Joint effects of stochastic machine failure, backorder of permissible shortage, rework, and scrap on stock replenishing decision

Article history: Received March 28 2018 Received in Revised Format June 14 2018 Accepted June 14 2018 Available online June 14 2018 With the intention of addressing product quality, machine reliability, and acceptable service level issues in real fabrication systems, this paper studies joint effects of stochastic breakdown, backorder of permissible shortage, rework, and scrap on the optimal stock replenishing policy. A decision model is developed to solve the problem, which consists of mathematical modeling, formulations, and optimization method in order to help analyze the problem, derive the system cost function, and find the optimal replenishment cycle length decision. Applicability of the research results are demonstrated by a numerical example. The proposed decision model enables production managers to not only determine the optimal stock replenishing policy, but also reveal individual impact and/or joint effects of stochastic breakdown, defective rate, backorder of allowable shortage, rework, and scrap on the replenishing decision. With such an in-depth study, diverse system characteristics become available for managerial decision-making. © 2019 by the authors; licensee Growing Science, Canada


Introduction
The present work explores joint effects of stochastic machine failure, backorder of permissible shortage, rework, and scrap on stock replenishing decision.Traditional economic production quantity (EPQ) model (Taft, 1918) employed a mathematical model to calculate total relevant production cost per unit time, balance stock holding and setup costs to determine the best lot size for a fabrication cycle.Perfect fabrication process with no permitted stock-out situation is the simple assumption of classic EPQ model.However, due to unpredictable issues in the manufacturing practices, machine failures and imperfect products are inevitable.Bielecki and Kumar (1988) analyzed and discussed values of several relevant variables in an unreliable fabrication system, which contributed to the optimality of zero-inventory policy.Groenevelt et al. (1992) presented two different inventory control policies to deal with equipment failure occurrence, namely an abort-resume (or AR) discipline and a no resumption (or NR) policy.The former conditionally resumes the interrupted batch after completion of repair of the equipment, and the latter unconditionally abandons the interrupted lot at the time a failure occurs.Separate analyses and results were conducted and demonstrated.Boone et al. (2000) explored a fabrication system with defective products and machine failure.Mathematical model was developed to help expose the effects of these imperfect factors in fabrication on the optimal cycle length.As a result, important guidelines for selecting suitable fabrication cycle time were presented for managerial decision makings.Chiu (2010) employed mathematical modeling, theorems of renewal reward and convexity, and a recursive algorithm to help derive the optimal run time for a fabrication system with random breakdowns under AR policy and an imperfect rework process.Results and applicability were demonstrated via a numerical example.Extra studies that investigated the effect of machine failures on manufacturing systems can also be referred elsewhere (Giri & Dohi, 2005;Chakraborty et al., 2008;Rivera-Gómez et al., 2013;Chiu et al., 2016;Zhang et al., 2016;Luong & Karim, 2017).
With the purpose of reducing quality cost in production to lower overall fabrication relevant cost, production managers often adopt rework policy to repair the imperfect products.Rosenblatt and Lee (1986) explored the effect of imperfect fabrication process on the optimal manufacturing cycle length.A constant portion of defective products is generated when an in-control status of fabrication process turns into the out-of-control status.Approximate solutions were used for finding the most economic batch size in their study.Zargar (1995) proposed two distinct rework strategies to investigate their impacts on manufacturing cycle length.Queuing models for these separate rework strategies were also constructed, and simulation techniques were employed to expose individual characteristics of these rework strategies as well as their effects on cycle length.Teunter and Flapper (2003) derived the optimal batch size for a single stage manufacturing system with defective products.Upon completion of all-unit screenings, the perfect quality items, repairable nonconforming products, and scrapped items are separated.Regular production process will be switched to rework mode when the accumulation of repairable defective products reaches a predetermined quantity 'N', and a linear cost and time is assumed for the reworked items.Accordingly, they derived a clear equation for average profit for any given N as well as the optimal N based on this equation.Eroglu and Ozdemir (2007) examined an economic order quantity (EOQ) model by considering defective items and shortages backordered.All-unit screening separates good products, imperfect items and scraps.Effect of defective rate on optimal ordering quantity was explored together with numerical illustrations of research results.Pasandideh et al. (2013) studied a specific multi-item EPQ model featuring limited orders and rework of repairable nonconforming products, with the objectives of minimizing both the needed storage space and total inventory cost, simultaneously.Metaheuristic algorithms (i.e., NSGA-II and MOPSO) were used to solve the proposed bi-objective nonlinear problem.Solution and performance of these algorithms (including CPU time) were confirmed and assessed via two-sample t-tests to demonstrate their accuracy and efficiency.Diverse aspects of fabrication systems by considering defective items and/or their quality improvement issues were also explored elsewhere (Abilash & Sivapragash, 2016;Balaji et al., 2016;Chiu et al., 2016;Jawla & Singh, 2016;Mičieta et al., 2016;Buckova et al., 2017;Chiu et al., 2017a,b;Khanna et al., 2017).
Moreover, in the situations of consolidate orders from the internal demands or intra-supply chain members, or other operating strategy viewpoints, backordering of permissible shortages can be an effective policy to lower overall operating cost and/or smooth fabrication schedule.However, to avoid excessive stock-out duration in a cycle, setting a minimum service level is required to retain customer satisfaction in terms of stock availability.Ramani and Kutty (1985) explored the effects of service level constraints on total inventory costs for a multiproduct multiple group stock system.306 spare components of real data were collected from a transport firm for their study.Based on multi-criteria these data were first categorized into 9 subgroups and each received a specific service level constraint.Total costs for inventories are calculated based on weighted/combined service levels.Accordingly, the local and global minimal costs were found and analyzed to provide an insight into management of groups of components.Rabinowitz et al. (1995) examined a stochastic (r, Q) ordering policy inventory model with a partial backorder strategy.A controlling parameter 'b' is assumed in their model to represent the maximal permissible level of shortage that is backordered in a given replenishment cycle time.The controlling parameter makes their proposed inventory system different from the policies of lost sales and 100% backlogging.Solution procedure was proposed to calculate the optimal values for reorder point r, fixed order quantity Q, and maximal allowable backlog level b, which minimize the expected system expenses.Chiu et al. (2007) investigated a fabrication system with allowable backlogging under service level constraint, scraps, and rework of defective products.Their objective was to determine the optimal batch size for the proposed system in terms of system cost minimization.They demonstrated that their obtained result is better than the same system when the backordering is not permitted.Furthermore, they discussed the so-called "imputed backorder cost" and its relationship with the allowable backlogging level and its impact on the proposed system cost.Jha and Shanker (2013) determined the optimal production-inventory policy for a multiple buyer fabrication system, wherein a product is manufactured by a producer to meet independent demands from multiple customers.Lead times required by different customers can be shortened at extra expedited charge and all unsatisfied demand of customers can be backordered.A service level constraint for each customer is assumed in their proposed system.With a help from an analytical model and the Lagrangian multiplier technique, they were able to simultaneously derive the optimal order quantity, lead time, and number of shipments to each customer per cycle.Additional studies (Fergany, 2016;Rakyta et al., 2016;Jindal & Solanki, 2016;Jaggi et al., 2016;Qu & Ji, 2016;Salemi, 2016;Oblak et al., 2017;van Rhyn & Hancke, 2017) have also been conducted to explore the effects of permitted backlogging and service level constraint on fabrication systems.Nevertheless, little attention has been paid to exploration of joint effects of machine failure, backorder of permissible shortage, rework and scrap on stock replenishing decision, and the present work is set to fill this gap.

