μ_{2} t}, t > 0$

$γ_{1} + γ_{2} = 1$
and
$γ_{1} / μ_{1} + γ_{2} / μ_{2} = 1 / μ$
, then we know that (see e.g., Kao 1997)
$α = (γ_{1}, γ_{2})$
and

Hence
$S = (\begin{matrix} - μ_{1} & 0 \\ 0 & - μ_{2} \end{matrix}) .$

$(λ_{i} I - S) = (\begin{matrix} μ_{1} + λ_{i} & 0 \\ 0 & μ_{2} + λ_{i} \end{matrix}),$

$(λ_{i} I - S)^{- 1} = (\begin{matrix} 1 / (μ_{1} + λ_{i}) & 0 \\ 0 & 1 / (μ_{2} + λ_{i}) \end{matrix}),$
and
$S^{0} = (\begin{matrix} μ_{1} \\ μ_{2} \end{matrix}),$
from which we obtain
$α (λ_{i} I - S)^{- 1} S^{0} = \frac{γ_{1} μ_{1}}{μ_{1} + λ_{i}} + \frac{γ_{2} μ_{2}}{μ_{2} + λ_{i}} .$

Corollary would then lead to
$\frac{(1 - λ / μ) δ_{i}}{1 - (1 - δ_{i}) {[γ_{1} μ_{1} / (μ_{1} + λ_{i}) + γ_{2} μ_{2} / (μ_{2} + λ_{i})]}^{- 1}} \geq \frac{π_{i}}{h_{i} + π_{i}} .$

Corollary
For a production system with multiple heterogenous items, when M has a Gamma(μ,k) distribution, stockless operation is optimal, if and only if for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[1 + λ_{i} / (μ k)]}^{k}} \geq \frac{π_{i}}{h_{i} + π_{i}} .$

Proof of Corollary
Combining equations Equations and yields
$Ψ_{N_{i}} (s) = \frac{δ_{i} (1 - λ / μ) (1 - s)}{1 - [1 - δ_{i} (1 - s)] {[1 + (λ_{i} (1 - s) / μ k)]}^{k}} .$

The basic property
$p_{N_{i}} (0) = Ψ_{N_{i}} (0)$
and Equation lead to
$p_{N_{i}} (0) = \frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[1 + λ_{i} / (μ k)]}^{k}} .$

Equation and Theorem produce the inequality in .

Inspection of the expression in shows the condition for the optimality of stockless production is clearly dependent on both design variables: manufacturing speed (μ) and manufacturing consistency (k). In addition, the managerial implication of Corollary confirms our intuition: given the terms (1 − λ/μ) > 0 and
$- [1 + λ_{i} / (μ k {)]}^{k} < 0$
, it is easy to see that as manufacturing speed or manufacturing consistency or both increase, so does the suitability of stockless production. This implication is the result, as expected, of the probability of zero orders outstanding being an increasing function of manufacturing speed and manufacturing consistency. We conjecture that, in addition to the Gamma(μ,k) distribution, this result should also hold for all other reasonable probability distributions of unit manufacturing times.

In the remainder of this section we will establish two propositions that should be of value to production system designers. Let
$θ = \max_{i = 1, \dots, n} (π_{i} / h_{i}) .$

Proposition
Consider a production system with multiple heterogenous items, where M has a Gamma(μ,k) distribution. Let
$\underline{μ} = λ (1 + θ),$
and let
$\bar{μ}$
denote the maximum value of μ such that for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) e^{λ_{i} / μ}} \leq \frac{π_{i}}{h_{i} + π_{i}} .$

If the production system is designed to operate at a manufacturing speed μ greater than μ , then it is optimal for the production system to be stockless, regardless of manufacturing consistency k.

If the production system is designed to operate at a manufacturing speed μ lower than
$\bar{μ}$
, then it is not optimal for the production system to be stockless, regardless of manufacturing consistency k.

Proof of Proposition
We know from Theorem that it is optimal for the production system to be stockless, if and only if for each item i
$p_{N_{i}} (0) \geq \frac{π_{i}}{h_{i} + π_{i}} .$

From basic principles we have that for any item i
$p_{N_{i}} (0) \geq 1 - λ / μ .$

Thus if for each item i
$1 - λ / μ \geq \frac{π_{i}}{h_{i} + π_{i}},$
then it must be optimal for the production system to be stockless. Part (i) immediately follows after some algebra, the definition of θ, and the observation that condition is independent of the manufacturing consistency k.

