Keywords

1 Introduction

In this chapter, we develop extension of the classical newsvendor model. We model a single, unreliable machine that repetitively produces a set of parts in batches subject to shortage and overage (inventory-holding) costs. Our model makes the following assumptions. First, we assume that there is only a single demand point for all parts, and that it occurs at the end of a finite production horizon. Second, the demand for each part is a random variable with a known distribution, where the uncertainty in the demand quantity is not resolved until the demand point. Lastly, we assume a fixed production sequence. Under these assumptions, our model determines an optimal production quantity for each part. The development of this model is motivated by applications at metal stamping plants (Kletter 1994). This model could be used as part of a manufacturing control system, embedded in a software tool that would receive data in real time from the shop floor and assist plant management in decision making.

This chapter is structured as follows. Section 2 presents a review of the literature. A model is then formulated in Sect. 3 as an extension of a classical newsvendor problem. In Sect. 4, we derive properties of the objective function that are exploited to develop a solution algorithm, presented in Sect. 5 and that takes advantage of the special structure of the model. In Sect. 6, we show numerical results from exercise of the model. Finally, in Sect. 7, three extensions to the model are presented, including the incorporation of options to run overtime.

2 Literature Review

We briefly review the literature that is related to our model. We will divide our literature review into two parts: those that model the problem of planning production quantities on an unreliable machine, and those that use a newsvendor model for problems closely related to the one we study here.

2.1 Unreliable Machine

The presence of machine unreliability in a manufacturing system has been studied in a variety of different contexts, including problems of sequencing, scheduling, and lot sizing. We briefly review each of these areas.

We first discuss sequencing of jobs on an unreliable machine. The earliest work is that of Glazebrook (1984) who models the problem as a rather general cost-discounted Markov decision process. He shows the conditions under which the optimal policy is of an index type (i.e., the job to be processed is the one with the smallest Gittins index; see Gittins 1979). Pinedo and Rammouz (1988) find the optimal nonpreemptive policies for several objective functions in the case of a Poisson failure process. For a general failure process and a discrete time model, Birge and Glazebrook (1988) find bounds on the error of following the strategy that is optimal when the failure process is memoryless. Birge et al. (1990) study in greater detail the problem of minimizing weighted flow-time and obtain results that are consistent with and complementary to Pinedo and Rammouz. Epstein et al. (2010) analyze optimal sequencing on an unreliable machine where the machine may slow or stop completely. For a detailed and current overview of this research area, see the surveys contained in Lee (2004), Pinedo (2008), Diedrich et al. (2009), and Racke (2009).

There is also a significant body of work on lot sizing on an unreliable machine. Yano and Lee (1995) provide a broad review of the literature in this area. See Schmidt (2000) for an overview of the scheduling literature in cases where the machine is continuously available for processing, for example, when there is incomplete information about when the machine may change availability. Al-Salamah and Abudari (2011) model a production process with failures: however, failures don’t result in the stoppage of the machine but instead produces nonconforming items. El-Ferik (2008) determines optimal production quantities under an assumption that the production facility is subject to random failure, where preventative maintenance schedules need to be balanced with production. Halim et al. (2010) show the effect on optimal lot sizing when an unreliable machine is subject to fuzzy demand and repair time. Giri et al. (2005) model a two-stage production system where the upstream stage is subject to failures but the downstream stage is not. They assume that after a machine failure, production of the affected lot is not resumed.

In the case of an infinite planning horizon, Giri and Dohi (2004) derive optimal lot sizes under a net present value (NPV) approach.

Of particular note is the work of Groenevelt et al. (1992a, 1992b), who extend the basic economic manufacturing quantity (EMQ) model to incorporate the effects of machine breakdowns. The first paper assumes that repairs are instantaneous but bear a fixed cost. The second paper assumes (as we do) that repairs are not instantaneous but instead consume machine time. This model permits any repair time distribution, but assumes that the time between failures is exponentially distributed. Under the assumption of lost sales, the authors seek an optimal lot size and safety stock level to minimize cost subject to a constraint on the service level. They require, however, some awkward assumptions regarding safety stock to achieve separability in the optimization of the lot sizes and safety stock level. The authors do not explore the impacts of multiple parts sharing the same machine.

Other authors, such as Sethi and Zhang (1994) have approached similar problems from a control theoretic perspective. These authors consider the problem of finding an optimal setup schedule (a sequence of parts and the times at which the changeovers will occur) for an unreliable machine. They show that in the limit (as the length of the horizon tends to infinity), the stochastic problem can be reduced to a deterministic problem, and show how to obtain the optimal control policy. The authors also cite many other similar works.

