The Impact of Imperfect Processes on Production Run Times
The Impact of Imperfect Processes on Production Run Times
Tonya Boone School of Business, College of William and Mary Williamsburg, VA 23187-8795. email: [email protected] Web: http://faculty.wm.edu/Tonya.Boone
Ram Ganeshan School of Business, College of William and Mary Williamsburg, VA 23187-8795. email: [email protected] Web: http://business.wm.edu/Ram.Ganeshan
Yuanming Guo Advanta Corporation Spring House, PA 19477. email: [email protected]
J. Keith Ord School of Business, Georgetown University Washington, DC, 20057. email: [email protected]
Abstract This paper investigates the interaction between the economics of production and imperfections in the production process. SpeciÞcally, this paper is the Þrst to devise a model in an attempt to provide managers with guidelines to choose the appropriate production run times to buffer against both the production of defective items and stoppages occurring due to machine breakdowns. In addition to providing several structural properties of the model, we show that a manager will always incur a cost penalty when (s)he uses the results of two oft-cited models — the EM Q (Economic Order/Manufacturing Quantity) and the NR-E (No-Resumption, Exponential machine breakdown) — to determine production run times. KeyW ords: Inventory Models; Logistics; Probability Models
2
1
Introduction
Since its inception in Harris (1913), the EM Q (Economic Manufacturing Quantity or Economic Order Quantity (EOQ)) model has spawned research efforts primarily attempting to relax many of the idealistic assumptions built into it. Some of the more oft-cited extensions since 1970 include the incorporation of variable demand (Ritche and Tsado, 1986; Snyder, 1977); a Þnite production horizon (Schwarz, 1977); learning effects (Ax¨aster and Elmaghraby, 1981; Chand, 1989); deteriorating items (Shah, 1977); setup reduction (Porteus, 1986); and the modeling of imperfect processes (Porteus, 1986; Groenevelt, Pintelon, and Seidmann, 1992a & b). Our primary focus in this paper is to incorporate imperfections in the production process into the classical EM Q model. Recent research analyzing the relationship between production lot sizes or run times and imperfect production processes have largely centered around two key issues: process deterioration and yield; and machine breakdowns and repairs. Porteus (1986) was perhaps the Þrst to model the relationship between production lot size and process deterioration. In his model, the production process can go “outof-control” (i.e., start producing defective items) each time it produces an
3
item. In a related paper, Rosenblatt and Lee (1986), assume that the duration of the “in-control” state is a random variable with an exponential failure time distribution. Both these seminal papers conclude that managers should use a smaller lot size or production run times since these lead to fewer defective items. Several researchers have since built upon these two initial studies. Notable among them are Chand (1989), who validates Porteus’s (1986) model when learning effects are present. In two papers, Cheng (1989, 1991) extends the classic EMQ problem by assuming, Þrst, that the production costs are a function of reliability (proportion of defectives) and demand, and second, that they are a function of reliability and setup costs. Meanwhile, another stream of research has focused on the impact of machine breakdowns and preventive maintenance on production run times. Groenevelt et al. (1992a) show that when machine breakdown times follow an exponential distribution, the optimal lot size will always be greater than one in the classical EM Q model (intuitively this should be true for all breakdown time distributions). They later extend their model (1992b) to consider the problem of simultaneously determining lot size and safety stocks when repair times are signiÞcant. Kun-Jen (1997) extends the Groenevelt et al. (1992a) model to provide bounds on the optimal lot size. With these bounds, a simple algorithm to Þnd the optimal lot size is also developed. Finally Liu
4
and Cao (1999) model a (m, M) production system (setup when inventory level is m, and shutoff when inventory is built up to M ) that is subject to stochastic breakdowns. This model is similar in spirit to the classic (s, S) inventory model, i.e., Þnd the optimal reorder point (s), and order up to level (S) that minimizes inventory costs. Liu and Cao’s objective, however, is to Þnd the right levels of m and M that minimize the production system costs. Although the various studies reviewed above provide considerable insight into how lot size is affected by either deterioration of the process or machine breakdowns, none of them incorporate both of these aspects into a lot-sizing model. Our goal in this paper is to model a scenario where there is both process deterioration and an eventual breakdown of the production process. The rest of the paper is organized as follows. The next section explains our notation and modeling environment. The third section further develops the model and analyzes the model structure in order to gain insight into the model and its solution. The fourth section presents detailed numerical analysis and managerial insights. Finally, in the Þfth section, we summarize the paper and suggest directions for some future research.
5
2
Model Development
Following is the notation we use to develop our model. We defer the speciÞc deÞnitions of some of the variables until they are used. D = demand rate per unit time, h = the holding cost per unit per unit time, K = setup cost for each production run, M = the cost of corrective maintenance, N = random variable representing the number of defective items, R = production rate per unit time, s = cost incurred by producing a defective item, T = random variable denoting time-to-breakdown, U =Uniform distribution, θ = actual production time for each run, θ∗ = optimal production run time, τ = time taken to transition from an in-control to an out-of-control state, f (.) = the probability distribution function (pdf) denoting the transition from an in-control to an out-of-control state, g(x) = the rate at which the production process produces defectives at time x, given that it is in an out-of-control state (in our model, g(x) = α), φ(.) = pdf of T ,
6
Φ(.) = cumulative density function (cdf) of T , NR-E = No-Resumption, Exponential machine breakdown policy, and NR-E/U/α = NR-E policy with uniform failure time and a constant defective rate, α. We model a production system manufacturing a single item on a single machine in batches. The production rate (R) and the demand rate (D) are assumed to be constant and the production process is able to produce items at a rate faster than it is demanded (R > D). The machine requires a setup, costing $K, and is susceptible to stochastic breakdowns, with the time to breakdown described by a random variable T with probability density function φ(.). Additionally, the nature of the breakdown process follows the “No-Resumption, General machine breakdown” (NR-G, Groenevelt et al., 1992a) policy. Under this policy, “...when a break-down takes place, the interrupted lot is aborted and a new one is to be started only when all available inventory is depleted. Maintenance of the production unit is carried out after a failure or a pre-determined amount of time, which ever comes Þrst. Each maintenance action restores the system to the same initial working conditions.” As have researchers in the past, we assume that the setup and the cor7
rective maintenance (costing $M ) do not take up any time. According to Groenevelt et al. (1992a), examples where such an assumption is justiÞed include ßexible manufacturing system environments where setup times are rather small, or in production systems with a modular design where maintenance is fast and easy. In order to realistically incorporate the effects of process deterioration, we modify the NR-G policy the following way: at the beginning of a production lot, the process is “in-control,” and only items of acceptable quality are produced. In those lots where a breakdown occurs, the machine goes “out of control,” or, in other words, starts producing defective items prior to the breakdown. For the purposes of this study, the “out of control” state does not cause the breakdown, rather it precedes total machine failure. From a modeling perspective, we assume that the elapsed time, τ, until the production process shifts from the “in-control” to the “out of control” state is a random variable with pdf f (τ ) and cumulative density function F (τ ) with f (τ ) > 0, 0 < τ ≤ T. For the purposes of exposition, we call f(τ ) the “failure time” distribution (Barlow and Proschan, 1965). Additionally, we assume that the system produces defects at a rate g(x) at any given time x after the shift to the out-of-control state. We assume that g(x) is positive and non-decreasing, that is, g(x) > 0 and g 0 (x) ≥ 0 for x ≥ 0. The premise
8
is that once the system is in the out-of-control state, it produces defective items at a rate that is either a constant or increases with the time the system is in the out-of-control state. If N is the random variable representing the number of defective items produced in a production cycle where a breakdown has occurred, and if θ is the actual production time of the production run, it is easy to see that the expected number of defective items, E(N) is E(N ) = R
Z
θ
0
Z
t
f(τ ) 0
Z
t−τ
g(x)dxdτ φ(t)dt. 0
Our ultimate objective is to Þnd the production run length θ that minimizes the total cost per unit time ET C(θ). SpeciÞcally, the total cost is the sum of setup, maintenance, linear holding costs, and the cost of producing defective items. Since our production process has a renewal epoch at the start of every production run, we apply the renewal reward theorem (Ross 1981) to obtain the average total cost per unit time by taking the ratio of the expected cost per renewal cycle to its duration.