Description and formulation of the proposed fabrication system
Consider a fabrication system has annual manufacturing rate P1 to meet annual product demand rate λ and it is subject to stochastic machine failure with a mean of β per year where β follows Poisson distribution.In addition, its process is not perfect for a certain portion of scrap and rework-able items may randomly be produced.Also, the system permits backorder of shortage under a service level constraint (1 -α)% where α denotes the percentage of stock-out time in a replenishment cycle length, so the stock-out situation can be retained at an acceptable level.
An abort/resume policy is adopted when machine failure occurs.Under this policy, machine repair starts right away and the interrupted lot is resumed instantly when the repair task is accomplished.A constant machine repair time tr is assumed, and if the practical repair time exceeds tr, a spare machine will replace the broken one at tr.During the manufacturing process, an x ratio of nonconforming items may be produced randomly at rate d1, hence d1 = P1x.To guarantee the status of positive on-hand inventories, (P1 -d1 -λ) > 0 must satisfy.Nonconforming items are screened and the scrap (i.e., a θ portion of nonconforming items) and the rework-able (i.e., the other 1 -θ portion) products are separated.At the end of uptime t1', a rework process is used right away to repair the rework-able, at a rate of P2 and with additional rework cost CR per item reworked.During the reworking time t2', there is a failure in repair portion θ1 and they are scrapped at a unit disposal cost of CS.Hence, the overall scrap rate in the proposed system is φ = θ + (1 -θ)θ1.Additional parameters used in this study are listed in Nomenclature in Appendix A. Due to randomness of machine failure occurring in uptime T1, the following conditions have to be considered, respectively:

