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Robust planning in optimization for production system subject to random machine breakdown and failure in rework
Computers & Operations Research 37 (2010) 899 -- 908
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Robust planning in optimization for production system subject to random machine breakdown and failure in rework Singa Wang Chiu ∗ Department of Business Administration, Chaoyang University of Technology, 168 Jifong East Road, Wufong, Taichung 413, Taiwan
A R T I C L E
I N F O
Available online 31 March 2009 Keywords: Production Breakdowns Failure in rework Abort/resume policy Robust planning
A B S T R A C T
This study is concerned with robust planning in optimization, specifically in determining the optimal run time for production system that is subject to random breakdowns under abort/resume (AR) control policy and failure in rework. In most real-life production processes, generation of defective items and breakdowns of manufacturing equipment are inevitable. In this study, random defective rate is assumed and all manufactured items are screened. The perfect quality, reworkable and scrap items are identified and separated; failure-in-rework is assumed. The system is also subject to random machine breakdown; and when it occurs, the AR policy is adopted. Under such policy, the production of the interrupted lot will be immediately resumed when the machine is restored. Mathematical modeling and derivation of the production-inventory cost functions for both systems with/without breakdowns are presented. The renewal reward theorem is used to cope with the variable cycle length when integrating cost functions. The long-run average cost per unit time is obtained. Theorems on convexity and on bounds of production run time are proposed and proved. A recursive searching algorithm is developed for locating the optimal run time that minimizes the expected production-inventory costs. A numerical example with sensitivity analysis is provided to give insight into the optimal operational control of such an unreliable system. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The economic order quantity (EOQ) model was first introduced several decades ago [1]. It is a mathematical model that can be used to derive an optimal order quantity that assists corporations in minimizing total inventory costs. In manufacturing sector, when products are produced in-house instead of being acquired from outside suppliers, the economic production quantity (EPQ) model (also known as economic manufacturing quantity model) is often used to deal with the non-instantaneous replenishment rate in order to minimize the expected production-inventory costs per unit time [2,3]. Despite the simplicity of EOQ and EPQ models, they are still applied industry-wide today [4,5]. The classical EPQ model assumes that all items produced are of perfect quality. However, in real-life production systems, due to process deterioration or other factors, generation of imperfect quality items is inevitable. Hence, a considerable amount of research has since been carried out to enhance the classic EPQ model by addressing the issue of defective items produced [6–12].
∗ Tel.: +886 4 2332 3000x4379; fax: +886 4 2374 2331. E-mail address: [email protected]. 0305-0548/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2009.03.016
Sometimes defective items produced randomly can be reworked and repaired to reduce overall production-inventory costs [13–18]. Examples of papers that studied the effect of rework on EPQ model are surveyed as follows. Hayek and Salameh [13] assumed that all of the defective items produced are repairable and derived an optimal operating policy for EPQ model under the effect of rework of all defective items. Chiu et al. [14] considered a finite production model with random defective rate, imperfect rework process, and scrap items produced in both regular and rework processes. They derived the optimal lot-size that minimizes total costs. Jamal et al. [15] studied the optimal manufacturing batch size with rework process at a single-stage production system. Cases of rework being completed within the same production cycle as well as rework being done after N cycles are examined. They developed mathematical models for each case, and derived total system costs and the optimal batch sizes accordingly. Another critical reliability factor that can be very disruptive when happening—particularly in a highly automated production environment is the breakdowns of production equipments. Groenevelt et al. [19] first studied two production control policies that deal with stochastic machine breakdowns. The first one assumes that the production of the interrupted lot is not resumed (called no resumption or NR policy) after a breakdown. The second policy considers that the production of the interrupted lot will be immediately resumed
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(called abort/resume or AR policy) after the breakdown is fixed and if the current on-hand inventory is below a certain threshold level. In their articles, both policies assume the repair time is negligible and they studied the effects of machine breakdowns and corrective maintenance on economic lot size decisions. Studies have since been carried out to address the issue of production system with machine breakdowns (see for instance [20–32]). Examples of papers that studied the effect of breakdowns on production systems are surveyed below. Abboud [20] considered an EMQ model with Poisson machine failures and random machine repair time. A simple approximation model was developed to describe the behavior of such systems, and specific formulations were derived for the cases where the repair times are exponential and constant. Kim et al. [22] considered an extended optimal lot sizing model with an unreliable machine. The production-inventory cost function was derived when the machine has an arbitrarily distributed time between failures and constant time to repair. Under the assumption that time between failures is exponentially distributed, the optimal lot size was determined and some properties of the proposed model were derived in their study. Chung [24] studied bounds for production lot sizing with machine breakdown. He obtained the upper and lower bounds of the optimal lot sizes for the aforementioned two extensions (i.e. NR and AR policies) to the EPQ model proposed by Groenevelt et al. [19]. Dohi et al. [26] studied the minimal repair policies for an economic manufacturing process. Two models with and without an infinite number of minimal repairs were formulated; and the optimal EMQ policies which minimize the expected costs were derived, respectively. Liu and Cao [28] analyzed a production-inventory model under the assumptions that demand follows a compound Poisson process and the machine is subject to random breakdowns. Boone et al. [29] investigated the impact of imperfect processes on the production run time. They built a model in an attempt to provide managers with guidelines to choose the appropriate production runtimes to cope with both the defective items and stoppages occurring due to machine breakdowns. Giri and Dohi [31] presented the exact formulation of stochastic EMQ model for an unreliable production system. Their EMQ model is formulated based on the net present value (NPV) approach and by taking limitation on the discount rate the traditional long-run average cost model is obtained. They also provided the criteria for the existence and uniqueness of the optimal production time and computational results showing that the optimal decision based on the NPV approach is superior to that based on the long-run average cost approach. Lin and Kroll [32] examined an EMQ model for an imperfect production process that is subject to random machine breakdowns. In their study, the time-to-shift and time-to-breakdown are two random variables that follow different exponential distributions. The objective of their paper was to find an optimal production lot size that minimizes the expected (long-run) total cost per unit time. Several models are investigated and a numerical approach is developed to obtain an optimal production lot size. This study considers the robust planning in optimization (with the aim of minimizing the expected production-inventory cost) for an imperfect production system that is subject to random breakdowns under abort/resume control policy and failure in rework. For the reason that little attention have been paid to the area of investigating the joint effects of random breakdowns (under AR policy) and failurein-rework on the practical production systems, this paper intends to serve this purpose. The proposed solution methodology/procedure for finding the optimal run time could be used by the practitioners in the manufacturing sector, for example, for production processes in printed circuit board assembly (PCBA), in plastic injection molding, or in other industries such as chemical, textiles, metal components, etc., where robust planning for unreliable production systems
is critical in terms of level of product quality and cost minimization.
2. Problem description, modeling, and formulation Assume that an imperfect manufacturing system has a production rate P and it is much larger than the demand rate . This production process may randomly produce x portion of defective items at a rate d. Hence, the production rate of defective items d can be expressed as d = Px. Also, during the production uptime, machine breakdowns may take place randomly and the abort/resume policy is adopted. Under such policy, when a breakdown takes place the machine is under repair immediately. The interrupted lot will be resumed right after the restoration of machine and the repair time is assumed to be constant. During the manufacturing process, all items are screened and the inspection cost per item is included in the unit production cost C. A portion of defective items is considered to be scrap and was discarded before the reworking of the other (1 − ) portion of the imperfect quality items. The rate of rework is P1 and it starts immediately after the regular production process ends. Failure-in-rework is assumed and a 1 portion (where 0 1 1) of the reworked items fails and becomes scrap. Let d1 denote the production rate of scrap items during the rework process (where subscript means it is for the scrap generation), then d1 can be expressed as: d1 = P1 1 . Cost parameters considered in the proposed model include setup cost K, holding cost h, repairing cost CR and holding cost h1 for each reworked item, disposal cost CS per scrap, and the cost for repairing and restoring machine M. Additional notation used in this study is listed below.
t1
t tr
t2 t3
H1 H] H2 H4 H5 H3 t2 t3 Q T
the optimal production time (i.e. production uptime) to be determined for the proposed EPQ model (in terms of years), production time before a random breakdown occurs (in years), time required for repairing the machine, a constant and it is not part of the production uptime t1 ; however, tr is part of the production cycle time T , time needed for reworking of defective items in the case of machine breakdown takes place, time needed for consuming all available perfect quality items in the case of breakdown takes place, number of breakdowns per year, the mean of a random variable following a Poisson distribution, the maximum level of on-hand inventory in units when regular production process ends, the maximum level of on-hand inventory in units when rework process finishes, the level of on-hand inventory when machine breakdown occurs, the level of on-hand inventory when machine is repaired and restored, the level of on-hand inventory when machine is restored and the remaining production uptime is accomplished, the maximum level of on-hand inventory when machine is restored and the reworking of defective items is completed, time required for reworking of defective items in the case of breakdown does not occur, time required for depleting all available perfect quality items in the case of breakdown does not occur, production lot size for each cycle, cycle length in the case of breakdown does not occur,
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
901
Fig. 2. On-hand inventory of defective items in EPQ model with random breakdown under the AR policy.
Fig. 1. On-hand inventory of perfect quality items in EPQ model with random breakdown under the AR policy.
T T TC1 (t) TC2 (t1 ) TCU(t1 )
cycle length in the case of machine breakdown takes place, cycle length whether a machine breakdown or not; it can be obtained from Eq. (26), the total inventory costs per cycle in the case of machine breakdown takes place, the total inventory costs per cycle in the case of machine breakdown does not occur, the total inventory costs per unit time whether a breakdown takes place or not.