E(cost/cycle) = = =
Z Z
Z
∞
E(cost/cycle|T = t)φ(t)dt
0
θ
E(cost/cycle)|T = t)φ(t)dt + 0
θ
θ 0
Z
∞
(1)
E(cost/cycle)|T = t)φ(t)dt
Z t Z t−τ h(R − D)Rt2 + sR f(τ ) {K + M + g(x)dxdτ }φ(t)dt 2D 0 0 Z ∞ h(R − D)Rθ 2 + {K + }φ(t)dt (2) 2D θ
9
= K + MΦ(θ) + sR
Z
θ
φ(t)
0
+
Z
t
f (τ )
0
Z
0
h(R − D)R 2 {θ (1 − Φ(θ)) + 2D
E(Duration of a cycle) =
Z
∞
t−τ
g(x)dxdτ dt Z
θ
t2 φ(t)dt}, and
(3)
0
E(Duration of a cycle|T = t)φ(t)dt
0
Z ∞ R Rθ tφ(t)dt + φ(t)dt D 0 D θ Rθ RZθ tφ(t)dt + = (1 − Φ(θ)) D 0 D
=
Z
θ
(4)
Applying the renewal reward theorem , we get the long-term cost per unit time as ET C(θ) =
E(cost/cycle) E(Duration of a cycle) K + M Φ(θ) + +
=
Rθ 0
t2 φ(t)dt} + sR Rθ (1 D
h(R−D)R {θ 2 (1 2D
Rθ 0
φ(t)
− Φ(θ)) +
Rt 0
R D
f (τ ) Rθ 0
− Φ(θ))
R t−τ 0
g(x)dxdτ dt
tφ(t)dt
. (5)
The optimal production run time, θ∗ , is determined by minimizing the cost expression (5) with respect to (wrt) θ.
3
The NR-E/U/α model
In general, it is not easy to derive a closed expression for θ∗ in (5). To make the analysis tractable but still allow us to gain sufficient insight, we 10
assign speciÞc distributional forms for φ(.), f (.), and g(.). We assume for all future analysis that the time-to-breakdown, T , is exponentially distributed (Barlow and Proschan 1965) with mean µ. Furthermore, we assume that the failure time distribution is uniformly distributed between 0 and T , or f (τ ) = 1/t, T = t. The key idea behind this assumption is that the process is equally likely to fail anytime between [0, T ]. Additionally, once the process is in the “out-of-control” state, it produces defectives at a constant rate, i.e., for a given breakdown time T = t, and a time to failure τ, g(x) = α, t − τ ≤ x ≤ t (Rosenblatt and Lee, 1986; Guo, 1996). The NR-G policy with exponential time-to-breakdown is often described by the acronym “NR-E” policy. For the purposes of exposition, we call our model the NR-E/U/α indicating that the NR-E policy has been expanded to include the uniform failure time distribution and a constant defective rate, α. Substituting the speciÞc distributional forms of φ(.), f (.), and g(.) into the total cost equation (5) and simplifying yields
ET C(θ) =
DµK MDµ Dsα(eµθ − µθ − 1) + + R(1 − e−µθ ) R 2(eµθ − 1) h(R − D)(eµθ − µθ − 1) . + µ(eµθ − 1)
(6)
To solve for θ, we take the derivative of ET C(θ) with respect to θ and set it equal to zero. After some algebra, we show that the optimal solution, 11
θ∗ , satisÞes the equation e−µθ + µθ − 1 =
2Dµ2 K . R(Dsαµ + 2h(R − D))
(7)
2
2Dµ K Let V (θ) = e−(µθ) +µθ −1− R(Dsαµ+2h(R−D)) . Clearly, V (θ) is continuous, and 0
V (θ) = µ(1 − e−µθ ) > 0, implying that V (θ) is an increasing function wrt θ. Note that V (0) = −(1 +
2Dµ2 K ) R(Dsαµ+2h(R−D))
< 0, and Limitθ→∞ V (θ) → ∞.
Hence, there is a unique point θ ∗ > 0 such that V (θ ∗ ) = 0. For different distributional forms of φ(.), f(.), and g(.), one can conceivably have multiple solutions to the equation
∂ET C(θ) ∂θ
= 0. The solution to
such a scenario would be to choose the θ that produces the lowest ET C(θ).
Observations and Analysis In this section, we analyze the structure of the model and the optimal production run time in order to gain insight into how the production run times are inßuenced by process deterioration and machine breakdowns. The proofs of the following propositions are available in the Appendix. Proposition 1 The minimum long range expected total cost ET C(θ ∗ ) is given by
M Dµ +h(R R
− D)θ ∗ + Dsαµ θ∗ , where θ∗ satisÞes equation (7) 2
We note here that ET C(θ ∗ ) is the average long-run minimum cost in a cycle and hence is a deterministic estimate of the minimum possible cost per 12
cycle. Therefore, it is possible for production cycles of length θ∗ to have a cost greater than ET C(θ∗ ). This proposition suggests that the holding and the defective costs are linear functions of the optimal lot size. Therefore, as the optimal lot size increases, so does the minimum long-range cost in a linear manner. Intuitively, an increase in the defective production rate would tend to reduce the optimal production run time. On the other hand, an increase in the failure rate would increase the production run length. We address this balance between the defective production rate and the failure rate and its impact on the production run length in the next set of propositions.
Proposition 2 (a) The optimal production run size is a decreasing function of α. (b) The minimum long-range cost per unit time is an increasing function of α. (c) The optimal production run size is an increasing function of µ as long as 2π − Dsαµπ 2 ≥
θ∗ , µ
where π =
∗
[e−(µθ ) +µθ∗ −1] . (1−e−µθ∗ )
(d) A sufficient condition for the minimum long-range cost to increase with µ is that 2π − Dsαµπ2 ≥
θ∗ . µ
Proposition 2(a) is intuitive — as α increases for a given failure rate, the cost of producing defective items increases and therefore smaller production runs are preferred. Although the optimal production run time decreases with 13
α, Proposition 2(b) shows that the total long-range optimal cost increases with α. The clear implication is that the penalty for ignoring process deterioration (as do the EMQ and the NR-E models) increases as the rate of deterioration increases. The condition in Proposition 2(c) is easily satisÞed for low values of α relative to µ. A simple and intuitive explanation would be that as the failure rate increases, it may be the best strategy to wait until the machine fails, i.e., to perform the maintenance and the setups together. When α is large, however, its effect is large enough to start reducing the optimal production run in order to control the cost of defective items. Proposition 2(d) gives the condition under which the long-range optimal costs increase with the failure rates. The intuitive explanation of the condition is similar to the one provided for Proposition 2(c).
Proposition 3 (a) θ∗ is always less than the NR-E production run time (b) θ∗ is greater than that for the classic EM Q only if e−µθe + µθe − 1 ≤ 2Dµ2 K , R(Dsαµ+2h(R−D))
where θe =
q
2sD/hR(R − D).