Reiman and Wein (1998) study a two customer class, single server system with setups. The authors use heavy traffic diffusion approximations to analyze a system with a renewal arrival process, general service times, and either setup costs or setup times. They solve a control problem to minimize a linear function of the queue length plus setup costs, if any. Within these heavy traffic diffusion approximations, one could model the unreliability of the machine within the service time distribution.

2.2 Related Newsvendor Models

The classic “newsvendor” model has been the subject of many extensions that are similar to those we consider here. For many decades, researchers have considered models with multiple items (Evans 1967, Smith et al. 1980). Rose (1992) considers uncertain replenishment, but assumes demand is deterministic. Others have studied multiple time periods for production (Bitran et al. 1986; Matsuo 1990; Ciarallo et al. 1994). Jain and Silver (1995) model uncertainty in supply and permit the option to reserve reliable capacity for a premium charge. Dada et al. (2007) consider a newsvendor that purchases a single item provided by multiple suppliers, some of which are unreliable, and develop a model for optimal supplier selection. Huggins and Olsen (2010) extend the basic newsvendor problem for a single item to permit expediting for unmet demand.

3 Formulation

The mathematical structure of our model will closely parallel that of the classic newsboy model, which we now briefly describe. The following table (Table 12.1) lists the notation that we will use throughout this section.

Table 12.1 Notation for formulation
Full size table

For simplicity we will first state the formulation as a single part formulation, dropping the index ifrom our notation. The problem is then to choose an order-up-to quantity yto minimize the expected purchase, holding, and shortage costs. Mathematically, we can state the problem as

$${C}^{{_\ast}}(x) ={ \min }_{ y\geq x}c(y - x) + p \int \limits _{y}^{\infty }(t - y)g(t)\mathrm{d}t + h \int \limits _{0}^{y}(y - t)g(t)\mathrm{d}t.$$
(12.1)

The problem is solved by finding the value of ysuch that ∂C(x) ∕ ∂yis zero. To find this partial derivative, we need to employ Leibnitz’s theorem for differentiation of an integral:

$$\frac{\partial } {\partial y}\int \limits _{p(y)}^{q(y)}f(x,y)\mathrm{d}x = \int \limits _{p(y)}^{q(y)}\frac{\partial f(x,y)} {\partial y} \mathrm{d}x+\frac{\partial q(y)} {\partial y} f(q(y),y)-\frac{\partial p(y)} {\partial y} f(p(y),y)$$
(12.2)

(Beyer 1987). We will use this extensively in our analysis. From this rule, it is easy to see that the optimal solution y  ∗ to the newsboy model occurs at the point where \(G({y}^{{_\ast}}) = (p - c)/(p + h)\), unless this implies y  ∗  < x, in which case it is optimal not to order.

We now extend this basic single part model to our multiple part, unreliable production model, for now ignoring overtime opportunities. The problem is to find the optimal order-up-to levels to minimize the sum of purchasing, holding, and shortage costs over all parts. Let y, x, c, p, h, and g( ⋅) retain the same meanings as above, except now we add a subscript i, for each part i = 1, …, N. We assume without loss of generality that the parts are indexed in the order in which they will be produced. In practice, the order-up-to strategy would be implemented as follows: produce the first part until the inventory level reaches the optimal y 1, then the production switches to the part 2 until its inventory level reaches the optimal y 2, etc.

Let Tdenote the amount of time available for production, and the time available after setups as \({T}_{i} = T - {s}_{1} {-}^{\ldots } - {s}_{i}\). If we are already setup to produce part 1, then we set s 1 = 0. We assume for simplicity that each part is produced at the same rate when the machine is working. We now introduce machine unreliability into the formulation by including the PDF f(t; T). Kletter (1996) provides a variety of formulas for this distribution in the case where the interarrival time of machine failures and repairs are exponentially distributed. However, the results below are independent of the form of this distribution.