To prove part (ii), we know from Corollary that for the production system under study, it is not optimal to be stockless if and only if for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[1 + λ_{i} / (μ k)]}^{k}} \leq \frac{π_{i}}{h_{i} + π_{i}} .$

But since
$lim_{k \to \infty} {(1 + \frac{λ_{i}}{μ k})}^{k} = e^{λ_{i} / μ},$
a moment's reflection reveals that if for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) e^{λ_{i} / μ}} \leq \frac{π_{i}}{h_{i} + π_{i}},$
then it will not be optimal for the production system to be stockless, regardless of manufacturing consistency k.

Proposition tells us that when the manufacturing speed is “fast enough,” that is, μ exceeds the threshold μ defined by expression , then stockless production is optimal regardless of the level of manufacturing consistency that can be achieved. At the same time, Proposition tells us that when the manufacturing speed is “too slow,” that is, μ falls below the threshold
$\bar{μ}$
defined by expression , then stockless production would not be optimal even if unit manufacturing times were constant.

Of special interest is the fact that μ is completely defined by the total demand rate λ (and not the specific value of n nor the individual demand rates (
$λ_{i}$
's)) and the item with the highest
$π_{i} / h_{i}$
ratio. To illustrate, consider for example Company B which has a production system with two heterogeneous items:

$λ_{1} = 20, h_{1} = 100, π_{1} = 400$
.

$λ_{2} = 30, h_{1} = 200, π_{1} = 600$
.

Using Proposition we obtain that μ = 250 and
$\bar{μ} = 132$
(where for simplification we assume only integer values of μ are possible). Thus regardless of the level of manufacturing consistency achievable, stockless production is always optimal when μ is ≥ 250, and at the same time never optimal when μ is ≤132.

Although for the above example Proposition provides guidance when the manufacturing speed is ≥ 250 or
$\leq 132$
, it does not address whether stockless production is optimal when the manufacturing speed falls between 132 and 250 or more generally between
$\bar{μ}$
and μ . The following proposition addresses this lacuna.

Proposition
If a production system with multiple heterogeneous items is designed to operate at a manufacturing speed μ in the range
$\bar{μ} < μ < \underline{μ}$
, then for a stockless production system to be optimal, a manufacturing consistency of at least
$k^{}$
is required, where
$k^{}$
is the minimum value of k such that for each item i
${(1 + \frac{λ_{i}}{μ k})}^{k} \geq (\frac{1}{1 - δ_{i}}) [1 - δ_{i} (1 - λ / μ) (\frac{h_{i} + π_{i}}{π_{i}})] .$

Proof of Proposition
We know from Corollary that it is optimal for the production system to be stockless, if and only if for each item i
$\frac{δ_{i} (1 - λ / μ)}{1 - (1 - δ_{i}) {[1 + λ_{i} / (μ k)]}^{k}} \geq \frac{π_{i}}{h_{i} + π_{i}} .$

We also know from Equation that the LHS of inequality is always positive. In turn,
$δ_{i} = λ_{i} / λ > 0$
, and as a consequence of the basic queueing theory principle λ/μ < 1, we have that (1 − λ/μ) > 0. Hence the numerator of inequality is also always positive. This implies that the denominator of inequality will always be positive as well. Expression can now be derived from expression via simple algebraic manipulations.

Returning to our example of Company B, Proposition tells us that for a production system with for instance a manufacturing speed μ = 140, where
$\bar{μ} = 132 < μ = 140 < \underline{μ} = 250$
, stockless production is optimal when a manufacturing consistency of at least
$k^{} = 1.00$
is achieved.

Production system designers can utilize Propositions and to generate “design tables” that list
$(μ, k^{})$
pairs for which stockless operation is optimal as well as technologically feasible. For Company B, we already know that system designers only need to consider designs with μ values in the range 133 ≤ μ ≤ 250 (where again for simplification we assume only integer values of μ are possible). For brevity, let us assume that after some analysis, system designers have narrowed their choices to the six designs in Table below. Note that Design 1 represents a “slow” but “consistent” system, Design 5 represents a “fast” but “inconsistent” system, and Design 6 represents a system that is “so fast” that manufacturing consistency does not matter.