Proposition 3(a) has an intuitive explanation. The NR-E model ignores the cost of process deterioration. A positive α tends to decrease the optimal production run time since shorter run times yield fewer defectives. Proposition 3(b), meanwhile, provides a condition for θ∗ to be greater than θEM Q the production run time for the EM Q. Unlike the NR-E model, a relatively 14
large α compared to µ would force smaller lot sizes, and hence lead to situations where θ∗ < θEM Q . On the other hand, a high µ relative to α would result in θ∗ > θEM Q , much like the NR-E model. Proposition 4 (a) There is always a cost penalty of using θNR−E for θ∗ (b) The above cost penalty is always increasing in α. Proposition 4(a) shows that there is always a cost penalty for using the optimal θNR−E for θ∗ . The importance of process deterioration costs is substantiated by Proposition 4(b) — as the defective production rate increases, so does the penalty. Additionally, Groenevelt et al. (1992a) show that using the EM Q production run time instead of the NR-E run time incurs a penalty of about 2% of the total costs, indicating that using the EM Q production run time in an environment where both process deterioration and machine breakdowns exist further exacerbates the cost penalty.
4
Numerical Computation and Managerial Insights
Assuming K = $450, M = $1, 000, h = $10, D = 30 items/time unit, R = 35 items/time unit, and s = $100, we perform a numerical analysis to illustrate key insights from the model. Figure 1 shows how θ ∗ varies with α for different values of µ. Each point is obtained by solving for θ in equation (7) with the corresponding values of α and µ. As expected, the optimal production 15
run time decreases with increasing α. From a practical perspective, α can represent the type of deterioration to the production process, i.e., a wornout tool will have a lower value of α than a broken one. Such information can be used to adjust the run times correspondingly. However, this warrants constructing a database of the most frequent quality problems in order to get an estimate of the value of α. Figure 2, meanwhile, shows how the optimal production run time varies with µ for different values of α. The solutions of the NR-E model (α = 0) and the EM Q (α = 0, µ = 0) model are also shown in the Þgure. As in Proposition 3(a), the NR-E/U/α model always recommends a lower production run time than the NR-E model, and, in this case, is greater than the EM Q. However, contrary to the NR-E model, the NR-E/U/α solution can be smaller than the EM Q model solution, depending on the condition prescribed in Proposition 3(b). Table 1(a) gives the cost of misspecifying the NR-E/U/α model with the optimal solution of the NR-E model. When α → 0, the NR-E/U/α model solution approaches that of the NR-E model. At low values of α, one would therefore expect the defective, holding, setup, and consequently expected total costs of both the models to be almost identical. For a given µ, the holding and the defective costs are increasing functions of α. Therefore,
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as α increases, so does the cost of misspeciÞcation of the holding and the defective costs. Furthermore, it is easy to see that the penalty also increases with demand rate (D), and unit defective cost (s). Another observation from Table 1(a) is that the cost of misspeciÞcation decreases for higher values of µ. This is explained by virtue of the fact that when machine failures occur frequently, the actual production run time is smaller than the optimal one. Additionally, the time the production process is “out-of-control” also decreases, diminishing the effect of α on the total cost solution. On the other hand, since θNR−E ≥ θ∗ and the time between setups decreases as α increases, the setup costs are higher when using θ∗ . However as Table 1(a) shows, the misspeciÞcation results in a cost penalty when all the cost components are considered. Table 1(b) meanwhile gives the cost of misspecifying the NR-E/U/α model by the optimal solution of the EM Q model. The errors of misspeciÞcation, as one would expect, are much higher. In this case, the penalty decreases with increasing α and µ. Although counterintuitive, the decreasing penalty with higher α has a simple explanation. From Figure 2, as α increases, the optimal NR-E/U/α model solution is closer in magnitude to the EM Q solution than when compared to the NR-E model. One can therefore expect the cost penalty to be lower using the EM Q solution. It should be
17
noted that the magnitude of the penalty in using the EM Q model is higher than for the NR-E/U/α model for every µ and α combination. Another insight is that for high values of α and µ, i.e., when the process has high failure rates and defectives, there is a smaller cost penalty of misspeciÞcation because of smaller production runs, and all models have high ET C 0 s. However, in a relatively reliable process, the cost of misspeciÞcation is very high, illustrating the importance of including process deterioration in planning production run times.
5
Summary and Conclusions
Our primary objective was to study the interaction between the economics of production and imperfections in the production process. Our key contribution is the inclusion of both process deterioration and machine breakdown into our model. In particular, we attempt to provide production managers with guidelines to choose the appropriate run times to buffer against the production of defective items and stoppages occurring due to machine breakdown. There are two primary Þndings from our model. First, when failure rates are signiÞcantly higher than machine breakdown rates, our model suggests that a manager make the run times lower than the EM Q run times. Mean-
18
while, a relatively high machine breakdown rate would prompt an upward adjustment (relative to EM Q run times) in the run times, much like the NR-E model. With exchange curves such as in Figures 1 and 2, a manager can quickly zero in on the appropriate run times with a reasonable estimate of the failure and breakdown rates. Second, we show that there is always a cost of misspecifying the production run time with the solution of either the EM Q or the NR-E models. Given that many managers still use the EM Q as a benchmark, such a misspeciÞcation of run times might result in poor resource allocation decisions. Although our model integrates deterioration and breakdowns effectively, it is not without its shortcomings. Thus, we see future research in the following directions: (i) most of our results are for speciÞc distributional forms of failure and breakdown distributions. They need to be extended to more general distributional forms. (ii) Our model can be extended to include inspection schedules. And (iii) future studies could include the impact of setup cost reduction on production run times, in addition to process deterioration and machine breakdown.
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References Ax¨aster, S. and Elmaghraby, S. E. 1981. A note on EMQ under learning and forgetting, AIIE Transactions, 13: 86-90. Barlow, R. E. and Proschan, F. 1965. Mathematical theory of reliability, Wiley: New York. Chand, S. 1989. Lot sizes and setup frequency with learning in setups and process quality. European Journal of Operations Research, 42: 190-202. Cheng, T. C. E. 1989. An economic production quantity model with ßexibility and reliability considerations. European Journal of Operations Research, 39: 174-179. Cheng, T. C. E. 1991. Economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Transactions, 23: 23-28. Groenevelt, H., Pinetelon, L., and Siedmann. 1992a. Production lot sizing with machine breakdowns. Management Science, 38, 1: 104-123. Groenevelt, H., Pinetelon, L., and Siedmann. 1992b. Production batching with machine breakdowns and safety stocks. Operations Research, 40: 959-971.
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Harris, F. (1913). How many parts to make at once. Factory, The Magazine of Management, 10: 135-136; 152. Kun-Jen, C. 1997. Bounds for production lot sizing with machine breakdowns. Computers and Industrial Engineering, 32: 139-144 . Liu, B. and Cao, J. 1999. Analysis of a production-inventory system with machine breakdowns and shutdowns. Computers and Operations Research, 26, 1: 73-91. Porteus E. L. 1986. Optimal lot-sizing, process quality improvement and setup cost reduction. Operations Research, 34: 137-144. Ritche, E. and Tsado, A. 1986. The penalties of using the EOQ: A comparison of lot-sizing rules for linearly increasing demand. Production and Inventory Management Journal, First Quarter: 12-19. Ross, S. M. 1981. Introduction of probability models, Academic Press:New York. Rosenblatt, M. J. and Lee, H. L. 1986. Economic Production Cycles with imperfect production processes, IIE Transactions, 18: 48-55. Schwarz, L. B. 1977. A note on the near optimality of “5-EOQ’s” worth forecast horizons. Operations Research, 25: 533-536. Shah, Y. K. 1977. An order-level lot size model for deteriorating items. AIIE Transactions, 9:108-112.