Condition 1: when t is less than t5'
The first condition means that a stochastic machine failure takes place in the backlog being gradually satisfied time (i.e., t5').When machine failure occurs, an AR policy is used and fabrication of the interrupted lot is resumed right after the machine is repaired and restore.The on-hand inventory status including backordering is depicted in Fig. 1.It can be seen that at the time of machine breakdown taking place, the backlogging level is at H0, and it stays at H0 for a period of time tr until the machine is restored.In the end of t5', the on-hand inventory level of finished products turns into positive, and it continues to grow to level H1, when fabrication uptime ends.Then, the reworking of rework-able nonconforming products starts in t2', which brings the on-hand inventory level of finished products to H, when rework process ends.All finished products are depleted in the end of t3', and inventory status turns into negative (i.e., shortage occurs) during t4', until it reaches maximal allowable backlog level B (i.e., which is set by the service level constraint) in the end of t4'.Then, the fabrication uptime of the next cycle starts.The following expressions can be directly observed from Fig. 1: The on-hand inventory levels of nonconforming products and scrap items are illustrated in Fig. 2 and Fig. 3, respectively.Note that the level of on-hand defective items (see Fig. 2) is at d1t when a machine failure occurs, and after the machine is fixed the on-hand defective items continues to increase to d1T1 at the end of uptime.
  Similarly, the level of on-hand scrap items stops at (d1tθ) when a breakdown occurs (Fig. 3), and after the machine is fixed the on-hand scrap items continues to increase to (d1T1θ) in the end of uptime.Then, during reworking time t2', the on-hand inventory of scrap items continues to grow, and at the end of rework process, the scrap items reaches the maximum level at (d1T1θ + d2t2'), where Total relevant cost per cycle when a breakdown takes place in t5', TRC1(T1) comprises the fabrication setup cost, variable manufacturing and reworking costs, holding cost for reworked items, disposal cost for scrap items, backordering cost for shortage, holding cost during t1', t2', and t3', fixed machine repairing cost, holding and purchasing cost for safety stock (to be used during tr), and delivery cost for finished products.Hence, TRC1(T1) is Replacing fabrication lot-size Q with T1P1, Eq. ( 13) becomes Substituting Eq. ( 1) to Eq. ( 12) in Eq. ( 14) and applying the expected values of x (for dealing with its randomness), yield E[TRC1(T1)] as shown in Eq. (B-1) in Appendix B. Upon obtaining the total cost per cycle in the case that a breakdown takes place in t5', we then examine the second situation in the following sub-section.

Condition 2: when t is less than T1, but it is greater than t5'
The second condition stands when a stochastic machine failure takes place in the positive stock growing stage (see Fig. 4).Note that the on-hand level of perfect quality products arrives at H2 when a machine failure occurs, and after the machine is fixed, the on-hand perfect quality products continues to climb to H1 at the end of uptime, and it reaches H at the end of rework process.Then, at the end of t3' all perfect quality items are consumed.During t4', the stock-out situation begins and it continues until the end of t4' when shortages reach the maximal permissible level B (determined according to a pre-set service level constraint).In situation 2, Eq.(1) to Eq. ( 7) and Eq. ( 9) to Eq. ( 12) exposed in previous subsection are still in effect.The following new equation is added in analysis of situation 2: Fig. 4. The on-hand inventory status including the backlogging in the proposed fabrication system when a breakdown taking place in t1' The on-hand inventory levels of nonconforming products and scrap items are depicted in Fig. 5 and Fig. 6, respectively.Note that the level of on-hand defective items (see Fig. 5) is at d1t when a machine failure occurs, and after the machine is repaired and restored, the on-hand defective items continues to increase to d1T1.
During the reworking time t2', the reworkable nonconforming products [d1T1(1 -θ)] are depleting at a rate of P2.Similarly, the level of on-hand scrap items stops at (d1tθ) when a breakdown occurs (Fig. 6), and after the machine is repaired and restored, the on-hand scrap items continues to increase to (d1T1θ) at the end of uptime.Then, in the end of rework process, the scrap items reaches maximum level at (d1T1θ + d2t2') or φ(xQ) (see Eq. ( 12)).Total relevant cost per cycle in the case that a breakdown takes place in t1', TRC2(T1) comprises the fabrication setup cost, variable manufacturing and reworking costs, holding cost for reworked items, disposal cost for scrap items, backordering cost for shortage, holding cost during t1', t2', and t3', fixed machine repairing cost, holding and purchasing cost for safety stock, and delivery cost for finished products.Therefore, TRC2(T1) is Replacing fabrication lot-size Q with T1P1, Eq. ( 16) yields Substituting Eq. (1) to Eq. ( 7), Eq. ( 9) to Eq. ( 12), and Eq. ( 15) in Eq. ( 17) and applying the expected values of x (for dealing with its randomness), with extra derivations yields E[TRC2(T1)] as displayed in Eq. (B-2) in Appendix B. Finally, we examine the third situation in the following sub-section.