Table 1
Design Table for Company B

Design μ $k^{}$

1 133 27.57

2 135 3.65

3 140 1.00

4 150 0.31

5 180 0.03

6 250 —

In order to determine the production system design to adopt, system designers would factor in the capital and operating expenditures associated with each option in the design table. This is precisely what system designers did in our consulting work with minor variations depending on project idiosyncracies.
3.2 Homogeneous Items

Design	μ	$k^{*}$
1	133	27.57
2	135	3.65
3	140	1.00
4	150	0.31
5	180	0.03
6	250	—

In this subsection we consider manufacturing systems that produce multiple homogeneous items. An example of such a system can be found in a firm that engages in the production of both pink and blue baby shoes. In such a setting it is not unreasonable to expect that the cost and demand parameters for both colors would be almost identical.

To facilitate a comparison between heterogeneous and homogeneous multi‐item production systems, in this subsection the subindex i = 1 will be used to denote each and every one of the n homogeneous items. In the following corollary, we establish the role that the variables manufacturing speed and manufacturing consistency play in the design of multi‐item production systems with homogeneous items.

Corollary
For a production system with multiple homogeneous items, when M has a Gamma(μ,k) distribution, stockless operation is optimal, if and only if
$\frac{1 / n - λ_{1} / μ}{1 - [(n - 1) / n] {[1 + λ_{1} / (μ k)]}^{k}} \geq \frac{π_{1}}{h_{1} + π_{1}} .$
26a

Moreover,
$p_{N_{1}} (0) = \frac{1 / n - λ_{1} / μ}{1 - [(n - 1) / n] {[1 + λ_{1} / (μ k)]}^{k}},$
26b

where
$p_{N_{1}} (0)$
is the probability for each of the n homogeneous items, that there are no outstanding production orders.

Proof of Corollary
Observe that when items are homogeneous,
$λ_{i} = λ_{1}$
,
$λ = n λ_{1}$
, and
$δ_{i} = δ_{1} = 1 / n$
. Appropriately substituting these equalities in expressions and leads to expressions and .

Two managerial implications arise from Corollary that seemingly run counter to the principle of risk pooling. We have that as demands for items are combined or pooled, the likelihood of stockless production being optimal is reduced. In addition, whether or not demand pooling actually affects the optimality of stockless production depends on several factors. These two managerial implications are illustrated next via an example and then formalized in a proposition.

Consider Company C, which is designing a production system with homogeneous items that come in two basic colors (blue and red), and in turn each color comes in four modalities (1, 2, 3, and 4). Thus we have a total of eight homogenous items, where for instance, B1 denotes blue in modality 1 and R4 denotes red in modality 4. The parameters of these eight homogenous items are as follows:
$λ_{ij}$
= 20,
$h_{ij}$
= 150, and
$π_{ij}$
= 300, where i = B,R and j = 1,2,3,4. If the production system is designed to operate at manufacturing speed μ = 240 and manufacturing consistency k = 2, then the LHS and RHS in Equation are equal to 0.824 and 0.667, respectively. Hence stockless production is optimal.

If Company C then decides, for each color, to pool the demands of modalities 1 and 2 into modality 5, and pool the demands of modalities 3 and 4 into modality 6, we have the following parameters for the resulting four homogeneous items:
$λ_{ij}$
= 40,
$h_{ij}$
= 150, and
$π_{ij}$
= 300, where i = B,R and j = 5,6. The LHS in Equation is now equal to 0.696, the RHS remains equal to 0.667, and consequently stockless production is still optimal after the pooling. However, with the decrease of the LHS in Equation from 0.824 to 0.696, the likelihood that stockless production is optimal has been reduced.

Finally, if Company C decides for each color to pool the demands of modalities 5 and 6 into a single modality 7, the resulting parameter values of the two homogeneous items are
$λ_{i 7} = 80$
,
$h_{i 7} = 150$
, and
$π_{i 7} = 300$
, where i = B,R. As the LHS in Equation is now only 0.522, this second round of pooling makes stockless production no longer optimal.

In short, sometimes pooling will preserve the optimality of stockless production and sometimes it will not. Put another way, we now know that while pooling will always be desirable for companies where it is optimal to have inventories, the same cannot be said for firms that want to achieve stockless production. This reality was discovered in the consulting project with the oil and gas company mentioned in section 1 of this paper.