To avoid shortages from occurring, the following two assumptions must be made: (1) the production rate of perfect quality items must always be greater than or equal to the sum of the demand rate and the production rate of defective items. That is: (P − d − ) 0 or (1 − x − /P) 0; (2) a safety stock of the amount up to tr must be provided to meet the demand required during the time tr (that the machine is being repaired) when a breakdown occurs at the early stage of a production run. Unit holding cost h2 applies to each safety stocks in computation of the total production-inventory costs. Let t denotes production time before a breakdown taking place. Then because during production uptime t1 machine breakdown may occur randomly; the following two situations must be investigated, respectively.
cycle length T : H2 = (P − d − )t
(1)
H4 = H2 − tr = (P − d − )t − g
(2)
H5 = H4 + (P − d − )(t1 − t)
(3)
H3 = H5 + t2 (P1 − d1 − )
(4)
t1 =
Q P
therefore Q = t1 P
T = t + tr + (t1 − t) + t2 + t3
(5) (6)
where tr = gand d = Px. The on-hand inventory of defective items when a random breakdown occurs during t1 is illustrated in Fig. 2. One notices that defective items produced during the production time t (before a breakdown takes place) is dt and total defective items produced during production uptime t1 can be computed as shown in Eq. (7). A portion (where 0 1) of the imperfect quality items is assumed to be scrap (totaled dt1 or t1 Px). The other (1− ) portion is reworked right after the production uptime t1 ends. The time needed for reworking of defective items t2 and the time needed for depleting all on-hand perfect quality items t3 can also be computed as shown in Eqs. (8) and (9): d · t1 = x · Q = x · t1 · P
2.1. EPQ model with random breakdown under AR policy and failure in rework
t2 =
The first case considers that when random breakdown occurs during production uptime t1 . That is when t < t1 . The abort/resume policy is adopted when random breakdown takes place. Under such a policy, production of the interrupted lot will be immediately resumed when the breakdown is fixed. The on-hand inventory level of perfect quality items is depicted in Fig. 1. For the following mathematical derivation, this paper employs the solution procedure that is similar to the one presented by Hayek and Salameh [13]. From Fig. 1, the following parameters can be obtained directly: the level of on-hand inventory when machine breakdown occurs H2 , the level of inventory at the time when machine is repaired H4 , the maximum level of on-hand inventory when machine is restored and the remaining production uptime is accomplished H5 , the level of on-hand inventory when the reworking of defective items are completed H3 , the production uptime t1 , and the
t3 =
dt1 (1 − ) Pxt1 (1 − ) = P1 P1 H3
= T − t1 − tr − t2
(7) (8)
(9)
Because the rework process is assumed to be imperfect, a portion
1 of the reworked items fails and becomes scrap items. Refer to Fig. 3, the production rate d1 in producing scrap items during rework process can be written as Eq. (10) and total scrap items produced when rework process finishes, can be computed as Eq. (11): d1 = P1 · 1
where 0 1 < 1
d1 · t2 = 1 · [(1 − ) · x · Q] = 1 · [(1 − )x · t · P]
(10) (11)
During the production uptime t1 there are (xQ) scrap items produced and in the rework process there are [1 (1 − )xQ] scrap items produced. Let denote the overall scrap rate out of the defective
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
Fig. 3. On-hand inventory level of scrap in EPQ model with random breakdown under the AR policy.
Fig. 4. On-hand inventory of perfect quality items in EPQ model without breakdowns [14].
items (xQ), then = [ + 1 (1 − )]. The cycle length can be rewritten as
H = H1 + (P1 − d1 − )t2
T =
t1 P[1 − · x]
t3 = (12)
Total production-inventory cost per cycle in the case of random machine breakdown takes place (under AR policy) during production uptime t1 is TC 1 (t1 ) = CPt1 + K + M + h2 (tr )T + CR Pt1 x(1 − ) P1 t2 (t2 ) + CS Pt1 x + h1 2 H2 H4 + H5 H2 + H4 +h (t) + (tr ) + (t1 − t) 2 2 2 H3 dt1 H3 + H5 (t2 ) + (t3 ) + dt(tr ) + (t1 ) + 2 2 2
H
= T − t1 − t2
(17) (18)
T = t1 + t2 + t3
(19)
d1 = P1 · 1
(20)
where 0 1 < 1
The total scrap items produced when the rework process ends, can be computed as d1 · t2 = 1 · [(1 − ) · x · Q] = 1 · [(1 − )x · t1 · P]
(21)
Because during the rework process, there are [xQ] scrap items produced. Thus the cycle length can be rewritten as (13)
The proportion x of defective items is assumed to be a random variable with a known probability density function. In order to take the randomness of defective rate into account, one can use the expected values of x in the inventory cost analysis. Substituting all related parameters from Eqs. (1) to (12) in TC1 (t1 ), one obtains the expected production-inventory cost per cycle E[TC1 (t)] for the case of EPQ model with random breakdown under the AR policy and failurein-rework as follows: E[TC 1 (t)] = K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) + CS P E[x] − hPg + hPgE[x]]t1 2 hP hP +(hPg) · t+ [1−2E[x]+2 E[x2 ]]− +hP E[x] 2 2 2 P 2 (1 − ) E[x2 ] + [h1 − h(1 − 1 )] t12 (14) 2P1 2.2. EPQ model without random breakdown The second case considers that breakdown does not occur during production uptime t1 ; that is txt1 . Analysis and result of this case is the same as what was presented by Chiu et al. [14], with the difference on decision variable used, i.e. t1 is utilized in this paper and the production lot size Q was used by [14]. Fig. 4 depicts the on-hand inventory level of perfect quality items when a random breakdown does not occur during production uptime. From Fig. 4, one can obtain the production uptime t1 as shown in Eq. (5), total defective items produced during the production uptime t1 are the same as shown in Eq. (7), and the following variables: H1 = (P − d − )t1
(15)
dt1 (1 − ) xQ(1 − ) t2 = = P1 P1
(16)
T=
t1 P[1 − · x]
(22)
The total inventory costs per cycle when machine breakdown does not occur, TC2 (t1 ) is TC 2 (t1 ) = CPt1 + K + h2 (tr )T + CR [t1 Px(1 − )] + CS (t1 Px) H1 + dt1 P 1 t2 (H1 + H) H +h (t1 ) + (t2 ) + (t3 ) + h1 (t2 ) 2 2 2 2 (23) Again, to take the randomness of defective items into account, one can use the expected values of x in the inventory cost analysis. Substituting all related parameters from Eqs. (15) to (22) in TC2 (t1 ), one obtains the expected total production-inventory cost per cycle in the case of machine breakdown does not occur, E[TC2 (t1 )] as follows: E[TC2 (t1 )] = K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) + CS P E[x]]t1 2 hP hP + [1 − 2E[x] + 2 E[x2 ]] − + hP E[x] 2 2 P 2 (1 − )2 E[x2 ] + [h1 − h(1 − 1 )] t12 2P1
(24)
Eq. (24) yields the same result as was given by Chiu et al. [14], if one substitutes t1 = Q/P in the corresponding cost function E[TC2 (t1 )] in [14]. 3. Determining the optimal production run time 3.1. Integration of cost functions for EPQ models with/without breakdowns The expected production-inventory cost functions E[TC1 (t1 )] and E[TC2 (t1 )] for the cases of EPQ models with/without random breakdowns have been derived in Sections 2.1 and 2.2, respectively. Owing
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
to the assumptions of stochastic breakdown and random scrap rate (xQ), the cycle length in the proposed EPQ model is not a constant. This study employs the renewal reward theorem to cope with the variable cycle length, that is to compute the expected value of cycle length E[T] first. Let f (t) denote the probability density function of random production time t before breakdown occurs and let F(t) be the cumulative density function of t. Then the expected productioninventory cost per unit time (whether a breakdown takes place or not), E[TCU(t1 )] is
t1 0
E[TCU(t1 )] =
E[TC1 (t1 )]f (t) dt +
∞ t1
E[TC2 (t1 )]f (t) dt
E[T]
(25)
Substituting T and T from Eqs. (12) and (22), in Eq. (25), one obtains the expected cycle length E[T] as displayed in E[T] =
t1 0
E[T ]f (t) dt +
∞ t1
E[T]f (t) dt =
P[1 − E(x)]t1
(26)
Now, substituting E[TC1 (t)], E[TC2 (t1 )], and E[T] from Eqs. (14), (24), and (26), in Eq. (25), one obtains the expected inventory cost per unit time, E[TCU(t1 )] as follows:
t1 0
E[TCU(t1 )] =
E[TC1 (t1 )]f (t)dt +
∞ t1
E[TC2 (t1 )]f (t)dt
903
3.2. Mathematical analysis In order to find the optimal production run time t1∗ , two theorems are proposed in this study. Let (t1 ) be the following: 2[K + · (1 − e−t1 )]
(t1 ) =
2 [Pt21 hgE(x)
(29)
+ (2 + t1 )](e−t1 )
Theorem 1. E[TCU(t1 )] is convex if 0 < t1 < (t1 )
(30)
Refer to Appendix B for proof. In order to minimize the expected overall costs E[TCU(t1 )], Eq. (30) must be satisfied. To find optimal value of t1∗ that yields minimum cost, one can set the first derivative of E[TCU(t1 )] (i.e. Eq. (28)) equal to 0 as follows: −K dE[TCU(t1 )] = + hgE(x)(e−t1 ) + 2[1 − E(x)] [1 − E(x)] Pt21 dt1 e−t1 M hg −(1 − e−t1 ) + + + =0 (31) P t1 t12