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Snyder, R. D. 1977. The classical economic order quantity formula. Operations Research Quarterly, 24: 125-127.
Appendix: Proofs for Propositions 1-4. Proposition 1 Substituting the expression for e−µθ from Equation (7) in Equation (6) we can, after some tedious algebra, obtain the result. Proposition 2 0
0
(a) Let θα = ∂θ∗ /∂α and θµ = ∂θ∗ /∂µ. To prove (a), we need to show 0
that θα ≤ 0. Differentiating equation (7) wrt α yields: −(µθ∗ )
µe
2D2 µ3 Ks θα + µθα = − R(Dsαµ + 2h(R − D))2 0
0
Simplifying and manipulating gives 0
θα = −
2D2 µ2 Ks ≤ 0, R(Dsαµ + 2h(R − D))2 (1 − e−µθ∗ )
which proves (a). (b)
∂ET C(θ∗ ) ∂α
=
2Dµ2 K
µ(1−e−µθ∗ )R(Dsαµ+2h(R−D))
The required condition therefore follows. 22
−1 =
∗
[e−(µθ ) +µθ∗ −1] µ(1−e−µθ∗ )
−1 =
π µ
− 1.
(c) Differntiating equation (7) wrt µ and manupulating the terms yields: 0
−µθ∗
µθµ (1−e
2D2 µ3 Ksα 4DµK ∗ − )= −θ∗ (1−e−µθ ) 2 R(Dsαµ + 2h(R − D)) R(Dsαµ + 2h(R − D))
Using Equation (7), we can rewrite the above expression as: 0
∗
∗
∗
∗
µ2 θµ (1−e−µθ ) = 2[e−(µθ ) +µθ∗ −1]−Dsαµ[e−(µθ ) +µθ∗ −1]2 −θ∗ µ(1−e−µθ ) If π =
∗
[e−(µθ ) +µθ∗ −1] , (1−e−µθ∗ )
0
then θµ = 2π − Dsαµπ 2 −
0
that for θµ ≥ 0, 2π − Dsαµπ 2 ≥
θ∗ . µ
It therefore follows
θ∗ . µ
(d) This is a straight-forward result in lieu of (c) Proposition 3 (a) Let ω(z) = e−µz + µz − 1. It is easy to see that ω(z) is an increasing function and hence if z1 > z2 , ω(z1 ) > ω(z2 ) and vice-versa. We know that the NR-E production run time, θNR−E is a solution to ω(θNR−E ) = Note that ω(θ∗ ) =
2Dµ2 K R(Dsαµ+2h(R−D))
2Dµ2 K . 2hR(R−D))
≤ ω(θNR−E ). It therefore follows that
θ∗ ≤ θNR−E . (b) As before, recognizing that ω(z) is an increasing function, it is easy to see that for θ∗ ≥ θe , ω(θ∗ ) =
2Dµ2 K R(Dsαµ+2h(R−D))
≥ ω(θe ) = e−µθe + µθe − 1.
Proposition 4 The long range cost of using the NR-E production run time in the NRE/U/α model can be shown as
µθ
NR−E −µθ M Dµ N R−E −1) +h(R−D)θNR−E + Dsα(e 2(eµθN R−E . R −1)
23
The cost penalty is therefore proves (a). Additionally,
Dsα(eµθN R−E −µθNR−E −1) , 2(eµθNR−E −1)
Dsα(eµθN R−E −µθN R−E −1) 2(eµθNR−E −1)
α, which proves (b).
24
a positive value, which
is an increasing function in
Figure 1: How the Optimal Run Time (θ *) Varies with Failure Rate (α ) 9
µ =0.9
Optimal Run Time (θ )
8
µ values
µ =0.7
7
0.1
µ =0.5
6
0.3 0.5
µ =0.3
5
0.7 0.9
4 3
µ =0.1
2 0
0.02
0.03
0.04
0.05
0.06
Failure Rate (α α)
0.07
0.08
0.09
0.1
Figure 2: Comparing EMQ, NR-E , and the NR-E/U/ α Optimal Solution for Varying µ
Optimal Run Time ( * )
9
NR-E (α =0)
8
α = .03
7
α = .05
6
α values
5
0
4
0.03
3 2
0.05
θEMQ=1.851
1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Breakdown Rate (µ )
0.8
0.9
α = .07
0.07
α = .09
0.09
1
Table 1(a): Misspecification Cost as % of the NR-E/U/α model cost for using the NR-E model solution
µ values
0.1 0.2 0.3 0.4
0 0.00% 0.00% 0.00% 0.00%
0.02 0.03% 0.08% 0.12% 0.14%
0.03 0.06% 0.17% 0.26% 0.31%
0.5 0.6 0.7 0.8 0.9 1
0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
0.14% 0.12% 0.09% 0.05% 0.03% 0.01%
0.31% 0.27% 0.20% 0.13% 0.08% 0.04%
α values 0.04 0.11% 0.28% 0.43% 0.52%
0.05 0.16% 0.43% 0.65% 0.77%
0.06 0.23% 0.59% 0.90% 1.07%
0.07 0.31% 0.78% 1.17% 1.39%
0.08 0.39% 0.99% 1.47% 1.74%
0.09 0.49% 1.21% 1.79% 2.11%
0.1 0.59% 1.45% 2.13% 2.51%
0.52% 0.46% 0.35% 0.24% 0.15% 0.08%
0.78% 0.69% 0.54% 0.38% 0.24% 0.15%
1.08% 0.96% 0.76% 0.55% 0.37% 0.23%
1.41% 1.26% 1.01% 0.74% 0.51% 0.34%
1.77% 1.59% 1.29% 0.97% 0.68% 0.47%
2.15% 1.95% 1.60% 1.21% 0.87% 0.61%
2.56% 2.33% 1.93% 1.48% 1.09% 0.78%
Table 1(b): Misspecification Cost as % of the NR-E/U/α model cost for using the EMQ model solution
µ values
0.1 0.2 0.3 0.4 0.5
0 21.27% 16.83% 13.92% 11.75% 9.98%
0.02 19.83% 14.89% 11.83% 9.67% 8.04%
0.03 19.16% 14.03% 10.92% 8.79% 7.22%
0.6 0.7 0.8 0.9 1
8.48% 7.17% 6.03% 5.05% 4.22%
6.74% 5.66% 4.76% 4.00% 3.36%
6.00% 5.01% 4.20% 3.53% 2.97%
α values 0.04 18.52% 13.22% 10.08% 7.99% 6.48%
0.05 17.90% 12.46% 9.31% 7.26% 5.81%
0.06 17.31% 11.75% 8.61% 6.60% 5.21%
0.07 16.73% 11.08% 7.95% 5.99% 4.66%
0.08 16.18% 10.45% 7.35% 5.44% 4.17%
0.09 15.65% 9.86% 6.79% 4.94% 3.72%
0.1 15.14% 9.31% 6.27% 4.47% 3.31%
5.33% 4.43% 3.70% 3.10% 2.60%
4.73% 3.90% 3.24% 2.70% 2.26%
4.19% 3.42% 2.82% 2.34% 1.95%
3.71% 3.00% 2.45% 2.02% 1.67%
3.27% 2.61% 2.11% 1.72% 1.42%
2.88% 2.27% 1.81% 1.46% 1.19%
2.52% 1.96% 1.54% 1.23% 0.99%
Tonya Boone School of Business, College of William and Mary Williamsburg, VA 23187-8795. email: [email protected] Web: http://faculty.wm.edu/Tonya.Boone
Ram Ganeshan School of Business, College of William and Mary Williamsburg, VA 23187-8795. email: [email protected] Web: http://business.wm.edu/Ram.Ganeshan
Yuanming Guo Advanta Corporation Spring House, PA 19477. email: [email protected]
J. Keith Ord School of Business, Georgetown University Washington, DC, 20057. email: [email protected]
Abstract This paper investigates the interaction between the economics of production and imperfections in the production process. SpeciÞcally, this paper is the Þrst to devise a model in an attempt to provide managers with guidelines to choose the appropriate production run times to buffer against both the production of defective items and stoppages occurring due to machine breakdowns. In addition to providing several structural properties of the model, we show that a manager will always incur a cost penalty when (s)he uses the results of two oft-cited models — the EM Q (Economic Order/Manufacturing Quantity) and the NR-E (No-Resumption, Exponential machine breakdown) — to determine production run times. KeyW ords: Inventory Models; Logistics; Probability Models
2
1
Introduction
Since its inception in Harris (1913), the EM Q (Economic Manufacturing Quantity or Economic Order Quantity (EOQ)) model has spawned research efforts primarily attempting to relax many of the idealistic assumptions built into it. Some of the more oft-cited extensions since 1970 include the incorporation of variable demand (Ritche and Tsado, 1986; Snyder, 1977); a Þnite production horizon (Schwarz, 1977); learning effects (Ax¨aster and Elmaghraby, 1981; Chand, 1989); deteriorating items (Shah, 1977); setup reduction (Porteus, 1986); and the modeling of imperfect processes (Porteus, 1986; Groenevelt, Pintelon, and Seidmann, 1992a & b). Our primary focus in this paper is to incorporate imperfections in the production process into the classical EM Q model. Recent research analyzing the relationship between production lot sizes or run times and imperfect production processes have largely centered around two key issues: process deterioration and yield; and machine breakdowns and repairs. Porteus (1986) was perhaps the Þrst to model the relationship between production lot size and process deterioration. In his model, the production process can go “outof-control” (i.e., start producing defective items) each time it produces an
3