The dynamic between pooling and stockless production, illustrated above, is captured in the following proposition.

Let
$n_{BP}$
and
$n_{AP}$
denote the number of items before and after pooling, respectively.

Proposition
Consider a production system with multiple homogeneous items, where M has a Gamma(μ,k) distribution. Define
$\hat{n}$
as the minimum value of n such that the inequality in is satisfied.
Risk pooling always works against the optimality of stockless operation.

If
$\hat{n} > n_{BP}$
, then it is not optimal to have a stockless operation before risk pooling and it will never be optimal after risk pooling.

If
$1 - \frac{λ}{μ} \geq \frac{π_{1}}{h_{1} + π_{1}},$
then it will always be optimal to have a stockless operation regardless of the amount of risk pooling done.

Otherwise, risk pooling will affect the optimality of stockless production if and only if
$n_{AP} < \hat{n}$
.

Proof of Proposition
We will first prove the result in (i). In Online Appendix S3 we demonstrate that the LHS of inequality is an increasing function of n for n ≥ 1. At the same time, risk pooling induces a reduction of n, which decreases the LHS of inequality . Given that risk pooling decreases the LHS of inequality but leaves unchanged the RHS of inequality , means that risk pooling always works against the optimality of stockless operation.

Combining the definition of
$\hat{n}$
and Corollary with the property that the LHS of inequality is an increasing function of n for n ≥ 1, leads to the result in (ii).

In turn, substituting n = 1 in the LHS of inequality yields 1 − λ/μ. Therefore, because the LHS of inequality is an increasing function of n for n ≥ 1, we have that
$\frac{1 / n - λ_{1} / μ}{1 - [(n - 1) / n] {[1 + λ_{1} / (μ k)]}^{k}} \geq 1 - \frac{λ}{μ} .$

Hence if
$1 - \frac{λ}{μ} \geq \frac{π_{1}}{h_{1} + π_{1}},$
we arrive at
$\frac{1 / n - λ_{1} / μ}{1 - [(n - 1) / n] {[1 + λ_{1} / (μ k)]}^{k}} \geq 1 - \frac{λ}{μ} \geq \frac{π_{1}}{h_{1} + π_{1}},$
which implies that
$\hat{n} = 1$
and thus stockless operation will be optimal for any value of n ≥ 1. This establishes the result in (iii).

If
$\hat{n} \leq n_{BP}$
, then stockless operation is optimal for
$n_{BP}$
items. At the same time, if
$1 - \frac{λ}{μ} < \frac{π_{1}}{h_{1} + π_{1}},$
then from inequality and the fact that the LHS of inequality (26) is an increasing function of n for n ≥ 1, we know that
$\hat{n} > 1$
. Thus based on the definition of
$\hat{n}$
, we conclude that the optimality of stockless production is affected if and only if
$n_{AP} < \hat{n}$
, which corresponds to the result in (iv).

The predictive power of Proposition can be seen by revisiting the Company C example. From the inequality in a we determine
$\hat{n} = 3$
. Because
$\hat{n} = 3 n_{BRP} = 8$
, we know (ii) does not apply. Next, finding
$1 - λ / μ = 1 - 160 / 240 = 0.333 π_{1} / (h_{1} + {pi}_{1}) = 300 / (150 + 300) = 0.667,$
we know (iii) does not apply. This leaves (iv), from which we know that pooling will affect the optimality of stockless production if and only if
$n_{AP} < \hat{n} = 3$
. In the first round of pooling undertaken
$n_{BP} = 8$
and
$n_{AP} = 4$
; as
$n_{AP} = 4 \hat{n} = 3$
pooling should have no effect on the optimality of stockless operation and we found there was none. However, in the second round of pooling,
$n_{BP}$
= 4 and
$n_{AP}$
= 2; as
$n_{AP} = 2 < \hat{n} = 3$
the optimality of stockless operation should be affected and we found that was the case.

The intuition behind the impact of risk pooling on the optimality of stockless production, that emanates from Corollary and is captured in Proposition , is as follows:
We know from Corollary that the optimality of stockless operation depends on the relationship between
$p_{N_{1}} (0)$
and
$π_{1} / (h_{1} + π_{1})$
, specifically on
$p_{N_{1}} (0)$
being at least equal to
$π_{1} / (h_{1} + π_{1})$
.