E[T] ⎧ ⎫ ⎧ ⎫ K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +C P E[x] − hPg + hPg E[x]]t + (hPg) · t ⎪ ⎪ 1 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ hP t ⎪ ⎪ 1 2 E[x2 ]] ⎪ ⎪ f (t) dt + [1 − 2 E[x] + ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 (1 − ) E[x2 ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP P ⎪ ⎪ ⎨ ⎬ 2 ⎪ ⎪ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP ⎪ ⎪ ⎪ ⎪ 2 2 ⎨ ⎬ ⎪ ⎪ +C P E[x]]t + [1 − 2 E[x] + E[x ]] 1 S ⎪ ⎪ ∞ ⎪ 2 ⎪ ⎪ f (t) dt ⎪ + t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP P (1 − ) E[x ] ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 = Pt1 [1 − E(x)]
(27)
This study assumes that random number of breakdowns per unit time follows a Poisson distribution, with mean equals to per unit time. Thus, time-to-breakdown should obey an exponential distribution with density function, f (t) = e−t , and cumulative density function F(t) = 1 − e−t . To solve the integration of mean-time-tobreakdown in the expected cost function E[TCU(t1 )] in Eq. (27), one obtains the following (see Appendix A for detailed computations): E[TCU(t1 )] =
{C + CR E[x](1 − ) + CS E[x] − hg} + h2 g [1 − E(x)] · t1 K + +hgE(x)(1−e−t1 ) + 2[1−E(x)] [1−E(x)] Pt1
+
M hg + P
(1 − e−t1 ) t1
(28)
= hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x] PE[x2 ](1 − )2 P1
[h1 − h(1 − 1 )]
1 [1 − E(x)] ×
2
+
−(1 − e−t1 ) t12
+
−K Pt21
+ hgE(x)(e−t1 ) +
e−t1
M hg + P
=0
t1
(32)
1 >0 [1 − E(x)] M hg −K −t1 + hg E(x)( e ) + + + therefore 2 P Pt21
since
×
where
+
or
−(1 − e−t1 ) t12
+
e−t1 t1
=0
(33)
To find bounds for the optimal production run time, we let = [M + Phg] and ∗ = t1U
2[K + · P
−1
]
(34)
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
∗ t1L = the positive root of ⎫ ⎧ ⎨ − ± 2 + [2hgE(x)P + P −1 −1 ] · 2K ⎬ −1 ⎭ ⎩ 2hgE(x)P + P
(35)
4. Numerical example
∗ < t∗ < t∗ . Theorem 2. t1L 1 1U
To prove Theorem 2, recall Eq. (33) and multiply it by (2Pt21 ):
(Pt21 ) + 2{−K + hgE(x)(Pt21 2 e−t1 ) + [M + Phg][−1 + e−t1 + t1 e−t1 ]} = 0
(36)
Let = [M + Phg], then Eq. (36) becomes {(P )t12 − 2(K + ) + 2(e−t1 ){[hgE(x)P ]t12 + ( · )t1 + }} = 0 2
(37) or {[2(e−t1 )[hgE(x)P ] + (P )]t12 + [2(e−t1 )( · )]t1 2
+ 2[(e−t1 − 1) − K ]} = 0
(38)
t1∗ = the positive root of ⎫ ⎧ ⎨ −(e−t1 ) ± (e−t1 )2 2 −[2(e−t1 )hgE(x)P+P −1 −1 ] · 2[(e−t1 −1)−K ] ⎬ ⎩
2(e−t1 )hgE(x)P +P
⎭
−1
(39) Eq. (38) can be rearranged as follows: 2(e−t1 ){[hgE(x)P ]t12 +( · )t1 +} = 2(K +)−(P )t12 2
therefore (e−t1 ) =
2(K + ) − (P )t12 2
2{[hgE(x)P
]t12
+ ( · )t1 + }
(40) (41)
Since e−t1 is the complement of the cumulative density function F(t1 ) = 1 − e−t1 and 0 F(t1 ) 1, hence 0 e−t1 1. Let zL and zU denote the bounds for e−t1 then from Eqs. (38) to (39) one obtains ∗ t1U = the positive root of ⎧ ⎫ ⎨−zL · ± z2 2 −[2zL hgE(x)P+P −1 −1 ] · 2[(zL −1)−K ]⎬ L
⎩
−1
2zL hgE(x)P +P
⎭ (42)
= the positive root of ⎧ ⎫ ⎨−zU · ± z2 2 −[2zU hgE(x)P+P −1 −1 ] · 2[(zU −1)−K ]⎬ U ⎩
−1
2zU hgE(x)P +P
⎭ (43)
∗ < t∗ < t∗ . and t1L 1 1U Further, because 0 e−t1 1 and if we let zL = 0 and zU = 1, then Eqs. (42) and (43) become
∗ t1U
Suppose the demand of a manufactured product is 4000 units per year and the production rate of this item is 10,000 units per year. During the production uptime, machine is subject to a random breakdown that follows a Poisson distribution with mean = 0.5 times per year. Abort/resume policy is used when a random breakdown takes place; under such policy, the interrupted lot will be resumed right after restoration of the machine. The percentage of defective items produced is x that follows a uniform distribution over the interval [0, 0.2]. Among the imperfect quality items, a portion = 0.05 is considered to be scrap and the other portion can be reworked. The rework process starts when regular production process ends, at a rate P1 = 600 units per year. Failure in repair is assumed, a portion 1 = 0.05 of the reworked items fails and becomes scrap. Additional parameters considered by this example are given as follows: g
then
∗ t1L
searching algorithm (see Appendix C) based on the existence of bounds for et1 and t1∗ .
=
−1
2[K + · P
]
∗ = the positive root of t1L ⎧ ⎫ ⎨ − ± 2 + [2hgE(x)P + P −1 −1 ] · 2K ⎬ −1 ⎩ ⎭ 2hgE(x)P + P
(34)
(35)
Hence, the proof of Theorem 2 is completed. Although the optimal run time t1∗ cannot be expressed in a closed form, it can be located through the use of a proposed recursive
M K C h h1 h2 CR CS
0.018 years, a constant time needed to repair and restore the machine, $500 repair cost for each breakdown, $450 for each production run, $2 per item, $0.6 per item per unit time, $0.8 per item reworked per unit time, $0.6 per item of safety stocks per unit time, $0.5 repaired cost for each item reworked, $0.3 disposal cost for each scrap item.
For convexity of E[TCU(t1 )] (i.e. Eq. (28)), using both upper and lower bounds of t1∗ in Eq. (30), one finds out that it holds. A further investigation utilizing different values to test for satisfaction of Eq. (30) is shown in Table 1 (see Appendix D). From Eqs. (34) and (28), ∗ =0.5011 years and E[TCU(t ∗ )]=$9712.74. Also, from one obtains t1U 1U ∗ = 0.2751 years and E[TCU(t ∗ )] = Eqs. (35) and (28) one obtains t1L 1L $9593.97 (see Table 1, in Appendix D). In this example, because one concludes that the expected cost function E[TCU(t1 )] is convex and ∗ , t∗ ] (based on optimal run time t1∗ falls within the interval of [t1L 1U Theorems 1 and 2 proved in Section 3). Then by using the proposed recursive searching algorithm (see Appendix C), one can locate the optimal run time t1∗ . The step-by-step iterations and their results are displayed in Table 2 (see Appendix D) for both = 0.5 and 1.0, respectively. As result, one notices that in this example (when = 0.5) the optimal run time t1∗ = 0.3140 years and the optimal expected costs per unit time E[TCU(t1∗ )] = $9583.82 as depicted in Fig. 5. The behavior of E[TCU(t1∗ )] with respect to (1/ ) and x is illustrated in Fig. 6. One notices that as the mean-time-between-breakdowns (1/ ) decreases, the value of E[TCU(t1∗ )] increases; also as defective rate x increases, the E[TCU(t1∗ )] goes up significantly too. Fig. 7 shows the behavior of the optimal expected cost function E[TCU(t1∗ )] with respect to defective rates and values. Variation of mean-time-between-breakdowns (1/ ) effects on the long-run expected costs E[TCU(t1∗ )] is depicted in Fig. 8. One notices that as approaches to 0 (i.e. the chance of breakdown is close to zero), the optimal expected costs per unit time E[TCU(t1∗ )] becomes $9453 [14]. 5. Concluding remarks Stochastic machine breakdowns and random defective rate are two common and inevitable reliability factors that trouble the production planners and practitioners most. To effectively manage and
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
905
Fig. 5. The behavior of E[TCU(t1 )] with respect to production run time t1 .
Fig. 7. The behavior of E[TCU(t1∗ )] with respect to x and .
Fig. 6. The behavior of E[TCU(t1∗ )] with respect to (1/ ) and x.
control the disruption and minimize overall production costs become the primary goals of most manufacturing firms. It is no wonder that determination of optimal lot-size (or run time) for such an imperfect production system has received attentions from researchers in recent decades [7–12,19–29]. Also, in consideration of further reduction of the total production-inventory costs, the rework process sometimes can be employed by the manufacturing firms to deal with certain repairable items [13–18]. This study examines an imperfect manufacturing system that is subject to random breakdowns under the abort/resume control policy and failure in rework. For the reason that little attention has been paid to the area of investigating the joint effects of machine breakdowns (under AR policy) and failure-in-rework on the practical production systems, this paper intends to serve this purpose. Upon accomplishment of the present work, a complete solution procedure (including the mathematical modeling, derivations of total cost functions for EPQ models with/without breakdown, integration of the long-run expected average cost function, proof of theorems on convexity and on bounds of the run time, development of a recursive searching algorithm) and a numerical example are
Fig. 8. Variation of 1/ effects on E[TCU(t1∗ )].
presented to confirm that the optimal run time for the proposed model is obtainable (see Sections 2–4 for details). It is noted that without an in-depth investigation and robust analysis on such a realistic system, one cannot obtain the optimal run time solution that minimizes the long-run average productioninventory costs, nor can one gain insight into effects of the system parameters (as depicted in Figs. 6–8). For future research, one interesting topic will be to investigate the effect of no resumption (NR) control policy on the same model.
Acknowledgments The author appreciates National Science Council (NSC) of Taiwan for supporting this research under Grant no. NSC-96-2416-H-324004. In addition, the author greatly expresses her thankfulness to the Editor Professor Larsen for his time and efforts and kind assistance in processing this paper and to three anonymous reviewers for their valuable comments and suggestions to the present paper.