We can now write the problem as

$${C}^{{_\ast}}(x) = \min \limits_{{ y}_{1}\geq {x}_{1},\ldots,{y}_{N}\geq {x}_{N}}C(y,x),$$
(12.3)

where

$$C(y,x) = \sum \limits_{i=1}^{N}{C}_{ i}(y,x),$$
(12.4)

and

$$\begin{array}{rcl}{ C}_{i}(y,x)& =& {c}_{i} \int \limits _{{x}_{i}}^{{y}_{i} }(t - {x}_{i})f\left (\sum \limits _{\mathrm{j}=1}^{i-1}{y}_{ j} - {x}_{j} + t - {x}_{i};{T}_{i}\right )\mathrm{d}t + {c}_{i}({y}_{i} - {x}_{i})\bar{F}\left ( \sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right ) \\ \qquad & & +{p}_{i} \int \limits _{{x}_{i}}^{{y}_{i} }\int \limits _{u}^{\infty }(t - u){g}_{ i}(t)\mathrm{d}tf\left (\sum \limits _{j=1}^{i-1}{y}_{ j} - {x}_{j} + u - {x}_{\mathrm{i}};{T}_{i}\right )\mathrm{d}u \\ \qquad & & +{p}_{i}\bar{F}\left (\sum \limits _{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )\int \limits _{{y}_{i}}^{\infty }(t - {y}_{ i}){g}_{i}(t)\mathrm{d}t \\ \qquad & & +{p}_{i}F\left (\sum \limits_{j=1}^{i-1}{y}_{ j} - {x}_{j};{T}_{i}\right )\int \limits _{{x}_{i}}^{\infty }(t - {x}_{ i}){g}_{i}(t)\mathrm{d}t \\ \qquad & & +{h}_{i} \int \limits _{{x}_{i}}^{{y}_{i} }\int \limits _{0}^{u}(u - t){g}_{ i}(t)\mathrm{d}tf\left (\sum \limits_{j=1}^{i-1}{y}_{ j} - {x}_{j} + u - {x}_{i};{T}_{i}\right )\mathrm{d}u \\ \qquad & & +{h}_{i}\bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )\int \limits _{0}^{{y}_{i} }({y}_{i} - t){g}_{i}(t)\mathrm{d}t \\ \qquad & & +{h}_{i}F\left (\sum \limits_{j=1}^{i-1}{y}_{ j} - {x}_{j};{T}_{i}\right )\int \limits _{0}^{{x}_{i} }({x}_{i} - t){g}_{i}(t)\mathrm{d}t, \end{array}$$
(12.5)

where the summations from 1 to i − 1 are taken to be null at i = 1.

Each C i (y, x) represents the expected purchasing, holding, and shortage costs incurred for part igiven a set of order-up-to levels y i . We have written C i (y, x) as the sum of eight terms. The first two terms express the expected purchasing cost, where the first term is the expected purchasing cost if the realized uptime of the machine is such that the available supply of the ith part is between the values of 0 and y i  − x i and the second term is the expected purchasing cost if the realized uptime of the machine is such that the available supply of the ith part is the desired value y i  − x i . There is no purchasing cost if the available supply of the ith part is not greater than zero. The next three terms represent the expected shortage costs. The first of these terms is the expected shortage cost if the available supply is between 0 and y i  − x i , the second term is the expected shortage cost if the available supply is y i  − x i , and the third is the expected shortage cost if the available supply is 0. Similarly, the last three terms represent the expected holding costs, where the first of these terms is the expected holding cost if the available supply of the ith part is between 0 and y i  − x i , the second term is the expected holding cost if the available supply is y i  − x i , and the third is the expected holding cost if the available supply is 0.

4 Properties of the Objective Function

To obtain the optimal order quantities, we wish to show that the total cost function is convex with respect to the order quantities. If this is so, we can find minimizing order quantities by finding where the partial derivative of the total cost function is zero.

4.1 First Order Optimality Condition

We begin by finding the partial derivative of the total cost function with respect to y N . Using Leibnitz’s rule, we obtain

$$\frac{\partial } {\partial {y}_{N}}C(y,x) = ({c}_{N} - {p}_{N}\bar{{G}}_{N}({y}_{N}) + {h}_{N}\bar{{G}}_{N}({y}_{N}))\bar{F}\left (\sum \limits_{j=1}^{N}{y}_{ j} - {x}_{j};{T}_{N}\right ).$$
(12.6)

When written as the product of two terms as we have done, this derivative has a nice interpretation. The first term is the derivative of the cost function for the classical newsboy problem. This term is multiplied by the probability that we can complete our production plan in the time available.

Because of this structure, the first order optimality condition is reduced to \({G}_{N}({y}_{N}) = ({p}_{N} - {c}_{N})/({p}_{N} + {h}_{N})\), the solution to the classical newsboy problem. As before, it is easy to show that if this implies y N  < x N , then the optimal y N is x N . The optimal y N should not be dependent on the other y i , because once we have produced parts 1, …, N − 1, all we can do is try to minimize the costs for part N. The optimal y N should not be dependent on the machine’s reliability, because the best thing to do is attempt to achieve the optimal order-up-to quantity exactly.