item. In a related paper, Rosenblatt and Lee (1986), assume that the duration of the “in-control” state is a random variable with an exponential failure time distribution. Both these seminal papers conclude that managers should use a smaller lot size or production run times since these lead to fewer defective items. Several researchers have since built upon these two initial studies. Notable among them are Chand (1989), who validates Porteus’s (1986) model when learning effects are present. In two papers, Cheng (1989, 1991) extends the classic EMQ problem by assuming, Þrst, that the production costs are a function of reliability (proportion of defectives) and demand, and second, that they are a function of reliability and setup costs. Meanwhile, another stream of research has focused on the impact of machine breakdowns and preventive maintenance on production run times. Groenevelt et al. (1992a) show that when machine breakdown times follow an exponential distribution, the optimal lot size will always be greater than one in the classical EM Q model (intuitively this should be true for all breakdown time distributions). They later extend their model (1992b) to consider the problem of simultaneously determining lot size and safety stocks when repair times are signiÞcant. Kun-Jen (1997) extends the Groenevelt et al. (1992a) model to provide bounds on the optimal lot size. With these bounds, a simple algorithm to Þnd the optimal lot size is also developed. Finally Liu
4
and Cao (1999) model a (m, M) production system (setup when inventory level is m, and shutoff when inventory is built up to M ) that is subject to stochastic breakdowns. This model is similar in spirit to the classic (s, S) inventory model, i.e., Þnd the optimal reorder point (s), and order up to level (S) that minimizes inventory costs. Liu and Cao’s objective, however, is to Þnd the right levels of m and M that minimize the production system costs. Although the various studies reviewed above provide considerable insight into how lot size is affected by either deterioration of the process or machine breakdowns, none of them incorporate both of these aspects into a lot-sizing model. Our goal in this paper is to model a scenario where there is both process deterioration and an eventual breakdown of the production process. The rest of the paper is organized as follows. The next section explains our notation and modeling environment. The third section further develops the model and analyzes the model structure in order to gain insight into the model and its solution. The fourth section presents detailed numerical analysis and managerial insights. Finally, in the Þfth section, we summarize the paper and suggest directions for some future research.
5
2
Model Development
Following is the notation we use to develop our model. We defer the speciÞc deÞnitions of some of the variables until they are used. D = demand rate per unit time, h = the holding cost per unit per unit time, K = setup cost for each production run, M = the cost of corrective maintenance, N = random variable representing the number of defective items, R = production rate per unit time, s = cost incurred by producing a defective item, T = random variable denoting time-to-breakdown, U =Uniform distribution, θ = actual production time for each run, θ∗ = optimal production run time, τ = time taken to transition from an in-control to an out-of-control state, f (.) = the probability distribution function (pdf) denoting the transition from an in-control to an out-of-control state, g(x) = the rate at which the production process produces defectives at time x, given that it is in an out-of-control state (in our model, g(x) = α), φ(.) = pdf of T ,
6
Φ(.) = cumulative density function (cdf) of T , NR-E = No-Resumption, Exponential machine breakdown policy, and NR-E/U/α = NR-E policy with uniform failure time and a constant defective rate, α. We model a production system manufacturing a single item on a single machine in batches. The production rate (R) and the demand rate (D) are assumed to be constant and the production process is able to produce items at a rate faster than it is demanded (R > D). The machine requires a setup, costing $K, and is susceptible to stochastic breakdowns, with the time to breakdown described by a random variable T with probability density function φ(.). Additionally, the nature of the breakdown process follows the “No-Resumption, General machine breakdown” (NR-G, Groenevelt et al., 1992a) policy. Under this policy, “...when a break-down takes place, the interrupted lot is aborted and a new one is to be started only when all available inventory is depleted. Maintenance of the production unit is carried out after a failure or a pre-determined amount of time, which ever comes Þrst. Each maintenance action restores the system to the same initial working conditions.” As have researchers in the past, we assume that the setup and the cor7
rective maintenance (costing $M ) do not take up any time. According to Groenevelt et al. (1992a), examples where such an assumption is justiÞed include ßexible manufacturing system environments where setup times are rather small, or in production systems with a modular design where maintenance is fast and easy. In order to realistically incorporate the effects of process deterioration, we modify the NR-G policy the following way: at the beginning of a production lot, the process is “in-control,” and only items of acceptable quality are produced. In those lots where a breakdown occurs, the machine goes “out of control,” or, in other words, starts producing defective items prior to the breakdown. For the purposes of this study, the “out of control” state does not cause the breakdown, rather it precedes total machine failure. From a modeling perspective, we assume that the elapsed time, τ, until the production process shifts from the “in-control” to the “out of control” state is a random variable with pdf f (τ ) and cumulative density function F (τ ) with f (τ ) > 0, 0 < τ ≤ T. For the purposes of exposition, we call f(τ ) the “failure time” distribution (Barlow and Proschan, 1965). Additionally, we assume that the system produces defects at a rate g(x) at any given time x after the shift to the out-of-control state. We assume that g(x) is positive and non-decreasing, that is, g(x) > 0 and g 0 (x) ≥ 0 for x ≥ 0. The premise
8
is that once the system is in the out-of-control state, it produces defective items at a rate that is either a constant or increases with the time the system is in the out-of-control state. If N is the random variable representing the number of defective items produced in a production cycle where a breakdown has occurred, and if θ is the actual production time of the production run, it is easy to see that the expected number of defective items, E(N) is E(N ) = R
Z
θ
0
Z
t
f(τ ) 0
Z
t−τ
g(x)dxdτ φ(t)dt. 0
Our ultimate objective is to Þnd the production run length θ that minimizes the total cost per unit time ET C(θ). SpeciÞcally, the total cost is the sum of setup, maintenance, linear holding costs, and the cost of producing defective items. Since our production process has a renewal epoch at the start of every production run, we apply the renewal reward theorem (Ross 1981) to obtain the average total cost per unit time by taking the ratio of the expected cost per renewal cycle to its duration.