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Appendix A
Let
Computational procedures of Eq. (28): recall Eq. (27) from = hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x] Section 3.1: ∞ t1 PE[x2 ](1 − )2 + [h1 − h(1 − 1 )] 0 E[TC1 (t1 )]f (t)dt + t1 E[TC2 (t1 )]f (t)dt P1 E[TCU(t1 )] = E[T] ⎧ ⎫ ⎧ ⎫ K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +C P E[x] − hPg + hPg E[x]]t + (hPg) · t ⎪ ⎪ 1 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ t1 ⎪ hP ⎪ ⎪ 2 2 ⎪ ⎪ f (t) dt + [1 − 2 E[x] + E[x ]] ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP (1 − ) E[x ] P ⎪ ⎪ ⎨ ⎬ ⎪ [h1 − h(1 − 1 )] t12 ⎪ + hP E[x] + ⎩ − ⎭ 2 2P1 ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP ⎪ ⎪ ⎪ ⎪ 2 E[x2 ]] ⎨ ⎬ ⎪ ⎪ +C P E[x]]t + [1 − 2 E[x] + ∞ 1 S ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ f (t) dt + ⎪ ⎪ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP (1 − ) E[x ] P ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 = (27) Pt1 [1 − E(x)]
(A.8)
Let
By substituting Eq. (A.8) in Eq. (A.7) one obtains the following:
1 = [CP + CR PE[x](1 − ) + CS PE[x]]
2 =
(A.1)
2
hP hP [1 − 2E[x] + 2 E[x2 ]] − + hP E[x] 2 2 P 2 (1 − )2 E[x2 ] + [h1 − h(1 − 1 )] 2P1
E[TCU(t1 )] =
{C+CR E[x](1−)+CS E[x]−hg} · t1 +h2 g+ [1−E(x)] 2[1−E(x)] K + hgE(x)(1 − e−t1 ) + [1 − E(x)] Pt1
(A.2)
+
M hg + P
(1 − e−t1 ) t1
(28)
then Eq. (27) becomes t K + h2 gP[1 − E[x]] · t1 + 1 t1 + 2 t12 + 01 [M − hPg[1 − E(x)] · t1 + (hPg)t]f (t)dt E[TCU(t1 )] = P[1 − E(x)]t1
t1
since 0
since 0
t1
−t1
f (t)dt = F(t1 ) = 1 − e
t · f (t)dt = −t1 e−t1 −
1
e−t1 +
(A.4) 1
Substituting Eqs. (A.4) and (A.5) in Eq. (A.3), one obtains 1 2 M(1 − e−t1 ) K E[TCU(t1 )] = + · + t1 + [1 − E(x)] Pt1 P P Pt1 hg −t1 −t1 −hg + hgE(x)(1 − e )+ (1 − e ) + h2 g t1 (A.6) Substituting Eqs. (A.1) and (A.2) in Eq. (A.6), one has
+
PE[x2 ](1 − )2
2[1 − E(x)]
PE[x2 ](1 − )2
P1
[h1 − h(1 − 1 )]
Theorem 1. E[TCU(t1 )] is convex if 0 < t1 < (t1 ). The first and the second derivatives of E[TCU(t1 )] are
+
hP[1 − 2E[x] + E[x ]] − h 2
P1
Appendix B
1
+2hE[x] +
= hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x]
dE[TCU(t1 )] = + 2[1 − E(x)] [1 − E(x)] dt1
{C + CR E[x](1 − ) + CS E[x] − hg} E[TCU(t1 )] = + h2 g [1 − E(x)] +
where
(A.5)
(A.3)
2
hg M + P
d2 E[TCU(t1 )]
[h1 − h(1 − 1 )] · t1
K + hgE(x)(1 − e−t1 ) [1 − E(x)] Pt1 hg (1 − e−t1 ) M + + P t1
2 dt1
= [1 − E(x)]
+
+ (A.7)
−(1 − e−t1 ) t12
2K Pt31
−K Pt21
+
+ hgE(x)(e−t1 )
e−t1
t1
(B.1)
+ hgE(x)(− e−t1 ) 2
2 e−t1 2e−t1 hg M 2(1 − e−t1 ) − − + · P t1 t12 t13 (B.2)
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
From Eq. (B.2), since the first term of the second derivative of E[TCU(t1 )] is greater than zero, it implies if ×
then
or if
2K
t13 d2 E[TCU(t1 )] 2
dt1
Pt31
−
2
Pt31 2(1 − e−t1 )
2K
M hg + P 2 −t1 e − >0 t1
+ hgE(x)(− e−t1 ) +
+
−
2e−t1 t12
(B.3)
M hg 2(1 − e−t1 ) 2 + · − hgE(x)( e−t1 ) P t13
2 e−t1 hg M 2e−t1 + >0 + · P t1 t12
or if [2K + 2(M + Phg) · (1 − e
)]
> (e−t1 ) · t1 [Pt21 hgE(x)2 + (M + Phg) · (2 + t1 )]
(B.5)
Let = [M + Phg] and (e−t1 ) · [Pt21 hgE(x) + (M + Phg) · (2 + t1 )] > 0; Eq. (B.5) becomes 2
therefore
therefore
[2K + 2 · (1 − e−t1 )] 2 [Pt21 hgE(x)
d2 E[TCU(t1 )]
if 0 < t1 <
2
dt1
+ · (2 + t1 )](e−t1 )
> t1
(B.6)
>0
[2K + 2 · (1 − e−t1 )] 2 [Pt21 hgE(x)
+ · (2 + t1 )](e−t1 )
= (t1 )
2(K + ) − (P )t12 2
2{[hgE(x)P ]t12 + ( · )t1 + }
As stated in Section 3, because e−t1 is the complement of cumulative density function, therefore, 0 e−t1 1. Let 2(K + ) − (P )t12 v(t1 ) = 2 2{[hgE(x)P ]t12 + ( · )t1 + } therefore 0 v(t1 ) 1
(B.4)
Multiplying both sides of Eq. (B.4) by P t13 , one obtains the following: −t1
As stated in Section 3, although the optimal run time t1∗ cannot be expressed in a closed form, it can be located through the use of the following searching algorithm based on the existence of bounds for e−t1 and t1∗ . Recall Eq. (41): (e−t1 ) =
>0
907
(B.7)
Hence, the proof of Theorem 1 is completed.
One can then use the following recursive searching techniques to find t1∗ : (1) Let v(t1 ) = 0 and v(t1 ) = 1 initially and compute the upper and lower bounds for t1∗ , respectively (i.e. to find the initial values of ∗ , t ∗ ]). [t1L 1U ∗ , t ∗ ] in e−t1 and calcu(2) Substitute the current values of [t1L 1U late the new bounds (denoted as zL and zU ) for e−t1 . Hence, zL < v(t1 ) < zU . (3) Let v(t1 ) = zL and v(t1 ) = zU and compute the new upper and lower bounds for t1∗ , respectively (i.e. to update the current val∗ , t ∗ ]). ues of [t1L 1U (4) Repeat steps 2 and 3, until there is no significant difference ∗ and t ∗ (or there is no significant difference in terms between t1L 1U of their effects on E[TCU(t1∗ )]). (5) Stop. The optimal production run time t1∗ is obtained. A step-by-step demonstration of the aforementioned recursive searching algorithm is presented in Table 2 (Appendix D) for two different values of 's (refer to Section 4 for details). Appendix D An investigation utilizing different values to test for satisfaction of Eq. (30) is shown in Table 1. The step-by-step iterations and their results are displayed in Table 2.
Appendix C A proposed recursive algorithm for searching t1∗ .
Table 1 ∗ ∗ ∗ ∗ , (t1U ), t1L , and (t1L ). Variations of effects on t1U
1/
∗ t1U
∗ (t1U )
∗ t1L
∗ (t1L )
12.00 11.00 10.00 9.00 8.00
0.08 0.09 0.10 0.11 0.13
0.4544 0.4546 0.4549 0.4551 0.4555
9.8088 7.2323 5.3946 4.0784 3.1324
0.0699 0.0755 0.0820 0.0896 0.0985
0.1973 0.2133 0.2319 0.2539 0.2804
7.00 6.00 5.00 4.00 3.00
0.14 0.17 0.20 0.25 0.33
0.4560 0.4566 0.4574 0.4587 0.4608
2.4514 1.9628 1.6171 1.3830 1.2471
0.1093 0.1224 0.1384 0.1583 0.1830
0.3126 0.3528 0.4043 0.4731 0.5715
2.00 1.00 0.50 . . . 0.01
0.50 1.00 2.00 . . . 100.00
0.4650 0.4773 0.5011 . . . 1.5907
1.2260 1.4567 1.9410 . . . 5.6050
0.2140 0.2526 0.2751 . . . 0.2994
0.7333 1.1011 1.6254 . . . 4.2876
Bold numbers represent results of the numerical example (Section 4) where = 0.5.
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
Table 2 Iterations of the recursive searching algorithm for optimal production run time t1∗ .
Iteration
zL = e−t1U
∗ t1U
zU = e−t1L
∗ t1L
Difference between ∗ ∗ and t1L t1U
∗ [U] E[TCU(t1U )]
∗ [L] E[TCU(t1L )]
Difference between [U] and [L]
0.5
Initial 2nd 3rd 4th 5th
0.000 0.778 0.847 0.854 0.855
0.5011 0.3333 0.3161 0.3142 0.3140
1.000 0.871 0.857 0.855 0.855
0.2751 0.3097 0.3135 0.3140 0.3140
0.226 0.024 0.003 0.000 0.0000
$9712.74 $9585.87 $9583.85 $9583.83 $9583.82
$9593.97 $9583.93 $9583.82 $9583.82 $9583.82
$118.77 $1.94 $0.03 $0.01 $0.00
1.0
Initial 2nd 3rd 4th 5th
0.000 0.620 0.709 0.723 0.726
0.4773 0.3444 0.3240 0.3206 0.3200
1.000 0.777 0.735 0.728 0.727
0.2526 0.3079 0.3178 0.3195 0.3198
0.225 0.036 0.006 0.001 0.0002
$9753.66 $9663.75 $9660.72 $9660.63 $9660.62
$9692.58 $9661.45 $9660.65 $9660.62 $9660.62
$61.08 $2.30 $0.07 $0.01 $0.00
Bold numbers represent results of the numerical example (Section 4) where = 0.5.