Condition 3: when t is greater than or equal to T1
The third condition means that no machine failure takes place during the fabrication uptime (see Fig. 7).
The following expressions can be directly obtained by observing Fig. 7: The on-hand inventory status including the backlogging in the proposed fabrication system when breakdown does not take place during fabrication uptime     Total relevant cost per cycle in the case that no breakdown takes place in uptime, TRC3(T1) comprises the fabrication setup cost, variable manufacturing and reworking costs, holding cost for reworked items, disposal cost for scrap items, backordering cost for shortage, holding cost during t1, t2, and t3, holding and purchasing cost for safety stock, and the delivery cost for finished products.Hence, TRC3(T1) is Replacing fabrication lot-size Q with T1P1, Eq. ( 28) yields Substituting Eq. ( 18) to Eq. ( 27) in Eq. ( 29) and applying the expected values of x (for dealing with its randomness), and with extra derivations yields E[TRC3(T1)] as given in Eq. (B-3) in Appendix B.

Determining the optimal fabrication uptime
Since this study considers service level constraints to prevent excessive shortage occurrences in any given replenishment cycle, the following equation (Chiu et al., 2007) is used to maintain the minimal service level at (1 -α)%: Moreover, since in this study a Poisson distributed machine failure rate (with β per year as the mean) is assumed, hence, the time to a breakdown occurrence obeys exponential distribution with f(t) = βe -βt and F(t) = (1 -e -βt ) as its density and cumulative density functions, respectively.Therefore, the expected total system cost E[TRCU(T1)] is where Substituting E[TRC1(T1)], E[TRC2(T1)], and E[TRC3(T1)] (from Appendix B, Eqs.(B-1) to (B-3)), B, f(t), and E[T] in Eq. ( 31), and with further derivations we obtain E[TRCU(T1)] as follows: where w 1 , w 2 , w 3 , w 4 , w 5 , z 1 , v, and s denote the following: (38)

Convexity
The first-and second-derivatives of E[TRCU( T1)] (i.e., Eq. ( 33)) can be obtained as follows: and From Eq. ( 40), it is noted the first term of its RHS (right-hand side), λ /(1-φE[x]) is positive, so if the second term on RHS of Eq. ( 40) is also positive, then 3.2 Solution procedure for finding optimal uptime When Eq. ( 41) holds, we are certain that there exists T1* yielding minimum cost of E[TRCU(T1)].By setting the first-derivative equal to zero, we have the following: With additional derivations, Eq. ( 42) becomes (43) where m2, m1, and m0 represent the following: 1) Applying the square-root solution on Eq. ( 43) we find T1* as follows: Rearrange Eq. ( 47) yields where Table 1 reconfirms that because E[TRCU(T1)] is convex and the optimal T1* falls within the interval [T1L, T1U], the proposed recursive solution procedure can help us find the optimal fabrication cycle time T1* = 0.3893 and E[TRCU(T1* = 0.3893)] = $9,699.33.The effect of fabrication cycle length T1 on E[TRCU(T1)] is demonstrated in Fig. 8. Impacts from different defective rates and overall scrap rates on the expected total system relevant cost per unit time E[TRCU(T1)] are investigated and the results are exhibited in Fig. 9.It can be seen that as either random defective rate x or overall scrap rate φ goes up, the expected system cost increases, accordingly.As the assumption in our numerical example that x = 0.2 and φ = 0.0975, we find the optimal E[TRCU(T1*)] = $9,699.33.Joint effects of diverse overall scrap rates and variations in time-to-failure on the expected total system cost E[TRCU(T1)] are explored and the results are illustrated in Fig. 10.It is noted that as time-to-machine-failure 1/β increases (i.e., number of breakdowns per year β declines), the expected system cost decreases accordingly, and as 1/β goes up and higher than 1 year, the E[TRCU(T1*)] reduces, significantly.As the assumption in our numerical example that 1/β = 2 we arrive the optimal E[TRCU(T1*)] = $9,699.33.Due to the proposed system permits backordering of shortages under service level constraint (1 -α)% (i.e., a maximal percentage α of stock-out time allowed in any given cycle length), the stock-out situation can be retained at an acceptable level.Analytical result of impact of various service levels on the expected total system cost E[TRCU(T1)] are studied and the result is depicted in Fig. 11.It can be seen that as the service level (1 -α) is set higher, the expected system cost E[TRCU(T1)] raises notably, and as the assumption in our numerical example that (1 -α) is set at 80%, E[TRCU(T1*)] climbs to $9,699.33.Further exploratory results of the effects of different service levels on the optimal cycle length T1*, maximal levels of stock holding H and the permissible backlog B, E[TRCU(T1*)] and its increase percentage, and the cost of setting service level at (1 -α)% are also exhibited in Table 2.It is noted that if the service level is 100%, the proposed model becomes the same as a fabrication model subjects to the stochastic breakdown, but without backlogging (Chiu, 2010).Combined impacts from variations in overall scrap rate φ and service level (1 -α) on the expected total system cost E[TRCU(T1)] are also investigated and the results are demonstrated in Fig. 12.It is noted that as either the service level is higher or overall scrap rate goes up, the E[TRCU(T1)] increases remarkably.