We now turn to the more difficult task of taking the partial derivative of the total cost function C(y, x) with respect to y i for i < N. After simplification, the result is

$$\begin{array}{rcl} \frac{\partial } {\partial {y}_{i}}C(y,x)& =& \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ \qquad & & +\sum \limits_{k=i+1}^{N}({p}_{ k} - {c}_{k})\left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ \qquad & & -\sum \limits_{k=i+1}^{N}({p}_{ k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }{G}_{k}(u)f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u,\end{array}$$
(12.7)

where the summations from i + 1 to Nare taken to be null at i = N. This expression is easier to interpret if we rewrite it as

$$\begin{array}{rcl} \frac{\partial } {\partial {y}_{i}}C(y,x)& =& \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ \qquad & & +\sum \limits_{k=i+1}^{N}({p}_{ k} - {c}_{k})\left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ & & \qquad +{ \sum }_{k=i+1}^{N}({p}_{ k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }\bar{{G}}_{k}(u)f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u \\ & & \qquad -\sum \limits_{(k)=i+1}^{N}({p}_{ k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u, \end{array}$$
(12.8)

and then simplify to obtain

$$\begin{array}{rcl} \frac{\partial } {\partial {y}_{i}}C(y,x)& =& \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ \qquad & & -\sum \limits_{k=i+1}^{N}({c}_{ k} + {h}_{k})\left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ \qquad & & +\sum \limits_{k=i+1}^{N}({p}_{ k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }\bar{{G}}_{k}(u)f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u.\end{array}$$
(12.9)

The first term is analogous to ∂C(y, x) ∕ ∂y N discussed above. The second two terms give the impact of the choice of y i on the parts \(k = i + 1,\ldots,N\). The first of these terms represents the marginal cost of machine time. The expression in square brackets is the probability that machine output is insufficient to produce up to y k but sufficient to start production of part k. As this probability increases, total cost decreases at rate c k  + h k , assuming that the units built are not sold. The final term is the marginal cost of lost sales. The integral represents the expected sales given that machine output is greater than zero but less than y k . As this increases, shortage costs are accrued at a rate p k and holding costs, which have already been charged in the second term, are avoided at a rate h k .

It can be seen from the first order condition that as Ttends to infinity, the optimal y i each approach their “newsboy point” y i N, that is, the point where \(G({y}_{i}) = ({p}_{i} - {c}_{i})/({p}_{i} + {h}_{i})\). However, we can prove a stronger result, as stated by the following:

Theorem 1.

The optimal y i are never greater than y i N, their respective newsboy points.

Proof.

We have already shown that the optimal y N is y N N, the newsboy point for part N. Suppose that we have shown that the optimal y k are not greater than y k Nfor \(k = i + 1,\ldots,N\). We will now show that the optimal y i is also less than or equal to y i N. We first require the following.

Lemma 1.

$$\frac{\partial } {\partial {y}_{i}}C(y,x) \geq \bar{ F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i}))$$

Proof.

$$\begin{array}{rcl} \frac{\partial } {\partial {y}_{i}}C(y,x)& =& \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ \qquad & & +\sum \limits_{k=i+1}^{N}({p}_{ k} - {c}_{k})\left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ \qquad & & -\sum \limits_{k=i+1}^{N}({p}_{ k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }{G}_{k}(u)f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u \\ \end{array}$$

(from equation (12.7))

$$\begin{array}{rcl} & & \geq \bar{ F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ & & \quad + \sum \limits_{k=i+1}^{N}({p}_{ k} - {c}_{k})\left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ & & \quad -\sum \limits_{k=i+1}^{N}({p}_{ k} + {h}_{k}){G}_{k}({y}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\mathrm{d}u \\ \end{array}$$

(because G k ( ⋅) is nondecreasing)

$$\begin{array}{rcl} & =& \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ \qquad & & -\sum \limits_{k=i+1}^{N}({c}_{ k} - {p}_{k}\bar{{G}}_{k}({y}_{k}) + {h}_{k}{G}_{k}({y}_{k})) \\ \quad & & \times \left [F\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - F\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ] \\ & \geq & \bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})).\end{array}$$
(12.10)

(because \({G}_{k}({y}_{k}) \leq ({p}_{k} - {c}_{k})/({p}_{k} + {h}_{k})\)for \(k = i + 1,\ldots,N\), by the induction hypothesis). □ 

Using this result, it immediately follows that for any y i  > y i N, ∂C(y, x) ∕ ∂y i is positive. Therefore, if x i  < y i N, the optimal y i lies between x i and y i N. If x i  ≥ y i N, then it is optimal not to produce (the optimal y i equals x i ).