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Contents lists available at ScienceDirect
Computers & Operations Research journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c o r
Robust planning in optimization for production system subject to random machine breakdown and failure in rework Singa Wang Chiu ∗ Department of Business Administration, Chaoyang University of Technology, 168 Jifong East Road, Wufong, Taichung 413, Taiwan
A R T I C L E
I N F O
Available online 31 March 2009 Keywords: Production Breakdowns Failure in rework Abort/resume policy Robust planning
A B S T R A C T
This study is concerned with robust planning in optimization, specifically in determining the optimal run time for production system that is subject to random breakdowns under abort/resume (AR) control policy and failure in rework. In most real-life production processes, generation of defective items and breakdowns of manufacturing equipment are inevitable. In this study, random defective rate is assumed and all manufactured items are screened. The perfect quality, reworkable and scrap items are identified and separated; failure-in-rework is assumed. The system is also subject to random machine breakdown; and when it occurs, the AR policy is adopted. Under such policy, the production of the interrupted lot will be immediately resumed when the machine is restored. Mathematical modeling and derivation of the production-inventory cost functions for both systems with/without breakdowns are presented. The renewal reward theorem is used to cope with the variable cycle length when integrating cost functions. The long-run average cost per unit time is obtained. Theorems on convexity and on bounds of production run time are proposed and proved. A recursive searching algorithm is developed for locating the optimal run time that minimizes the expected production-inventory costs. A numerical example with sensitivity analysis is provided to give insight into the optimal operational control of such an unreliable system. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The economic order quantity (EOQ) model was first introduced several decades ago [1]. It is a mathematical model that can be used to derive an optimal order quantity that assists corporations in minimizing total inventory costs. In manufacturing sector, when products are produced in-house instead of being acquired from outside suppliers, the economic production quantity (EPQ) model (also known as economic manufacturing quantity model) is often used to deal with the non-instantaneous replenishment rate in order to minimize the expected production-inventory costs per unit time [2,3]. Despite the simplicity of EOQ and EPQ models, they are still applied industry-wide today [4,5]. The classical EPQ model assumes that all items produced are of perfect quality. However, in real-life production systems, due to process deterioration or other factors, generation of imperfect quality items is inevitable. Hence, a considerable amount of research has since been carried out to enhance the classic EPQ model by addressing the issue of defective items produced [6–12].
∗ Tel.: +886 4 2332 3000x4379; fax: +886 4 2374 2331. E-mail address: [email protected]. 0305-0548/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2009.03.016
Sometimes defective items produced randomly can be reworked and repaired to reduce overall production-inventory costs [13–18]. Examples of papers that studied the effect of rework on EPQ model are surveyed as follows. Hayek and Salameh [13] assumed that all of the defective items produced are repairable and derived an optimal operating policy for EPQ model under the effect of rework of all defective items. Chiu et al. [14] considered a finite production model with random defective rate, imperfect rework process, and scrap items produced in both regular and rework processes. They derived the optimal lot-size that minimizes total costs. Jamal et al. [15] studied the optimal manufacturing batch size with rework process at a single-stage production system. Cases of rework being completed within the same production cycle as well as rework being done after N cycles are examined. They developed mathematical models for each case, and derived total system costs and the optimal batch sizes accordingly. Another critical reliability factor that can be very disruptive when happening—particularly in a highly automated production environment is the breakdowns of production equipments. Groenevelt et al. [19] first studied two production control policies that deal with stochastic machine breakdowns. The first one assumes that the production of the interrupted lot is not resumed (called no resumption or NR policy) after a breakdown. The second policy considers that the production of the interrupted lot will be immediately resumed
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
(called abort/resume or AR policy) after the breakdown is fixed and if the current on-hand inventory is below a certain threshold level. In their articles, both policies assume the repair time is negligible and they studied the effects of machine breakdowns and corrective maintenance on economic lot size decisions. Studies have since been carried out to address the issue of production system with machine breakdowns (see for instance [20–32]). Examples of papers that studied the effect of breakdowns on production systems are surveyed below. Abboud [20] considered an EMQ model with Poisson machine failures and random machine repair time. A simple approximation model was developed to describe the behavior of such systems, and specific formulations were derived for the cases where the repair times are exponential and constant. Kim et al. [22] considered an extended optimal lot sizing model with an unreliable machine. The production-inventory cost function was derived when the machine has an arbitrarily distributed time between failures and constant time to repair. Under the assumption that time between failures is exponentially distributed, the optimal lot size was determined and some properties of the proposed model were derived in their study. Chung [24] studied bounds for production lot sizing with machine breakdown. He obtained the upper and lower bounds of the optimal lot sizes for the aforementioned two extensions (i.e. NR and AR policies) to the EPQ model proposed by Groenevelt et al. [19]. Dohi et al. [26] studied the minimal repair policies for an economic manufacturing process. Two models with and without an infinite number of minimal repairs were formulated; and the optimal EMQ policies which minimize the expected costs were derived, respectively. Liu and Cao [28] analyzed a production-inventory model under the assumptions that demand follows a compound Poisson process and the machine is subject to random breakdowns. Boone et al. [29] investigated the impact of imperfect processes on the production run time. They built a model in an attempt to provide managers with guidelines to choose the appropriate production runtimes to cope with both the defective items and stoppages occurring due to machine breakdowns. Giri and Dohi [31] presented the exact formulation of stochastic EMQ model for an unreliable production system. Their EMQ model is formulated based on the net present value (NPV) approach and by taking limitation on the discount rate the traditional long-run average cost model is obtained. They also provided the criteria for the existence and uniqueness of the optimal production time and computational results showing that the optimal decision based on the NPV approach is superior to that based on the long-run average cost approach. Lin and Kroll [32] examined an EMQ model for an imperfect production process that is subject to random machine breakdowns. In their study, the time-to-shift and time-to-breakdown are two random variables that follow different exponential distributions. The objective of their paper was to find an optimal production lot size that minimizes the expected (long-run) total cost per unit time. Several models are investigated and a numerical approach is developed to obtain an optimal production lot size. This study considers the robust planning in optimization (with the aim of minimizing the expected production-inventory cost) for an imperfect production system that is subject to random breakdowns under abort/resume control policy and failure in rework. For the reason that little attention have been paid to the area of investigating the joint effects of random breakdowns (under AR policy) and failurein-rework on the practical production systems, this paper intends to serve this purpose. The proposed solution methodology/procedure for finding the optimal run time could be used by the practitioners in the manufacturing sector, for example, for production processes in printed circuit board assembly (PCBA), in plastic injection molding, or in other industries such as chemical, textiles, metal components, etc., where robust planning for unreliable production systems
is critical in terms of level of product quality and cost minimization.
2. Problem description, modeling, and formulation Assume that an imperfect manufacturing system has a production rate P and it is much larger than the demand rate . This production process may randomly produce x portion of defective items at a rate d. Hence, the production rate of defective items d can be expressed as d = Px. Also, during the production uptime, machine breakdowns may take place randomly and the abort/resume policy is adopted. Under such policy, when a breakdown takes place the machine is under repair immediately. The interrupted lot will be resumed right after the restoration of machine and the repair time is assumed to be constant. During the manufacturing process, all items are screened and the inspection cost per item is included in the unit production cost C. A portion of defective items is considered to be scrap and was discarded before the reworking of the other (1 − ) portion of the imperfect quality items. The rate of rework is P1 and it starts immediately after the regular production process ends. Failure-in-rework is assumed and a 1 portion (where 0 1 1) of the reworked items fails and becomes scrap. Let d1 denote the production rate of scrap items during the rework process (where subscript means it is for the scrap generation), then d1 can be expressed as: d1 = P1 1 . Cost parameters considered in the proposed model include setup cost K, holding cost h, repairing cost CR and holding cost h1 for each reworked item, disposal cost CS per scrap, and the cost for repairing and restoring machine M. Additional notation used in this study is listed below.
t1
t tr
t2 t3
H1 H] H2 H4 H5 H3 t2 t3 Q T
the optimal production time (i.e. production uptime) to be determined for the proposed EPQ model (in terms of years), production time before a random breakdown occurs (in years), time required for repairing the machine, a constant and it is not part of the production uptime t1 ; however, tr is part of the production cycle time T , time needed for reworking of defective items in the case of machine breakdown takes place, time needed for consuming all available perfect quality items in the case of breakdown takes place, number of breakdowns per year, the mean of a random variable following a Poisson distribution, the maximum level of on-hand inventory in units when regular production process ends, the maximum level of on-hand inventory in units when rework process finishes, the level of on-hand inventory when machine breakdown occurs, the level of on-hand inventory when machine is repaired and restored, the level of on-hand inventory when machine is restored and the remaining production uptime is accomplished, the maximum level of on-hand inventory when machine is restored and the reworking of defective items is completed, time required for reworking of defective items in the case of breakdown does not occur, time required for depleting all available perfect quality items in the case of breakdown does not occur, production lot size for each cycle, cycle length in the case of breakdown does not occur,
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
901
Fig. 2. On-hand inventory of defective items in EPQ model with random breakdown under the AR policy.
Fig. 1. On-hand inventory of perfect quality items in EPQ model with random breakdown under the AR policy.
T T TC1 (t) TC2 (t1 ) TCU(t1 )
cycle length in the case of machine breakdown takes place, cycle length whether a machine breakdown or not; it can be obtained from Eq. (26), the total inventory costs per cycle in the case of machine breakdown takes place, the total inventory costs per cycle in the case of machine breakdown does not occur, the total inventory costs per unit time whether a breakdown takes place or not.