E(cost/cycle) = = =
Z Z
Z
∞
E(cost/cycle|T = t)φ(t)dt
0
θ
E(cost/cycle)|T = t)φ(t)dt + 0
θ
θ 0
Z
∞
(1)
E(cost/cycle)|T = t)φ(t)dt
Z t Z t−τ h(R − D)Rt2 + sR f(τ ) {K + M + g(x)dxdτ }φ(t)dt 2D 0 0 Z ∞ h(R − D)Rθ 2 + {K + }φ(t)dt (2) 2D θ
9
= K + MΦ(θ) + sR
Z
θ
φ(t)
0
+
Z
t
f (τ )
0
Z
0
h(R − D)R 2 {θ (1 − Φ(θ)) + 2D
E(Duration of a cycle) =
Z
∞
t−τ
g(x)dxdτ dt Z
θ
t2 φ(t)dt}, and
(3)
0
E(Duration of a cycle|T = t)φ(t)dt
0
Z ∞ R Rθ tφ(t)dt + φ(t)dt D 0 D θ Rθ RZθ tφ(t)dt + = (1 − Φ(θ)) D 0 D
=
Z
θ
(4)
Applying the renewal reward theorem , we get the long-term cost per unit time as ET C(θ) =
E(cost/cycle) E(Duration of a cycle) K + M Φ(θ) + +
=
Rθ 0
t2 φ(t)dt} + sR Rθ (1 D
h(R−D)R {θ 2 (1 2D
Rθ 0
φ(t)
− Φ(θ)) +
Rt 0
R D
f (τ ) Rθ 0
− Φ(θ))
R t−τ 0
g(x)dxdτ dt
tφ(t)dt
. (5)
The optimal production run time, θ∗ , is determined by minimizing the cost expression (5) with respect to (wrt) θ.
3
The NR-E/U/α model
In general, it is not easy to derive a closed expression for θ∗ in (5). To make the analysis tractable but still allow us to gain sufficient insight, we 10
assign speciÞc distributional forms for φ(.), f (.), and g(.). We assume for all future analysis that the time-to-breakdown, T , is exponentially distributed (Barlow and Proschan 1965) with mean µ. Furthermore, we assume that the failure time distribution is uniformly distributed between 0 and T , or f (τ ) = 1/t, T = t. The key idea behind this assumption is that the process is equally likely to fail anytime between [0, T ]. Additionally, once the process is in the “out-of-control” state, it produces defectives at a constant rate, i.e., for a given breakdown time T = t, and a time to failure τ, g(x) = α, t − τ ≤ x ≤ t (Rosenblatt and Lee, 1986; Guo, 1996). The NR-G policy with exponential time-to-breakdown is often described by the acronym “NR-E” policy. For the purposes of exposition, we call our model the NR-E/U/α indicating that the NR-E policy has been expanded to include the uniform failure time distribution and a constant defective rate, α. Substituting the speciÞc distributional forms of φ(.), f (.), and g(.) into the total cost equation (5) and simplifying yields
ET C(θ) =
DµK MDµ Dsα(eµθ − µθ − 1) + + R(1 − e−µθ ) R 2(eµθ − 1) h(R − D)(eµθ − µθ − 1) . + µ(eµθ − 1)
(6)
To solve for θ, we take the derivative of ET C(θ) with respect to θ and set it equal to zero. After some algebra, we show that the optimal solution, 11
θ∗ , satisÞes the equation e−µθ + µθ − 1 =
2Dµ2 K . R(Dsαµ + 2h(R − D))
(7)
2
2Dµ K Let V (θ) = e−(µθ) +µθ −1− R(Dsαµ+2h(R−D)) . Clearly, V (θ) is continuous, and 0
V (θ) = µ(1 − e−µθ ) > 0, implying that V (θ) is an increasing function wrt θ. Note that V (0) = −(1 +
2Dµ2 K ) R(Dsαµ+2h(R−D))
< 0, and Limitθ→∞ V (θ) → ∞.
Hence, there is a unique point θ ∗ > 0 such that V (θ ∗ ) = 0. For different distributional forms of φ(.), f(.), and g(.), one can conceivably have multiple solutions to the equation
∂ET C(θ) ∂θ
= 0. The solution to
such a scenario would be to choose the θ that produces the lowest ET C(θ).
Observations and Analysis In this section, we analyze the structure of the model and the optimal production run time in order to gain insight into how the production run times are inßuenced by process deterioration and machine breakdowns. The proofs of the following propositions are available in the Appendix. Proposition 1 The minimum long range expected total cost ET C(θ ∗ ) is given by
M Dµ +h(R R
− D)θ ∗ + Dsαµ θ∗ , where θ∗ satisÞes equation (7) 2
We note here that ET C(θ ∗ ) is the average long-run minimum cost in a cycle and hence is a deterministic estimate of the minimum possible cost per 12
cycle. Therefore, it is possible for production cycles of length θ∗ to have a cost greater than ET C(θ∗ ). This proposition suggests that the holding and the defective costs are linear functions of the optimal lot size. Therefore, as the optimal lot size increases, so does the minimum long-range cost in a linear manner. Intuitively, an increase in the defective production rate would tend to reduce the optimal production run time. On the other hand, an increase in the failure rate would increase the production run length. We address this balance between the defective production rate and the failure rate and its impact on the production run length in the next set of propositions.
Proposition 2 (a) The optimal production run size is a decreasing function of α. (b) The minimum long-range cost per unit time is an increasing function of α. (c) The optimal production run size is an increasing function of µ as long as 2π − Dsαµπ 2 ≥
θ∗ , µ
where π =
∗
[e−(µθ ) +µθ∗ −1] . (1−e−µθ∗ )
(d) A sufficient condition for the minimum long-range cost to increase with µ is that 2π − Dsαµπ2 ≥
θ∗ . µ
Proposition 2(a) is intuitive — as α increases for a given failure rate, the cost of producing defective items increases and therefore smaller production runs are preferred. Although the optimal production run time decreases with 13
α, Proposition 2(b) shows that the total long-range optimal cost increases with α. The clear implication is that the penalty for ignoring process deterioration (as do the EMQ and the NR-E models) increases as the rate of deterioration increases. The condition in Proposition 2(c) is easily satisÞed for low values of α relative to µ. A simple and intuitive explanation would be that as the failure rate increases, it may be the best strategy to wait until the machine fails, i.e., to perform the maintenance and the setups together. When α is large, however, its effect is large enough to start reducing the optimal production run in order to control the cost of defective items. Proposition 2(d) gives the condition under which the long-range optimal costs increase with the failure rates. The intuitive explanation of the condition is similar to the one provided for Proposition 2(c).
Proposition 3 (a) θ∗ is always less than the NR-E production run time (b) θ∗ is greater than that for the classic EM Q only if e−µθe + µθe − 1 ≤ 2Dµ2 K , R(Dsαµ+2h(R−D))
where θe =
q
2sD/hR(R − D).