Inspection of
$Ψ_{N} (\cdot)$
in Online Appendix S2 reveals that
$Ψ_{N} (\cdot)$
is a function of μ, k, and λ. However, since
$λ = \sum_{i = 1}^{n} λ_{i}$
, as demand is pooled, while the individual values of
$λ_{i}$
, i = 1,…,n will change, λ will not change (μ and k are unaffected by risk pooling as well). Consequently,
$Ψ_{N} (\cdot)$
and hence the probability distribution of N will stay the same.

Recall from Online Appendix S1 that
$δ_{i}$
is the probability that an outstanding production order is from item i and that
$δ_{i} = λ_{i} / λ$
. In turn, we know from the proof of Corollary that for homogeneous items
$δ_{i} = δ_{1} = 1 / n$
.

Note that for homogeneous items, as demand is pooled, n decreases and thus
$δ_{1}$
increases. This means that due to the symmetry of homogeneous items, as demand is pooled, the probability that an outstanding production order is from one of the pooled items will uniformly increase for all the pooled items.

Given that as demand is pooled,
$Ψ_{N} (\cdot)$
stays the same,
$δ_{1}$
increases, and the probability that an outstanding production order is from one of the pooled items will uniformly increase for all the pooled items, then
$p_{N_{1}} (0)$
will have to decrease for each of the pooled items.

Because risk pooling reduces
$p_{N_{1}} (0)$
but does not affect
$π_{1} / (h_{1} + π_{1})$
, the implication from point 1 above is that risk pooling always works against the optimality of stockless operation, which is the result in (i).

The result in (ii) says that if it is not optimal for the initial homogeneous items to be stockless, then risk pooling will not help because it always works against the optimality of stockless operation.

The result in (iii) says that if it is optimal to have a stockless operation in the extreme case where the initial homogeneous items would be pooled into a single item, then any lesser amount of risk pooling would not affect the optimality of stockless operation.

The result in (iv) says that if it is optimal for the initial homogeneous items to be stockless, and it is not optimal to be stockless in the extreme case where the initial homogeneous items would be pooled into a single item, then there exists a “threshold” amount
$\hat{n}$
of risk pooling [below/above] which risk pooling [will/will not] affect the optimality of stockless operation.

4. Single‐Item Production Systems

Having investigated production systems with multiple items, we now turn our attention to production systems with a single item. Our motive is two‐fold. First, as we mentioned earlier, in the consulting project with multiple plants, one of the stockless production systems being designed at one plant involved only a single item. Second, as we shall demonstrate, stockless single‐item production systems permit results that are parsimonious, and at the same time, more general than the results in Theorem . Since there is only one item, the subscript i will be omitted.

Corollary
Consider a production system with a single item. For any distribution of unit manufacturing time M and any distribution of demand D, stockless operation is optimal, if and only if
$μ \geq λ (1 + π / h) .$

Proof of Corollary
Using similar arguments to those in the proof of Theorem , but where now M and D are generally distributed, we know that it is optimal for the single item to be stockless, if and only if
$K (S = 0) \leq K (S = 1),$
where
$K (S = 0) = π \sum_{x = 1}^{\infty} x p_{N} (x)$
and
$K (S = 1) = (h + π) p_{N} (0) + π \sum_{x = 1}^{\infty} x p_{N} (x) - π .$

We also know that when M and D are generally distributed, the processing of orders at the manufacturing facility corresponds to a G/G/1 queueing system. Recalling that for a G/G/1 queueing system
$p_{N} (0) = 1 - λ / μ$
, combined with expression (29) and Equations and leads to expression (28), which completes the proof.

Several research findings and practical implications follow from the result in Corollary . First, while for multiple‐item systems the expressions in , , and eluded an interpretation, an understanding of the conditions for a stockless design to be optimal in a production system with one item is within reach: The manufacturing rate must exceed the demand rate by an amount equal to λπ/h. In turn, this amount is directly related to the demand rate (λ) and to the ratio of the demand back ordering cost rate to the inventory holding cost rate (π/h). The values π and h are inconsequential; it is the ratio π/h that matters.