To avoid shortages from occurring, the following two assumptions must be made: (1) the production rate of perfect quality items must always be greater than or equal to the sum of the demand rate and the production rate of defective items. That is: (P − d − ) 0 or (1 − x − /P) 0; (2) a safety stock of the amount up to tr must be provided to meet the demand required during the time tr (that the machine is being repaired) when a breakdown occurs at the early stage of a production run. Unit holding cost h2 applies to each safety stocks in computation of the total production-inventory costs. Let t denotes production time before a breakdown taking place. Then because during production uptime t1 machine breakdown may occur randomly; the following two situations must be investigated, respectively.
cycle length T : H2 = (P − d − )t
(1)
H4 = H2 − tr = (P − d − )t − g
(2)
H5 = H4 + (P − d − )(t1 − t)
(3)
H3 = H5 + t2 (P1 − d1 − )
(4)
t1 =
Q P
therefore Q = t1 P
T = t + tr + (t1 − t) + t2 + t3
(5) (6)
where tr = gand d = Px. The on-hand inventory of defective items when a random breakdown occurs during t1 is illustrated in Fig. 2. One notices that defective items produced during the production time t (before a breakdown takes place) is dt and total defective items produced during production uptime t1 can be computed as shown in Eq. (7). A portion (where 0 1) of the imperfect quality items is assumed to be scrap (totaled dt1 or t1 Px). The other (1− ) portion is reworked right after the production uptime t1 ends. The time needed for reworking of defective items t2 and the time needed for depleting all on-hand perfect quality items t3 can also be computed as shown in Eqs. (8) and (9): d · t1 = x · Q = x · t1 · P
2.1. EPQ model with random breakdown under AR policy and failure in rework
t2 =
The first case considers that when random breakdown occurs during production uptime t1 . That is when t < t1 . The abort/resume policy is adopted when random breakdown takes place. Under such a policy, production of the interrupted lot will be immediately resumed when the breakdown is fixed. The on-hand inventory level of perfect quality items is depicted in Fig. 1. For the following mathematical derivation, this paper employs the solution procedure that is similar to the one presented by Hayek and Salameh [13]. From Fig. 1, the following parameters can be obtained directly: the level of on-hand inventory when machine breakdown occurs H2 , the level of inventory at the time when machine is repaired H4 , the maximum level of on-hand inventory when machine is restored and the remaining production uptime is accomplished H5 , the level of on-hand inventory when the reworking of defective items are completed H3 , the production uptime t1 , and the
t3 =
dt1 (1 − ) Pxt1 (1 − ) = P1 P1 H3
= T − t1 − tr − t2
(7) (8)
(9)
Because the rework process is assumed to be imperfect, a portion
1 of the reworked items fails and becomes scrap items. Refer to Fig. 3, the production rate d1 in producing scrap items during rework process can be written as Eq. (10) and total scrap items produced when rework process finishes, can be computed as Eq. (11): d1 = P1 · 1
where 0 1 < 1
d1 · t2 = 1 · [(1 − ) · x · Q] = 1 · [(1 − )x · t · P]
(10) (11)
During the production uptime t1 there are (xQ) scrap items produced and in the rework process there are [1 (1 − )xQ] scrap items produced. Let denote the overall scrap rate out of the defective
902
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
Fig. 3. On-hand inventory level of scrap in EPQ model with random breakdown under the AR policy.
Fig. 4. On-hand inventory of perfect quality items in EPQ model without breakdowns [14].
items (xQ), then = [ + 1 (1 − )]. The cycle length can be rewritten as
H = H1 + (P1 − d1 − )t2
T =
t1 P[1 − · x]
t3 = (12)
Total production-inventory cost per cycle in the case of random machine breakdown takes place (under AR policy) during production uptime t1 is TC 1 (t1 ) = CPt1 + K + M + h2 (tr )T + CR Pt1 x(1 − ) P1 t2 (t2 ) + CS Pt1 x + h1 2 H2 H4 + H5 H2 + H4 +h (t) + (tr ) + (t1 − t) 2 2 2 H3 dt1 H3 + H5 (t2 ) + (t3 ) + dt(tr ) + (t1 ) + 2 2 2
H
= T − t1 − t2
(17) (18)
T = t1 + t2 + t3
(19)
d1 = P1 · 1
(20)
where 0 1 < 1
The total scrap items produced when the rework process ends, can be computed as d1 · t2 = 1 · [(1 − ) · x · Q] = 1 · [(1 − )x · t1 · P]
(21)
Because during the rework process, there are [xQ] scrap items produced. Thus the cycle length can be rewritten as (13)
The proportion x of defective items is assumed to be a random variable with a known probability density function. In order to take the randomness of defective rate into account, one can use the expected values of x in the inventory cost analysis. Substituting all related parameters from Eqs. (1) to (12) in TC1 (t1 ), one obtains the expected production-inventory cost per cycle E[TC1 (t)] for the case of EPQ model with random breakdown under the AR policy and failurein-rework as follows: E[TC 1 (t)] = K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) + CS P E[x] − hPg + hPgE[x]]t1 2 hP hP +(hPg) · t+ [1−2E[x]+2 E[x2 ]]− +hP E[x] 2 2 2 P 2 (1 − ) E[x2 ] + [h1 − h(1 − 1 )] t12 (14) 2P1 2.2. EPQ model without random breakdown The second case considers that breakdown does not occur during production uptime t1 ; that is txt1 . Analysis and result of this case is the same as what was presented by Chiu et al. [14], with the difference on decision variable used, i.e. t1 is utilized in this paper and the production lot size Q was used by [14]. Fig. 4 depicts the on-hand inventory level of perfect quality items when a random breakdown does not occur during production uptime. From Fig. 4, one can obtain the production uptime t1 as shown in Eq. (5), total defective items produced during the production uptime t1 are the same as shown in Eq. (7), and the following variables: H1 = (P − d − )t1
(15)
dt1 (1 − ) xQ(1 − ) t2 = = P1 P1
(16)
T=
t1 P[1 − · x]
(22)
The total inventory costs per cycle when machine breakdown does not occur, TC2 (t1 ) is TC 2 (t1 ) = CPt1 + K + h2 (tr )T + CR [t1 Px(1 − )] + CS (t1 Px) H1 + dt1 P 1 t2 (H1 + H) H +h (t1 ) + (t2 ) + (t3 ) + h1 (t2 ) 2 2 2 2 (23) Again, to take the randomness of defective items into account, one can use the expected values of x in the inventory cost analysis. Substituting all related parameters from Eqs. (15) to (22) in TC2 (t1 ), one obtains the expected total production-inventory cost per cycle in the case of machine breakdown does not occur, E[TC2 (t1 )] as follows: E[TC2 (t1 )] = K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) + CS P E[x]]t1 2 hP hP + [1 − 2E[x] + 2 E[x2 ]] − + hP E[x] 2 2 P 2 (1 − )2 E[x2 ] + [h1 − h(1 − 1 )] t12 2P1
(24)
Eq. (24) yields the same result as was given by Chiu et al. [14], if one substitutes t1 = Q/P in the corresponding cost function E[TC2 (t1 )] in [14]. 3. Determining the optimal production run time 3.1. Integration of cost functions for EPQ models with/without breakdowns The expected production-inventory cost functions E[TC1 (t1 )] and E[TC2 (t1 )] for the cases of EPQ models with/without random breakdowns have been derived in Sections 2.1 and 2.2, respectively. Owing
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
to the assumptions of stochastic breakdown and random scrap rate (xQ), the cycle length in the proposed EPQ model is not a constant. This study employs the renewal reward theorem to cope with the variable cycle length, that is to compute the expected value of cycle length E[T] first. Let f (t) denote the probability density function of random production time t before breakdown occurs and let F(t) be the cumulative density function of t. Then the expected productioninventory cost per unit time (whether a breakdown takes place or not), E[TCU(t1 )] is
t1 0
E[TCU(t1 )] =
E[TC1 (t1 )]f (t) dt +
∞ t1
E[TC2 (t1 )]f (t) dt
E[T]
(25)
Substituting T and T from Eqs. (12) and (22), in Eq. (25), one obtains the expected cycle length E[T] as displayed in E[T] =
t1 0
E[T ]f (t) dt +
∞ t1
E[T]f (t) dt =
P[1 − E(x)]t1
(26)
Now, substituting E[TC1 (t)], E[TC2 (t1 )], and E[T] from Eqs. (14), (24), and (26), in Eq. (25), one obtains the expected inventory cost per unit time, E[TCU(t1 )] as follows:
t1 0
E[TCU(t1 )] =
E[TC1 (t1 )]f (t)dt +
∞ t1
E[TC2 (t1 )]f (t)dt
903
3.2. Mathematical analysis In order to find the optimal production run time t1∗ , two theorems are proposed in this study. Let (t1 ) be the following: 2[K + · (1 − e−t1 )]
(t1 ) =
2 [Pt21 hgE(x)
(29)
+ (2 + t1 )](e−t1 )
Theorem 1. E[TCU(t1 )] is convex if 0 < t1 < (t1 )
(30)
Refer to Appendix B for proof. In order to minimize the expected overall costs E[TCU(t1 )], Eq. (30) must be satisfied. To find optimal value of t1∗ that yields minimum cost, one can set the first derivative of E[TCU(t1 )] (i.e. Eq. (28)) equal to 0 as follows: −K dE[TCU(t1 )] = + hgE(x)(e−t1 ) + 2[1 − E(x)] [1 − E(x)] Pt21 dt1 e−t1 M hg −(1 − e−t1 ) + + + =0 (31) P t1 t12