Proposition 3(a) has an intuitive explanation. The NR-E model ignores the cost of process deterioration. A positive α tends to decrease the optimal production run time since shorter run times yield fewer defectives. Proposition 3(b), meanwhile, provides a condition for θ∗ to be greater than θEM Q the production run time for the EM Q. Unlike the NR-E model, a relatively 14
large α compared to µ would force smaller lot sizes, and hence lead to situations where θ∗ < θEM Q . On the other hand, a high µ relative to α would result in θ∗ > θEM Q , much like the NR-E model. Proposition 4 (a) There is always a cost penalty of using θNR−E for θ∗ (b) The above cost penalty is always increasing in α. Proposition 4(a) shows that there is always a cost penalty for using the optimal θNR−E for θ∗ . The importance of process deterioration costs is substantiated by Proposition 4(b) — as the defective production rate increases, so does the penalty. Additionally, Groenevelt et al. (1992a) show that using the EM Q production run time instead of the NR-E run time incurs a penalty of about 2% of the total costs, indicating that using the EM Q production run time in an environment where both process deterioration and machine breakdowns exist further exacerbates the cost penalty.
4
Numerical Computation and Managerial Insights
Assuming K = $450, M = $1, 000, h = $10, D = 30 items/time unit, R = 35 items/time unit, and s = $100, we perform a numerical analysis to illustrate key insights from the model. Figure 1 shows how θ ∗ varies with α for different values of µ. Each point is obtained by solving for θ in equation (7) with the corresponding values of α and µ. As expected, the optimal production 15
run time decreases with increasing α. From a practical perspective, α can represent the type of deterioration to the production process, i.e., a wornout tool will have a lower value of α than a broken one. Such information can be used to adjust the run times correspondingly. However, this warrants constructing a database of the most frequent quality problems in order to get an estimate of the value of α. Figure 2, meanwhile, shows how the optimal production run time varies with µ for different values of α. The solutions of the NR-E model (α = 0) and the EM Q (α = 0, µ = 0) model are also shown in the Þgure. As in Proposition 3(a), the NR-E/U/α model always recommends a lower production run time than the NR-E model, and, in this case, is greater than the EM Q. However, contrary to the NR-E model, the NR-E/U/α solution can be smaller than the EM Q model solution, depending on the condition prescribed in Proposition 3(b). Table 1(a) gives the cost of misspecifying the NR-E/U/α model with the optimal solution of the NR-E model. When α → 0, the NR-E/U/α model solution approaches that of the NR-E model. At low values of α, one would therefore expect the defective, holding, setup, and consequently expected total costs of both the models to be almost identical. For a given µ, the holding and the defective costs are increasing functions of α. Therefore,
16
as α increases, so does the cost of misspeciÞcation of the holding and the defective costs. Furthermore, it is easy to see that the penalty also increases with demand rate (D), and unit defective cost (s). Another observation from Table 1(a) is that the cost of misspeciÞcation decreases for higher values of µ. This is explained by virtue of the fact that when machine failures occur frequently, the actual production run time is smaller than the optimal one. Additionally, the time the production process is “out-of-control” also decreases, diminishing the effect of α on the total cost solution. On the other hand, since θNR−E ≥ θ∗ and the time between setups decreases as α increases, the setup costs are higher when using θ∗ . However as Table 1(a) shows, the misspeciÞcation results in a cost penalty when all the cost components are considered. Table 1(b) meanwhile gives the cost of misspecifying the NR-E/U/α model by the optimal solution of the EM Q model. The errors of misspeciÞcation, as one would expect, are much higher. In this case, the penalty decreases with increasing α and µ. Although counterintuitive, the decreasing penalty with higher α has a simple explanation. From Figure 2, as α increases, the optimal NR-E/U/α model solution is closer in magnitude to the EM Q solution than when compared to the NR-E model. One can therefore expect the cost penalty to be lower using the EM Q solution. It should be
17
noted that the magnitude of the penalty in using the EM Q model is higher than for the NR-E/U/α model for every µ and α combination. Another insight is that for high values of α and µ, i.e., when the process has high failure rates and defectives, there is a smaller cost penalty of misspeciÞcation because of smaller production runs, and all models have high ET C 0 s. However, in a relatively reliable process, the cost of misspeciÞcation is very high, illustrating the importance of including process deterioration in planning production run times.
5
Summary and Conclusions
Our primary objective was to study the interaction between the economics of production and imperfections in the production process. Our key contribution is the inclusion of both process deterioration and machine breakdown into our model. In particular, we attempt to provide production managers with guidelines to choose the appropriate run times to buffer against the production of defective items and stoppages occurring due to machine breakdown. There are two primary Þndings from our model. First, when failure rates are signiÞcantly higher than machine breakdown rates, our model suggests that a manager make the run times lower than the EM Q run times. Mean-
18
while, a relatively high machine breakdown rate would prompt an upward adjustment (relative to EM Q run times) in the run times, much like the NR-E model. With exchange curves such as in Figures 1 and 2, a manager can quickly zero in on the appropriate run times with a reasonable estimate of the failure and breakdown rates. Second, we show that there is always a cost of misspecifying the production run time with the solution of either the EM Q or the NR-E models. Given that many managers still use the EM Q as a benchmark, such a misspeciÞcation of run times might result in poor resource allocation decisions. Although our model integrates deterioration and breakdowns effectively, it is not without its shortcomings. Thus, we see future research in the following directions: (i) most of our results are for speciÞc distributional forms of failure and breakdown distributions. They need to be extended to more general distributional forms. (ii) Our model can be extended to include inspection schedules. And (iii) future studies could include the impact of setup cost reduction on production run times, in addition to process deterioration and machine breakdown.
19
References Ax¨aster, S. and Elmaghraby, S. E. 1981. A note on EMQ under learning and forgetting, AIIE Transactions, 13: 86-90. Barlow, R. E. and Proschan, F. 1965. Mathematical theory of reliability, Wiley: New York. Chand, S. 1989. Lot sizes and setup frequency with learning in setups and process quality. European Journal of Operations Research, 42: 190-202. Cheng, T. C. E. 1989. An economic production quantity model with ßexibility and reliability considerations. European Journal of Operations Research, 39: 174-179. Cheng, T. C. E. 1991. Economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Transactions, 23: 23-28. Groenevelt, H., Pinetelon, L., and Siedmann. 1992a. Production lot sizing with machine breakdowns. Management Science, 38, 1: 104-123. Groenevelt, H., Pinetelon, L., and Siedmann. 1992b. Production batching with machine breakdowns and safety stocks. Operations Research, 40: 959-971.
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Harris, F. (1913). How many parts to make at once. Factory, The Magazine of Management, 10: 135-136; 152. Kun-Jen, C. 1997. Bounds for production lot sizing with machine breakdowns. Computers and Industrial Engineering, 32: 139-144 . Liu, B. and Cao, J. 1999. Analysis of a production-inventory system with machine breakdowns and shutdowns. Computers and Operations Research, 26, 1: 73-91. Porteus E. L. 1986. Optimal lot-sizing, process quality improvement and setup cost reduction. Operations Research, 34: 137-144. Ritche, E. and Tsado, A. 1986. The penalties of using the EOQ: A comparison of lot-sizing rules for linearly increasing demand. Production and Inventory Management Journal, First Quarter: 12-19. Ross, S. M. 1981. Introduction of probability models, Academic Press:New York. Rosenblatt, M. J. and Lee, H. L. 1986. Economic Production Cycles with imperfect production processes, IIE Transactions, 18: 48-55. Schwarz, L. B. 1977. A note on the near optimality of “5-EOQ’s” worth forecast horizons. Operations Research, 25: 533-536. Shah, Y. K. 1977. An order-level lot size model for deteriorating items. AIIE Transactions, 9:108-112.