Concluding Remarks
A decision model is developed to investigate a realistic fabrication system incorporating diverse practical factors in production, such as random breakdown, backorder of permissible shortage, scrap, and rework of repairable items.This decision model consists of mathematical modeling, formulations, and optimization method in order to help analyze the problem, derive the system cost function, and find optimal replenishment cycle length decision, respectively.Applicability of the research results are demonstrated by a numerical example.The proposed decision model allows production managers to not only find the optimal stock replenishment policy (Fig. 8), but also thoroughly reveal individual impact and/or joint effects of stochastic breakdown, defective rate, backorder of allowable shortage, rework, and scrap on the replenishing decision (Figs.9-12).In addition, information on detailed cost components in the specific fabrication system (Fig. 13) and contributors to any particular cost component (e.g.quality cost in production (Fig. 14)) can now be available for managerial decision-making.For direction of future study, an interesting extension will be to consider a multi-shipment inventory issuing policy rather than the continuous one in the same problem.

Appendix -C
Several different values of β have been plugged in Eq. ( 41) to test for convexity of the expected total system relevant cost per unit time E[TRCU(T1)], and the results are shown in Table C-1.It reveals that the proposed model can derive the optimal runtime solution for a wide-ranging breakdown rate.That is the model is suitable solving real manufacturing system that is subject to random breakdown rate per year ranging from 0 to 8. In the particular example that β = 0.5 (as highlighted in Table C-1), Eq. ( 41) is confirmed, since T1U = 0.5491 < z(T1U) = 2.4394 and T1L = 0.3423 < z(T1L) = 2.1346.

Fig. 1 .
Fig. 1.The on-hand inventory status including the backlogging in the proposed fabrication system when a breakdown taking place in t5'

Fig. 2 .Fig. 3 .
Fig. 2. The on-hand inventory level of nonconforming products in the proposed fabrication system when a breakdown taking place in t5'

Fig. 5 .Fig. 6 .
Fig. 5.The on-hand inventory level of nonconforming products in the proposed fabrication system when a breakdown taking place in t1'

Fig. 8 .Fig. 9 .
Fig. 8.The effect of fabrication cycle length T1 on E[TRCU(T1)] Fig. 9. Impacts from diverse defective rates and overall scrap rates on the expected total system relevant cost per unit time E[TRCU(T1)]

Fig. 10 .
Fig. 10.Joint effects of diverse overall scrap rates and variations in time-to-failure on the expected total system cost E[TRCU(T1)]

Fig. 11 .
Fig. 11.Analytical result of impact of various service levels on the expected total system cost E[TRCU(T1)]

Table 2
Exploratory results of effects of different service levels on various system parameters ) and applying E[x] to cope with the randomness of nonconforming rate, and with further derivations one can obtain E[TRC2(T1)] as sho wn in Eq. (B-2).

Table C -1
The results of convexity tests for E[TRCU(T1)] by using different values of β