E[T] ⎧ ⎫ ⎧ ⎫ K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +C P E[x] − hPg + hPg E[x]]t + (hPg) · t ⎪ ⎪ 1 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ hP t ⎪ ⎪ 1 2 E[x2 ]] ⎪ ⎪ f (t) dt + [1 − 2 E[x] + ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 (1 − ) E[x2 ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP P ⎪ ⎪ ⎨ ⎬ 2 ⎪ ⎪ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP ⎪ ⎪ ⎪ ⎪ 2 2 ⎨ ⎬ ⎪ ⎪ +C P E[x]]t + [1 − 2 E[x] + E[x ]] 1 S ⎪ ⎪ ∞ ⎪ 2 ⎪ ⎪ f (t) dt ⎪ + t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP P (1 − ) E[x ] ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 = Pt1 [1 − E(x)]
(27)
This study assumes that random number of breakdowns per unit time follows a Poisson distribution, with mean equals to per unit time. Thus, time-to-breakdown should obey an exponential distribution with density function, f (t) = e−t , and cumulative density function F(t) = 1 − e−t . To solve the integration of mean-time-tobreakdown in the expected cost function E[TCU(t1 )] in Eq. (27), one obtains the following (see Appendix A for detailed computations): E[TCU(t1 )] =
{C + CR E[x](1 − ) + CS E[x] − hg} + h2 g [1 − E(x)] · t1 K + +hgE(x)(1−e−t1 ) + 2[1−E(x)] [1−E(x)] Pt1
+
M hg + P
(1 − e−t1 ) t1
(28)
= hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x] PE[x2 ](1 − )2 P1
[h1 − h(1 − 1 )]
1 [1 − E(x)] ×
2
+
−(1 − e−t1 ) t12
+
−K Pt21
+ hgE(x)(e−t1 ) +
e−t1
M hg + P
=0
t1
(32)
1 >0 [1 − E(x)] M hg −K −t1 + hg E(x)( e ) + + + therefore 2 P Pt21
since
×
where
+
or
−(1 − e−t1 ) t12
+
e−t1 t1
=0
(33)
To find bounds for the optimal production run time, we let = [M + Phg] and ∗ = t1U
2[K + · P
−1
]
(34)
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
∗ t1L = the positive root of ⎫ ⎧ ⎨ − ± 2 + [2hgE(x)P + P −1 −1 ] · 2K ⎬ −1 ⎭ ⎩ 2hgE(x)P + P
(35)
4. Numerical example
∗ < t∗ < t∗ . Theorem 2. t1L 1 1U
To prove Theorem 2, recall Eq. (33) and multiply it by (2Pt21 ):
(Pt21 ) + 2{−K + hgE(x)(Pt21 2 e−t1 ) + [M + Phg][−1 + e−t1 + t1 e−t1 ]} = 0
(36)
Let = [M + Phg], then Eq. (36) becomes {(P )t12 − 2(K + ) + 2(e−t1 ){[hgE(x)P ]t12 + ( · )t1 + }} = 0 2
(37) or {[2(e−t1 )[hgE(x)P ] + (P )]t12 + [2(e−t1 )( · )]t1 2
+ 2[(e−t1 − 1) − K ]} = 0
(38)
t1∗ = the positive root of ⎫ ⎧ ⎨ −(e−t1 ) ± (e−t1 )2 2 −[2(e−t1 )hgE(x)P+P −1 −1 ] · 2[(e−t1 −1)−K ] ⎬ ⎩
2(e−t1 )hgE(x)P +P
⎭
−1
(39) Eq. (38) can be rearranged as follows: 2(e−t1 ){[hgE(x)P ]t12 +( · )t1 +} = 2(K +)−(P )t12 2
therefore (e−t1 ) =
2(K + ) − (P )t12 2
2{[hgE(x)P
]t12
+ ( · )t1 + }
(40) (41)
Since e−t1 is the complement of the cumulative density function F(t1 ) = 1 − e−t1 and 0 F(t1 ) 1, hence 0 e−t1 1. Let zL and zU denote the bounds for e−t1 then from Eqs. (38) to (39) one obtains ∗ t1U = the positive root of ⎧ ⎫ ⎨−zL · ± z2 2 −[2zL hgE(x)P+P −1 −1 ] · 2[(zL −1)−K ]⎬ L
⎩
−1
2zL hgE(x)P +P
⎭ (42)
= the positive root of ⎧ ⎫ ⎨−zU · ± z2 2 −[2zU hgE(x)P+P −1 −1 ] · 2[(zU −1)−K ]⎬ U ⎩
−1
2zU hgE(x)P +P
⎭ (43)
∗ < t∗ < t∗ . and t1L 1 1U Further, because 0 e−t1 1 and if we let zL = 0 and zU = 1, then Eqs. (42) and (43) become
∗ t1U
Suppose the demand of a manufactured product is 4000 units per year and the production rate of this item is 10,000 units per year. During the production uptime, machine is subject to a random breakdown that follows a Poisson distribution with mean = 0.5 times per year. Abort/resume policy is used when a random breakdown takes place; under such policy, the interrupted lot will be resumed right after restoration of the machine. The percentage of defective items produced is x that follows a uniform distribution over the interval [0, 0.2]. Among the imperfect quality items, a portion = 0.05 is considered to be scrap and the other portion can be reworked. The rework process starts when regular production process ends, at a rate P1 = 600 units per year. Failure in repair is assumed, a portion 1 = 0.05 of the reworked items fails and becomes scrap. Additional parameters considered by this example are given as follows: g
then
∗ t1L
searching algorithm (see Appendix C) based on the existence of bounds for et1 and t1∗ .
=
−1
2[K + · P
]
∗ = the positive root of t1L ⎧ ⎫ ⎨ − ± 2 + [2hgE(x)P + P −1 −1 ] · 2K ⎬ −1 ⎩ ⎭ 2hgE(x)P + P
(34)
(35)
Hence, the proof of Theorem 2 is completed. Although the optimal run time t1∗ cannot be expressed in a closed form, it can be located through the use of a proposed recursive
M K C h h1 h2 CR CS
0.018 years, a constant time needed to repair and restore the machine, $500 repair cost for each breakdown, $450 for each production run, $2 per item, $0.6 per item per unit time, $0.8 per item reworked per unit time, $0.6 per item of safety stocks per unit time, $0.5 repaired cost for each item reworked, $0.3 disposal cost for each scrap item.
For convexity of E[TCU(t1 )] (i.e. Eq. (28)), using both upper and lower bounds of t1∗ in Eq. (30), one finds out that it holds. A further investigation utilizing different values to test for satisfaction of Eq. (30) is shown in Table 1 (see Appendix D). From Eqs. (34) and (28), ∗ =0.5011 years and E[TCU(t ∗ )]=$9712.74. Also, from one obtains t1U 1U ∗ = 0.2751 years and E[TCU(t ∗ )] = Eqs. (35) and (28) one obtains t1L 1L $9593.97 (see Table 1, in Appendix D). In this example, because one concludes that the expected cost function E[TCU(t1 )] is convex and ∗ , t∗ ] (based on optimal run time t1∗ falls within the interval of [t1L 1U Theorems 1 and 2 proved in Section 3). Then by using the proposed recursive searching algorithm (see Appendix C), one can locate the optimal run time t1∗ . The step-by-step iterations and their results are displayed in Table 2 (see Appendix D) for both = 0.5 and 1.0, respectively. As result, one notices that in this example (when = 0.5) the optimal run time t1∗ = 0.3140 years and the optimal expected costs per unit time E[TCU(t1∗ )] = $9583.82 as depicted in Fig. 5. The behavior of E[TCU(t1∗ )] with respect to (1/ ) and x is illustrated in Fig. 6. One notices that as the mean-time-between-breakdowns (1/ ) decreases, the value of E[TCU(t1∗ )] increases; also as defective rate x increases, the E[TCU(t1∗ )] goes up significantly too. Fig. 7 shows the behavior of the optimal expected cost function E[TCU(t1∗ )] with respect to defective rates and values. Variation of mean-time-between-breakdowns (1/ ) effects on the long-run expected costs E[TCU(t1∗ )] is depicted in Fig. 8. One notices that as approaches to 0 (i.e. the chance of breakdown is close to zero), the optimal expected costs per unit time E[TCU(t1∗ )] becomes $9453 [14]. 5. Concluding remarks Stochastic machine breakdowns and random defective rate are two common and inevitable reliability factors that trouble the production planners and practitioners most. To effectively manage and
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
905
Fig. 5. The behavior of E[TCU(t1 )] with respect to production run time t1 .
Fig. 7. The behavior of E[TCU(t1∗ )] with respect to x and .
Fig. 6. The behavior of E[TCU(t1∗ )] with respect to (1/ ) and x.
control the disruption and minimize overall production costs become the primary goals of most manufacturing firms. It is no wonder that determination of optimal lot-size (or run time) for such an imperfect production system has received attentions from researchers in recent decades [7–12,19–29]. Also, in consideration of further reduction of the total production-inventory costs, the rework process sometimes can be employed by the manufacturing firms to deal with certain repairable items [13–18]. This study examines an imperfect manufacturing system that is subject to random breakdowns under the abort/resume control policy and failure in rework. For the reason that little attention has been paid to the area of investigating the joint effects of machine breakdowns (under AR policy) and failure-in-rework on the practical production systems, this paper intends to serve this purpose. Upon accomplishment of the present work, a complete solution procedure (including the mathematical modeling, derivations of total cost functions for EPQ models with/without breakdown, integration of the long-run expected average cost function, proof of theorems on convexity and on bounds of the run time, development of a recursive searching algorithm) and a numerical example are
Fig. 8. Variation of 1/ effects on E[TCU(t1∗ )].
presented to confirm that the optimal run time for the proposed model is obtainable (see Sections 2–4 for details). It is noted that without an in-depth investigation and robust analysis on such a realistic system, one cannot obtain the optimal run time solution that minimizes the long-run average productioninventory costs, nor can one gain insight into effects of the system parameters (as depicted in Figs. 6–8). For future research, one interesting topic will be to investigate the effect of no resumption (NR) control policy on the same model.
Acknowledgments The author appreciates National Science Council (NSC) of Taiwan for supporting this research under Grant no. NSC-96-2416-H-324004. In addition, the author greatly expresses her thankfulness to the Editor Professor Larsen for his time and efforts and kind assistance in processing this paper and to three anonymous reviewers for their valuable comments and suggestions to the present paper.