4.2 Convexity of the Total Cost Function

In this section, we prove the following:

Theorem 2.

If x k ≤ y k ≤ y k N for \(k = i,i + 1,\ldots,N\), then ∂ y C(y,x)∕∂y i y is nonnegative.

Proof.

We once again use Leibnitz’s rule to take the second partial derivative with respect to y i to obtain

$$\begin{array}{rcl} \frac{{\partial }^{2}} {\partial {y}_{i}^{2}}C(y,x)& =& \left [\bar{F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({p}_{i}{g}_{i}({y}_{i}) + {h}_{i}{g}_{i}({y}_{i}))\right ] \\ \qquad & & +\left [-f\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i}))\right ] \\ \qquad & & +\left [\sum \limits_{k=i+1}^{N}\left \{({p}_{ k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right )\right.\right. \\ \qquad & & \left.\left.-({p}_{k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }{G}_{k}(u) \frac{\partial } {\partial {y}_{i}}\left \{f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\right \}\mathrm{d}u\right \}\right ]. \\ & & \end{array}$$
(12.11)

We now show that this second partial derivative is nonnegative. We have written the second partial derivative as the sum of three (square bracketed) terms. The first term can be seen to be nonnegative by inspection. The second square bracketed term is nonnegative if \(-{c}_{i} + {p}_{i}\bar{{G}}_{i}({y}_{i}) - {h}_{i}{G}_{i}({y}_{i})\)is nonnegative, which is true if \({G}_{i}({y}_{i}) \leq ({p}_{i} - {c}_{i})/({p}_{i} + {h}_{i})\), which is always true for y i  < y i N. Showing that the third bracketed term is nonnegative is slightly more difficult. We note that for each k,

$$\begin{array}{rcl} & & ({p}_{k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ) \\ & & \qquad - ({p}_{k} + {h}_{k})\int \limits _{{x}_{k}}^{{y}_{k} }{G}_{k}(u) \frac{\partial } {\partial {y}_{i}}\left \{f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\right \}\mathrm{d}u \\ & & \quad \geq ({p}_{k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j}\mathrm{;}{T}_{k}\right )\right ) \\ & & \qquad - ({p}_{k} + {h}_{k}){G}_{k}({y}_{k})\int \limits _{{x}_{k}}^{{y}_{k} } \frac{\partial } {\partial {y}_{i}}\left \{f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\right \}\mathrm{d}u \\ \end{array}$$

(because G k ( ⋅) is nondecreasing)

$$\begin{array}{rcl} & & \geq ({p}_{k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ) \\ & & \quad - ({p}_{k} + {h}_{k}) \frac{({p}_{k} - {c}_{k})} {({p}_{k} + {h}_{k})}\int \limits _{{x}_{k}}^{{y}_{k} } \frac{\partial } {\partial {y}_{i}}\left \{f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j} + u - {x}_{k};{T}_{k}\right )\right \}\mathrm{d}u \\ \end{array}$$

(because \({G}_{k}({y}_{k}) \leq ({p}_{k} - {c}_{k})/({p}_{k} + {h}_{k}))\)

$$\begin{array}{rcl} & & = ({p}_{k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ) \\ & & \quad - ({p}_{k} - {c}_{k})\left (f\left (\sum \limits_{j=1}^{k}{y}_{ j} - {x}_{j};{T}_{k}\right ) - f\left (\sum \limits_{j=1}^{k-1}{y}_{ j} - {x}_{j};{T}_{k}\right )\right ) = 0.\end{array}$$
(12.12)

 □ 

Given the other y j , j≠i, this result allows us to find the optimal y i by determining if ∃ y i  ∈ [x i , y i N] such that \(\partial C(y,x)/\partial {y}_{i} = 0\). If such a y i exists then it is optimal, otherwise, the optimal policy is not to order. Since ∂C(y, x) ∕ ∂y i is a nondecreasing function of y i over the range [x i , y i N] when y k  ≤ y k Nfor \(k = i + 1,\ldots,N\), the optimal y i can be found by simple binary search.

Given the above results, after we have found y N we can find the other y i by solving the above problem as a N − 1 dimensional unconstrained minimization problem on the interval \({x}_{i} \leq {y}_{i} \leq {y}_{i}^{N},i = 1,\ldots,N - 1\). For a discussion of general algorithms to solve such problems, see Bazaraa et al. (1993). Below we present a solution algorithm that exploits the special structure of the model.