Our second research finding concerns the already noted trade‐off faced by production system designers: manufacturing speed vs. manufacturing consistency. Corollary tells us that the optimality of stockless operation for a production system with a single item is independent of k and thus of manufacturing consistency. The practical implication of this research finding is surprising: manufacturing speed matters and manufacturing consistency does not. However, it is more surprising that this finding is entirely in agreement with our findings in the multi‐item case, where both manufacturing speed and manufacturing consistency do play a role. Recall that in a multi‐item production system the value of k is relevant when
$\bar{μ}$
< μ < μ , and irrelevant when μ ≤
$\bar{μ}$
or μ ≥ μ . Since by substituting n = 1 in Proposition one obtains that
$\bar{μ}$
= μ , it follows that in a single‐item production system, k will always be irrelevant because when
$\bar{μ}$
= μ , no μ can satisfy the conditions
$\bar{μ}$
< μ < μ , and at the same time, any μ will trivially meet the conditions μ ≤
$\bar{μ}$
or μ ≥ μ .

Our third research finding, one already mentioned in our statement of Corollary , is that the optimality of stockless operation in a single‐item production system is independent of the distributions of demand and unit manufacturing time. This finding in combination with the second one above yields the following general principle: In cases where a designer does not know the distribution of demand and unit manufacturing time, the designer only needs to focus on the variable manufacturing speed when working on the design of a stockless production system with one item.
5. Discussion and Application of Results

In this section we detail how the research results contained in this study were applied to an oil and gas company project. Unconventional oil is petroleum extracted using techniques other than the conventional (oil well) method. Oil industries and governments across the globe are investing in unconventional oil sources due to the increasing scarcity of conventional oil reserves. The Research & Development Division of Company Alpha has developed over several years two technologies for unconventional oil. We will refer to these technologies as technology Y and technology Z. The resulting product from each technology is a sophisticated, heavy, bulky, and expensive apparatus that would be deployed in the ground, by a so‐called deployment crew.

For brevity, the resulting products of each technology will simply be called item Y and item Z, respectively. With the exception of a vital feature involving intellectual property which cannot be described in this article, items Y and Z are identical. Even though it is an oversimplification, the reader may think of two almost identical automobiles, where in one of them the steering wheel is on the right and in the other one the steering wheel is on the left.

Company Alpha is interested in taking technologies Y and Z from Research & Development to large‐scale production. For this purpose, the firm launched a multi‐year project to design a supply chain that would make large‐scale production economically feasible. One of the strategic decisions that Company Alpha faced in the supply‐chain project was whether to take one of the two technologies to large‐scale production, from now called Strategy 1, or take both technologies to large‐scale production, from now on called Strategy 2. Under Strategy 2, the production and deployment of items Y and Z would be divided equally due to their symmetry.

Company Alpha started the supply‐chain design project by focusing on the last two supply chain stages: Final assembly and deployment. The tactical decisions for these two stages were the average deployment rate and the average final assembly rate that would make stockless operation optimal. Several final assembly plant designs were being proposed, but in all of them items would be assembled one at a time. Additionally, the average deployment rate would be the result of the average deployment time per unit in combination with the number of deployment crews. For example, if the average deployment time per unit is 12 hours and there are five deployment crews, then the resulting average deployment rate would be 10 units per day.

Let

T_{d}

denote the random deployment time of one unit by one deployment crew (in hours),

E (T_{d})

denote the expected value of

T_{d}

, N denote the number of deployment crews, D denote the deployment rate, and λ denote the average deployment rate. We know that

D = 24 N / T_{d}

and

λ = 24 N / E (T_{d}) .

The first tactical decision was then to select the value of λ, which follows from a selection of N once

E (T_{d})

is established. Sampling information on

E (T_{d})

and targets for λ suggested that N would be a very large number, whose value is omitted for confidentiality reasons. Let

λ_{Y}

and

λ_{Z}

denote the average deployment rate of items Y and Z, respectively. Recall that under Strategy 2, due to their equal production and deployment split,

λ_{Y} = λ_{Z} = λ / 2

Let

T_{a}

be the random final assembly time (in hours) of one unit,

E (T_{a})

be the expected value of

T_{a}

, A be the final assembly rate, and μ be the average final assembly rate. It follows that

A = 24 / T_{a}

and

μ = 24 / E (T_{a}) .

The second tactical decision was choosing the value of

E (T_{a})

, or equivalently, the value of μ. In turn, choosing the value of μ depended on the final assembly process design.