21
Snyder, R. D. 1977. The classical economic order quantity formula. Operations Research Quarterly, 24: 125-127.
Appendix: Proofs for Propositions 1-4. Proposition 1 Substituting the expression for e−µθ from Equation (7) in Equation (6) we can, after some tedious algebra, obtain the result. Proposition 2 0
0
(a) Let θα = ∂θ∗ /∂α and θµ = ∂θ∗ /∂µ. To prove (a), we need to show 0
that θα ≤ 0. Differentiating equation (7) wrt α yields: −(µθ∗ )
µe
2D2 µ3 Ks θα + µθα = − R(Dsαµ + 2h(R − D))2 0
0
Simplifying and manipulating gives 0
θα = −
2D2 µ2 Ks ≤ 0, R(Dsαµ + 2h(R − D))2 (1 − e−µθ∗ )
which proves (a). (b)
∂ET C(θ∗ ) ∂α
=
2Dµ2 K
µ(1−e−µθ∗ )R(Dsαµ+2h(R−D))
The required condition therefore follows. 22
−1 =
∗
[e−(µθ ) +µθ∗ −1] µ(1−e−µθ∗ )
−1 =
π µ
− 1.
(c) Differntiating equation (7) wrt µ and manupulating the terms yields: 0
−µθ∗
µθµ (1−e
2D2 µ3 Ksα 4DµK ∗ − )= −θ∗ (1−e−µθ ) 2 R(Dsαµ + 2h(R − D)) R(Dsαµ + 2h(R − D))
Using Equation (7), we can rewrite the above expression as: 0
∗
∗
∗
∗
µ2 θµ (1−e−µθ ) = 2[e−(µθ ) +µθ∗ −1]−Dsαµ[e−(µθ ) +µθ∗ −1]2 −θ∗ µ(1−e−µθ ) If π =
∗
[e−(µθ ) +µθ∗ −1] , (1−e−µθ∗ )
0
then θµ = 2π − Dsαµπ 2 −
0
that for θµ ≥ 0, 2π − Dsαµπ 2 ≥
θ∗ . µ
It therefore follows
θ∗ . µ
(d) This is a straight-forward result in lieu of (c) Proposition 3 (a) Let ω(z) = e−µz + µz − 1. It is easy to see that ω(z) is an increasing function and hence if z1 > z2 , ω(z1 ) > ω(z2 ) and vice-versa. We know that the NR-E production run time, θNR−E is a solution to ω(θNR−E ) = Note that ω(θ∗ ) =
2Dµ2 K R(Dsαµ+2h(R−D))
2Dµ2 K . 2hR(R−D))
≤ ω(θNR−E ). It therefore follows that
θ∗ ≤ θNR−E . (b) As before, recognizing that ω(z) is an increasing function, it is easy to see that for θ∗ ≥ θe , ω(θ∗ ) =
2Dµ2 K R(Dsαµ+2h(R−D))
≥ ω(θe ) = e−µθe + µθe − 1.
Proposition 4 The long range cost of using the NR-E production run time in the NRE/U/α model can be shown as
µθ
NR−E −µθ M Dµ N R−E −1) +h(R−D)θNR−E + Dsα(e 2(eµθN R−E . R −1)
23
The cost penalty is therefore proves (a). Additionally,
Dsα(eµθN R−E −µθNR−E −1) , 2(eµθNR−E −1)
Dsα(eµθN R−E −µθN R−E −1) 2(eµθNR−E −1)
α, which proves (b).
24
a positive value, which
is an increasing function in
Figure 1: How the Optimal Run Time (θ *) Varies with Failure Rate (α ) 9
µ =0.9
Optimal Run Time (θ )
8
µ values
µ =0.7
7
0.1
µ =0.5
6
0.3 0.5
µ =0.3
5
0.7 0.9
4 3
µ =0.1
2 0
0.02
0.03
0.04
0.05
0.06
Failure Rate (α α)
0.07
0.08
0.09
0.1
Figure 2: Comparing EMQ, NR-E , and the NR-E/U/ α Optimal Solution for Varying µ
Optimal Run Time ( * )
9
NR-E (α =0)
8
α = .03
7
α = .05
6
α values
5
0
4
0.03
3 2
0.05
θEMQ=1.851
1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Breakdown Rate (µ )
0.8
0.9
α = .07
0.07
α = .09
0.09
1
Table 1(a): Misspecification Cost as % of the NR-E/U/α model cost for using the NR-E model solution
µ values
0.1 0.2 0.3 0.4
0 0.00% 0.00% 0.00% 0.00%
0.02 0.03% 0.08% 0.12% 0.14%
0.03 0.06% 0.17% 0.26% 0.31%
0.5 0.6 0.7 0.8 0.9 1
0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
0.14% 0.12% 0.09% 0.05% 0.03% 0.01%
0.31% 0.27% 0.20% 0.13% 0.08% 0.04%
α values 0.04 0.11% 0.28% 0.43% 0.52%
0.05 0.16% 0.43% 0.65% 0.77%
0.06 0.23% 0.59% 0.90% 1.07%
0.07 0.31% 0.78% 1.17% 1.39%
0.08 0.39% 0.99% 1.47% 1.74%
0.09 0.49% 1.21% 1.79% 2.11%
0.1 0.59% 1.45% 2.13% 2.51%
0.52% 0.46% 0.35% 0.24% 0.15% 0.08%
0.78% 0.69% 0.54% 0.38% 0.24% 0.15%
1.08% 0.96% 0.76% 0.55% 0.37% 0.23%
1.41% 1.26% 1.01% 0.74% 0.51% 0.34%
1.77% 1.59% 1.29% 0.97% 0.68% 0.47%
2.15% 1.95% 1.60% 1.21% 0.87% 0.61%
2.56% 2.33% 1.93% 1.48% 1.09% 0.78%
Table 1(b): Misspecification Cost as % of the NR-E/U/α model cost for using the EMQ model solution
µ values
0.1 0.2 0.3 0.4 0.5
0 21.27% 16.83% 13.92% 11.75% 9.98%
0.02 19.83% 14.89% 11.83% 9.67% 8.04%
0.03 19.16% 14.03% 10.92% 8.79% 7.22%
0.6 0.7 0.8 0.9 1
8.48% 7.17% 6.03% 5.05% 4.22%
6.74% 5.66% 4.76% 4.00% 3.36%
6.00% 5.01% 4.20% 3.53% 2.97%
α values 0.04 18.52% 13.22% 10.08% 7.99% 6.48%
0.05 17.90% 12.46% 9.31% 7.26% 5.81%
0.06 17.31% 11.75% 8.61% 6.60% 5.21%
0.07 16.73% 11.08% 7.95% 5.99% 4.66%
0.08 16.18% 10.45% 7.35% 5.44% 4.17%
0.09 15.65% 9.86% 6.79% 4.94% 3.72%
0.1 15.14% 9.31% 6.27% 4.47% 3.31%
5.33% 4.43% 3.70% 3.10% 2.60%
4.73% 3.90% 3.24% 2.70% 2.26%
4.19% 3.42% 2.82% 2.34% 1.95%
3.71% 3.00% 2.45% 2.02% 1.67%
3.27% 2.61% 2.11% 1.72% 1.42%
2.88% 2.27% 1.81% 1.46% 1.19%
2.52% 1.96% 1.54% 1.23% 0.99%