5 Solution Algorithm

The difficulty in finding the optimal production quantities is that the first order condition tells us that N − 1 of the y i are mutually dependent. We now describe a solution procedure that exploits the special structure of these dependencies. In particular, consider the difference

$$\begin{array}{rcl} \hat{{C}}_{i+1}& =& \frac{\partial C(y,x)} {\partial {y}_{i+1}} -\frac{\partial C(y,x)} {\partial {y}_{i}} \\ & =& \bar{F}\left (\sum \limits_{j=1}^{i+1}{y}_{ j} - {x}_{j};{T}_{i+1}\right )({c}_{i+1} - {p}_{i+1}\bar{{G}}_{i+1}({y}_{i+1}) + {h}_{i+1}{G}_{i+1}({y}_{i+1})) \\ & & \quad -\bar{ F}\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i}\right )({c}_{i} - {p}_{i}\bar{{G}}_{i}({y}_{i}) + {h}_{i}{G}_{i}({y}_{i})) \\ & & \quad - ({p}_{i+1} - {c}_{i+1})\left [F\left (\sum \limits_{j=1}^{i+1}{y}_{ j} - {x}_{j};{T}_{i+1}\right ) - F\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j};{T}_{i+1}\right )\right ] \\ & & \quad + ({p}_{i+1} + {h}_{i+1})\int \limits _{{x}_{i+1}}^{{y}_{i+1} }{G}_{i+1}(u)f\left (\sum \limits_{j=1}^{i}{y}_{ j} - {x}_{j} + u - {x}_{i+1};{T}_{i+1}\right )\mathrm{d}u. \\ & & \end{array}$$
(12.13)

Note that if y i is optimal, ∂C(y, x) ∕ ∂y i is zero, so that \(\hat{{C}}_{i+1} = C(y,x)/{y}_{i+1}\). The reason that this is significant is because \(\hat{{C}}_{i+1}\)is a function only of y 1, …, y i . Therefore if the optimal y 1is known then \(\hat{{C}}_{2}\)can be used to find the optimal y 2, and then \(\hat{{C}}_{3}\)can be used to find the optimal y 3, and so forth.

Since the optimal y 1is not known, we must use a search technique to find it. We now prove three important properties that will be helpful in this regard.

Let the production quantities that result from the above procedure be denoted by \(\hat{{y}}_{i}\). We first show that \(\hat{{y}}_{N} = {y}_{N}^{N}\)iff \(\partial C(y,x)/\partial {y}_{1} = 0\). Observe that \(\hat{{C}}_{N}\)is exactly equal to \(\partial C(y,x)/\partial {y}_{N} - \partial C(y,x)/\partial {y}_{N-1}\), and thus \(\hat{{y}}_{N} = {y}_{N}^{N}\)iff \(\partial C(y,x)/\partial {y}_{N-1} = 0\). Further, for any i, \(\hat{{C}}_{i+1} = C(y,x)/{y}_{i+1}\)iff \(\partial C(y,x)/\partial {y}_{i} = 0\). Therefore, \(\hat{{y}}_{N} = {y}_{N}^{N}\)iff \(\partial C(y,x)/\partial {y}_{1} = 0\).

The second property is that if the guess for the optimal value of y 1is too large, \(\hat{{y}}_{N} >{y}_{N}^{N}\). We have shown above that if x k  ≤ y k  ≤ y i Nfor \(k = i,i + 1,\ldots,N\), then ∂ 2 C(y, x) ∕ ∂y i 2 ≥ 0. Accordingly, if the guess for the optimal value of y 1is too large, ∂C(y, x) ∕ ∂y 1 > 0, so that in order for \(\hat{{C}}_{2} = 0,\hat{{y}}_{2}\)must be chosen such that ∂C(y, x) ∕ ∂y 2 > 0, so that \(\hat{{y}}_{2}\)will be greater than the optimal y 2. Repeating this argument, we see that each \(\hat{{y}}_{i}\)will be greater than the optimal y i , and thus \(\hat{{y}}_{N} >{y}_{N}^{N}\). By analogous reasoning, we can conclude that if the guess for the optimal value of y 1is too small, \(\hat{{y}}_{N} < {y}_{N}^{N}\).