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
Appendix A
Let
Computational procedures of Eq. (28): recall Eq. (27) from = hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x] Section 3.1: ∞ t1 PE[x2 ](1 − )2 + [h1 − h(1 − 1 )] 0 E[TC1 (t1 )]f (t)dt + t1 E[TC2 (t1 )]f (t)dt P1 E[TCU(t1 )] = E[T] ⎧ ⎫ ⎧ ⎫ K + M + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +C P E[x] − hPg + hPg E[x]]t + (hPg) · t ⎪ ⎪ 1 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ t1 ⎪ hP ⎪ ⎪ 2 2 ⎪ ⎪ f (t) dt + [1 − 2 E[x] + E[x ]] ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP (1 − ) E[x ] P ⎪ ⎪ ⎨ ⎬ ⎪ [h1 − h(1 − 1 )] t12 ⎪ + hP E[x] + ⎩ − ⎭ 2 2P1 ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ K + h2 gP[1 − E[x]] · t1 + [CP + CR PE[x](1 − ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP ⎪ ⎪ ⎪ ⎪ 2 E[x2 ]] ⎨ ⎬ ⎪ ⎪ +C P E[x]]t + [1 − 2 E[x] + ∞ 1 S ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ f (t) dt + ⎪ ⎪ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ hP (1 − ) E[x ] P ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ [h1 − h(1 − 1 )] t1 ⎭ + hP E[x] + ⎩ − 2 2P1 = (27) Pt1 [1 − E(x)]
(A.8)
Let
By substituting Eq. (A.8) in Eq. (A.7) one obtains the following:
1 = [CP + CR PE[x](1 − ) + CS PE[x]]
2 =
(A.1)
2
hP hP [1 − 2E[x] + 2 E[x2 ]] − + hP E[x] 2 2 P 2 (1 − )2 E[x2 ] + [h1 − h(1 − 1 )] 2P1
E[TCU(t1 )] =
{C+CR E[x](1−)+CS E[x]−hg} · t1 +h2 g+ [1−E(x)] 2[1−E(x)] K + hgE(x)(1 − e−t1 ) + [1 − E(x)] Pt1
(A.2)
+
M hg + P
(1 − e−t1 ) t1
(28)
then Eq. (27) becomes t K + h2 gP[1 − E[x]] · t1 + 1 t1 + 2 t12 + 01 [M − hPg[1 − E(x)] · t1 + (hPg)t]f (t)dt E[TCU(t1 )] = P[1 − E(x)]t1
t1
since 0
since 0
t1
−t1
f (t)dt = F(t1 ) = 1 − e
t · f (t)dt = −t1 e−t1 −
1
e−t1 +
(A.4) 1
Substituting Eqs. (A.4) and (A.5) in Eq. (A.3), one obtains 1 2 M(1 − e−t1 ) K E[TCU(t1 )] = + · + t1 + [1 − E(x)] Pt1 P P Pt1 hg −t1 −t1 −hg + hgE(x)(1 − e )+ (1 − e ) + h2 g t1 (A.6) Substituting Eqs. (A.1) and (A.2) in Eq. (A.6), one has
+
PE[x2 ](1 − )2
2[1 − E(x)]
PE[x2 ](1 − )2
P1
[h1 − h(1 − 1 )]
Theorem 1. E[TCU(t1 )] is convex if 0 < t1 < (t1 ). The first and the second derivatives of E[TCU(t1 )] are
+
hP[1 − 2E[x] + E[x ]] − h 2
P1
Appendix B
1
+2hE[x] +
= hP[1 − 2E[x] + 2 E[x2 ]] − h + 2hE[x]
dE[TCU(t1 )] = + 2[1 − E(x)] [1 − E(x)] dt1
{C + CR E[x](1 − ) + CS E[x] − hg} E[TCU(t1 )] = + h2 g [1 − E(x)] +
where
(A.5)
(A.3)
2
hg M + P
d2 E[TCU(t1 )]
[h1 − h(1 − 1 )] · t1
K + hgE(x)(1 − e−t1 ) [1 − E(x)] Pt1 hg (1 − e−t1 ) M + + P t1
2 dt1
= [1 − E(x)]
+
+ (A.7)
−(1 − e−t1 ) t12
2K Pt31
−K Pt21
+
+ hgE(x)(e−t1 )
e−t1
t1
(B.1)
+ hgE(x)(− e−t1 ) 2
2 e−t1 2e−t1 hg M 2(1 − e−t1 ) − − + · P t1 t12 t13 (B.2)
S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
From Eq. (B.2), since the first term of the second derivative of E[TCU(t1 )] is greater than zero, it implies if ×
then
or if
2K
t13 d2 E[TCU(t1 )] 2
dt1
Pt31
−
2
Pt31 2(1 − e−t1 )
2K
M hg + P 2 −t1 e − >0 t1
+ hgE(x)(− e−t1 ) +
+
−
2e−t1 t12
(B.3)
M hg 2(1 − e−t1 ) 2 + · − hgE(x)( e−t1 ) P t13
2 e−t1 hg M 2e−t1 + >0 + · P t1 t12
or if [2K + 2(M + Phg) · (1 − e
)]
> (e−t1 ) · t1 [Pt21 hgE(x)2 + (M + Phg) · (2 + t1 )]
(B.5)
Let = [M + Phg] and (e−t1 ) · [Pt21 hgE(x) + (M + Phg) · (2 + t1 )] > 0; Eq. (B.5) becomes 2
therefore
therefore
[2K + 2 · (1 − e−t1 )] 2 [Pt21 hgE(x)
d2 E[TCU(t1 )]
if 0 < t1 <
2
dt1
+ · (2 + t1 )](e−t1 )
> t1
(B.6)
>0
[2K + 2 · (1 − e−t1 )] 2 [Pt21 hgE(x)
+ · (2 + t1 )](e−t1 )
= (t1 )
2(K + ) − (P )t12 2
2{[hgE(x)P ]t12 + ( · )t1 + }
As stated in Section 3, because e−t1 is the complement of cumulative density function, therefore, 0 e−t1 1. Let 2(K + ) − (P )t12 v(t1 ) = 2 2{[hgE(x)P ]t12 + ( · )t1 + } therefore 0 v(t1 ) 1
(B.4)
Multiplying both sides of Eq. (B.4) by P t13 , one obtains the following: −t1
As stated in Section 3, although the optimal run time t1∗ cannot be expressed in a closed form, it can be located through the use of the following searching algorithm based on the existence of bounds for e−t1 and t1∗ . Recall Eq. (41): (e−t1 ) =
>0
907
(B.7)
Hence, the proof of Theorem 1 is completed.
One can then use the following recursive searching techniques to find t1∗ : (1) Let v(t1 ) = 0 and v(t1 ) = 1 initially and compute the upper and lower bounds for t1∗ , respectively (i.e. to find the initial values of ∗ , t ∗ ]). [t1L 1U ∗ , t ∗ ] in e−t1 and calcu(2) Substitute the current values of [t1L 1U late the new bounds (denoted as zL and zU ) for e−t1 . Hence, zL < v(t1 ) < zU . (3) Let v(t1 ) = zL and v(t1 ) = zU and compute the new upper and lower bounds for t1∗ , respectively (i.e. to update the current val∗ , t ∗ ]). ues of [t1L 1U (4) Repeat steps 2 and 3, until there is no significant difference ∗ and t ∗ (or there is no significant difference in terms between t1L 1U of their effects on E[TCU(t1∗ )]). (5) Stop. The optimal production run time t1∗ is obtained. A step-by-step demonstration of the aforementioned recursive searching algorithm is presented in Table 2 (Appendix D) for two different values of 's (refer to Section 4 for details). Appendix D An investigation utilizing different values to test for satisfaction of Eq. (30) is shown in Table 1. The step-by-step iterations and their results are displayed in Table 2.
Appendix C A proposed recursive algorithm for searching t1∗ .
Table 1 ∗ ∗ ∗ ∗ , (t1U ), t1L , and (t1L ). Variations of effects on t1U
1/
∗ t1U
∗ (t1U )
∗ t1L
∗ (t1L )
12.00 11.00 10.00 9.00 8.00
0.08 0.09 0.10 0.11 0.13
0.4544 0.4546 0.4549 0.4551 0.4555
9.8088 7.2323 5.3946 4.0784 3.1324
0.0699 0.0755 0.0820 0.0896 0.0985
0.1973 0.2133 0.2319 0.2539 0.2804
7.00 6.00 5.00 4.00 3.00
0.14 0.17 0.20 0.25 0.33
0.4560 0.4566 0.4574 0.4587 0.4608
2.4514 1.9628 1.6171 1.3830 1.2471
0.1093 0.1224 0.1384 0.1583 0.1830
0.3126 0.3528 0.4043 0.4731 0.5715
2.00 1.00 0.50 . . . 0.01
0.50 1.00 2.00 . . . 100.00
0.4650 0.4773 0.5011 . . . 1.5907
1.2260 1.4567 1.9410 . . . 5.6050
0.2140 0.2526 0.2751 . . . 0.2994
0.7333 1.1011 1.6254 . . . 4.2876
Bold numbers represent results of the numerical example (Section 4) where = 0.5.
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S.W. Chiu / Computers & Operations Research 37 (2010) 899 -- 908
Table 2 Iterations of the recursive searching algorithm for optimal production run time t1∗ .
Iteration
zL = e−t1U
∗ t1U
zU = e−t1L
∗ t1L
Difference between ∗ ∗ and t1L t1U
∗ [U] E[TCU(t1U )]
∗ [L] E[TCU(t1L )]
Difference between [U] and [L]
0.5
Initial 2nd 3rd 4th 5th
0.000 0.778 0.847 0.854 0.855
0.5011 0.3333 0.3161 0.3142 0.3140
1.000 0.871 0.857 0.855 0.855
0.2751 0.3097 0.3135 0.3140 0.3140
0.226 0.024 0.003 0.000 0.0000
$9712.74 $9585.87 $9583.85 $9583.83 $9583.82
$9593.97 $9583.93 $9583.82 $9583.82 $9583.82
$118.77 $1.94 $0.03 $0.01 $0.00
1.0
Initial 2nd 3rd 4th 5th
0.000 0.620 0.709 0.723 0.726
0.4773 0.3444 0.3240 0.3206 0.3200
1.000 0.777 0.735 0.728 0.727
0.2526 0.3079 0.3178 0.3195 0.3198
0.225 0.036 0.006 0.001 0.0002
$9753.66 $9663.75 $9660.72 $9660.63 $9660.62
$9692.58 $9661.45 $9660.65 $9660.62 $9660.62
$61.08 $2.30 $0.07 $0.01 $0.00
Bold numbers represent results of the numerical example (Section 4) where = 0.5.
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