Further analysis required the specification of the probability distributions of the deployment rate D and the final assembly time per unit

T_{a}

. Unfortunately, these distributions were unknown due to two reasons. First, the probability distribution of D depends on the final design of items Y and Z, and those designs were not finalized. Second, the probability distribution of

T_{a}

is a function of the final assembly plant design, which was also not finalized.

Due to these circumstances, we first decided to model the deployment of one unit by one deployment crew as a point process. Point processes are one of the most general types of stochastic processes (Cox and Isham 1980). Subsequently, because D is equivalent to the superposition of the deployment processes of the N deployment crews, and N was estimated to be a very large number, based on the behavior of the superposition of a large number of points processes (Çinlar 1968), we proposed to model D as a Poisson process with parameter λ. Regarding

T_{a}

, Company Alpha supply‐chain designers ultimately opted to model

T_{a}

as having a gamma distribution for the reasons listed earlier.

Due to their symmetry, the unit production cost of items Y and Z, denoted by c, was the same. The value of c depended on the final design of items Y and Z and the market prices of several raw materials and metals including carbon steel, stainless steel, and copper. Letting h be the per unit inventory holding cost per period and r the per inventory holding cost per period expressed as a percentage, then h = rc. In addition, the cost of having a deployment crew waiting for an item to deploy was π per period, where the value of π was calculated from various oil prices and other economic variables.

Uncertainty about parameter values such as

E (T_{d})

, N,

E (T_{a})

, c, r, and π, led Company Alpha to adopt a scenario‐driven approach to assess Strategies 1 and 2. In the study conducted to assess Strategies 1 and 2, a scenario was defined as a particular combination of

E (T_{d})

, N, h, and π values. As we detail next, the results in this study guided Company Alpha in the design of the final assembly process under Strategies 1 and 2.

For Strategy 1, Corollary was used to determine the minimum value of μ that the final assembly process should be designed to have for stockless operation to be optimal. We also knew from Corollary that any variability in the assembly times resulting from a design of the final assembly process could be safely ignored.

For Strategy 2, based on Proposition we knew that the final assembly process should be designed to have a value of

μ > \bar{μ}

for stockless operation to be optimal. Moreover, if the final assembly process was designed to have a value of μ > μ , then as in Strategy 1 any variability of the assembly times resulting from the design could be safely ignored. Lastly, if the final assembly process was designed to have a value of μ in the range (

\bar{μ}

, μ ), a manufacturing consistency in assembly times of at least

k^{*}

needed to be achieved, where

k^{*}

is specified in Proposition .

Proposition was used to explain why a final assembly process design with its associated μ and k values that made a stockless operation optimal under Strategy 2, may or may not be optimal under Strategy 1.

6. Conclusion

We studied a multi‐item manufacturing setting with demand‐ and production‐related uncertainty, where the problem is to design a production system so that stockless operation will be optimal. We focused on gaining an understanding of the effect of two variables that are commonly encountered in the design of production systems: manufacturing speed and manufacturing consistency. We derived conditions on these variables that establish when stockless operation is optimal. These conditions can be used by production system designers along with information on equipment costs, information costs, and execution costs to arrive at an optimal production system design. We illustrated the application of some of our results in a real‐world project at an oil and gas company.

Our work also yielded several important research findings and managerial implications. First, for stockless production systems with multiple items, both design variables play a role, while for single‐item stockless production systems manufacturing speed matters but manufacturing consistency does not. A second research finding is that even though the optimal design of a multi‐item stockless production system depends on the distributional form of demand and unit manufacturing times, this is not the case for a single‐item stockless production system. A third finding pertains to stockless production systems with heterogeneous items: The optimality conditions for the design variables are only a function of the total demand; in other words, the optimality of one item being or not being stockless is independent of the number of other items in the production system. On the other hand with homogeneous items, a fourth research finding is that the likelihood of stockless production being optimal declines as the demands of different products are pooled, that is, there is essentially a reverse risk pooling effect. Demand pooling could also invalidate the optimality of a stockless production system.

Several avenues may be pursued to extend the preliminary results of the study. Although modeling demand as a Poisson process served our exploratory needs well, it would be of theoretical and practical interest to obtain results for multi‐item production systems when the Poisson process is generalized to a renewal process. Consideration of more complex production systems with multiple manufacturing stages should be worth investigating too. Additional demand back ordering cost structures may be of interest as well.