The third and final property that we wish to show is that \(\hat{{C}}_{i+1}\)is an increasing function of y i + 1. This property is particularly important, as it allows us to find \(\hat{{y}}_{i+1}\)by simple binary search. To prove this, we take the partial derivative of \(\hat{{C}}_{i+1}\)with respect to y i + 1and simplify to obtain

$$\begin{array}{rcl} \frac{\partial } {\partial {y}_{i+1}}\hat{{C}}_{i+1} =\bar{ F}\left (\sum \limits_{j=1}^{i+1}{y}_{ j} - {x}_{j};{T}_{i+1}\right )({p}_{i+1}{g}_{i+1}({y}_{i+1}) + {h}_{i+1}{g}_{i+1}({y}_{i+1}))\!,\ \ \ \ & &\end{array}$$
(12.14)

which is clearly nonnegative since each term is nonnegative, and thus the result is proven.

Using these properties, we are now ready to state the following:

The algorithm essentially performs a binary search on the guess for the optimal y 1by maintaining an upper and lower bound (Uand L) on the optimal value. The algorithm terminates when the current value of \(\hat{{y}}_{N}\)is within some small positive ε of y N N.

Because the properties that we have proven above are valid only if x i  ≤ y i  ≤ y i Nfor i = 1, …, N, we must take care to ensure that this remains true throughout the algorithm. We perform the test in Step 2 to ensure that we do not proceed if any y i  > y i N. We set L = x 1so that \(\hat{{y}}_{1} \geq {x}_{1}\). Lastly, in a preprocessing step we remove a part ifrom consideration if x i  > y i N. We can do this because, for any such part, the optimal \(\hat{{y}}_{i}\)is x i , and it is thus optimal not to produce that part. Since the part would not be produced, it has no effect on the other parts.

6 Numerical Results

In this section, we present numerical results from an implementation of the solution algorithm described in the previous section.

For simplicity we describe a two part (N = 2) system, with identical parameters for the two parts. The base case parameters used are summarized in Table 12.2. We assume that the demand distribution gis normally distributed with mean equal to 100 and standard deviation equal to 10. We also assume that f, the distribution of output of the machine over a horizon of length T, has a mean of Ta standard deviation of 0.lT, which equates to a coefficient of variation of 0.1.

In this two part example, we know that y 2  ∗ will equal its newsvendor point, which in this case is equal to 97.47. To find y 1  ∗ , we developed a simple spreadsheet model that computes \(\hat{{C}}_{i}\). The solution procedure then simply searches over values of y 1until \(\hat{{C}}_{2} = 0\). The solution procedure converges to a value equal to the optimal solution within four digits of precision in just 13 iterations. In this case, the value is y 1  ∗  = 85. 31.

Table 12.2 Base case parameters for experiments
Full size table

Next, we wish to show the effect of the time constraint on the optimal value for y 1. In Fig. 12.1we show the results of varying the value of T, which also impacts our production distribution f. As expected, the value of Thas a major effect on the production schedule, until Tbecomes sufficiently large, at which point y 1  ∗ approaches its newsvendor point.

Fig. 12.1
figure 1

Optimal production quantity of part 1 as a function of T

Full size image

Finally, in Fig. 12.2we show the effect of varying the coefficient of variation in the production schedule. This was set to 0.1 in the base case. We see that increased variability has the effect of causing greater levels of planned production as a hedge against this uncertainty.

Fig. 12.2
figure 2

Optimal production quantity of part 1 as a function of the variability in production

Full size image

7 Extensions to the Model

In this section, we present three simple extensions to the model: (1) a method for rescheduling within the planning horizon, (2) an extension to evaluate whether or not to purchase the option to run overtime within the planning horizon, and (3) an extension that permits the modeling of a machine that operates at a different speed depending on the part being produced.

7.1 Dynamic Rescheduling

In the development above, we have discussed how to determine a set of production quantities to minimize expected total cost. We have assumed up to this point that this plan, once established, is fixed. That is, the plan is implemented by producing the predetermined optimal quantity of each part, and then switching production to the next part. Of course, as this plan is implemented, the reliability of the machine may be much higher or much lower than expected. As a result, if we were given the opportunity to do so, we might adjust the production plan based on the actual realized reliability of the machine.

Suppose we are now permitted to modify our choice of production quantity for the current part. We propose a simple method that allows someone on the shop floor to determine when to stop production of the current part based on the part’s inventory level. We denote this production level as the critical inventory level. The method described below could produce a chart with two axes: the horizontal axis will be time, and the vertical axis will be the critical inventory level. In this sense, this method could produce a visual aide for a production manager on the factory floor.

Suppose for a particular future point in time t1we would like to determine the amount of inventory at or above which it is optimal to stop producing the current part and switch to the next part. We denote this production level as the critical inventory level, and determine it as follows. The first step is to update the horizon length in the equations above by replacing Twith T − t 1. This reflects the amount of time that would be remaining for production at time t 1. Next, search over values for x 1, at each iteration finding the optimal y i , until we identify the lowest x 1such that at the optimum, y 1 = x 1. This is the critical inventory level, since this is the inventory level at which is it optimal not to produce.

We now prove the existence of such a critical inventory level. Recall that we are only interested in the lowest x 1such that at the optimum \({y}_{1} - {x}_{1} = 0\), so the only question we must answer is whether or not such an x 1exists. But this is clearly so, since if we set \({x}_{1} = {G{}_{1}}^{-1}(({p}_{1} - {c}_{1})/({p}_{1} + {h}_{1}))\), we know \({y}_{1} \leq {G{}_{1}}^{-1}(({p}_{1} - {c}_{1})/({p}_{1} + {h}_{1}))\)and since we must constrain y 1to be at least x 1, y 1 = x 1.

We can vary the value of t 1and find the critical inventory levels at each point over the planning horizon. The optimal dynamic operating policy is therefore implemented on the shop floor by producing until the inventory level crosses this curve. Once this happens and production is switched to the next part, the model should be solved again to find the critical inventory level as a function of time for the next part.

7.2 Including Options to Run Overtime

In the development above, we purposely omitted any discussion of how to make optimal overtime decisions. Suppose now that there are m = 1, …, N OTopportunities over the horizon to run overtime, and for simplicity assume that they are each of duration OT at cost w m. Without loss of generality, we will assume that the opportunities are indexed in the order of increasing cost.

In the development above, we computed optimal production quantities ignoring overtime opportunities. This is equivalent to assuming that we choose not to run overtime, and the resulting expected cost is the expected cost of this strategy. Suppose instead that we decide that we are going to run overtime once. To evaluate the expected cost of this strategy we simply replace Tby T + OT and find the optimal production quantities to compute the minimum expected cost, and then add w m. Note that it does not matter where within the planning horizon that we run overtime, since all overtime opportunities occur before the demand point. As a result, we can find the optimal policy by simply running the solution algorithm above N OT + 1 times, with Ttaking on the values \(T,T + \mathrm{OT},T + 2\mathrm{OT},\ldots,T + {N}_{\mathrm{OT}}\)OT, and choosing the strategy with lowest expected cost. Intuitively, increasing the length of the horizon will have a non-increasing benefit. If this is true, which we leave as a conjecture, the evaluation of policies can be stopped when the total cost increases from the previous iteration.

7.3 Extension to Different Machine Speeds

For notational convenience, up to this point we have assumed that the machine operates at the same speed when producing different parts. If the speeds are different, then the requirements on the machine need to be expressed in common units, such as time, instead of parts. This can be accommodated easily, replacing all expressions such as \(F\left ({\sum \nolimits }_{j=1}^{i}{y}_{j} - {x}_{j};{T}_{i}\right )\)with \(F\left ({\sum \nolimits }_{j=1}^{i}\frac{{y}_{j}-{x}_{j}} {{P}_{j}} ;{T}_{i}\right )\), where P j is the speed at which the machine produces part j when it is working. Our solution procedure for finding the optimal y i is unchanged by this modification.

8 Conclusion and Future Research

In this chapter, we have extended the basic newsvendor model to an unreliable machine that must produce multiple parts in a given period of time. We have seen that with an infinite production horizon, the problem simply decomposes into a single item newsvendor problem for each part. However, under a time constraint, the optimal production quantities are reduced from those in the infinite horizon case. We showed that the optimal production quantities are mutually dependent on one another, but have discovered a special structure in these relationships, and also proven other important properties of the objective function and decision variables. These results allowed us to construct a simple algorithm that performs a binary search for the optimal value of the production quantity for the first part. In each iteration, we exploit the special structure of the problem to easily determine the production quantities for the other parts, and easily test if the overall solution is optimal.

Our formulation has assumed that the production sequence is fixed. This could be the result of sequence-dependent setup costs or setup times, or a function of the timing of arrivals of materials from upstream suppliers. A future research topic could include relax this assumption and allow the decision maker to change the sequence, possibly with penalty costs associated with changes. We have also assumed that demand is satisfied for all parts at the end of the horizon. Another future research topic could be an extension to multiple time periods, or allowing different parts to have demand pull from inventory at different points in time.