Satisfying multiproduct demand with a FPR-based inventory system featuring expedited rate and scraps

Article history: Received October 14 2018 Received in Revised Format October 28 2018 Accepted November 22 2018 Available online November 22 2018 Facing stiff competition in worldwide markets, capability of meeting timely demands of multiproduct and satisfying customer’s desired product quality are essential to present-day manufacturers. Motivated by achieving the aforementioned goals, this research intends to find most economic common cycle length for a multiproduct finite production rate (FPR)-based inventory system, wherein, imperfect production process with expedited fabrication rate and random scrap is assumed. Extra setup and unit costs are associated with the adjusted rate, and imperfect products are screened and scrapped. A mathematical model is cautiously constructed to examine and resolve the problem. A numerical illustration is employed to exhibit the applicability of the proposed method. Except finding the most economic common cycle time for the problem, core contribution of this study also is associated with the individual and combined impact(s) of important factors to the problem, and hence, enabling management of manufacturing firms to make efficient/cost-effective decision and gain competitive advantages. © 2019 by the authors; licensee Growing Science, Canada


Introduction
This paper aims to find the most economic cycle time for a multiproduct FPR-based inventory system featuring expedited rate and scraps.The FPR model, also known as economic production quantity-EPQ model (Taft, 1918), assumed a perfect fabrication condition and focused on deciding optimal lot size which minimizes overall fabrication related costs.However, with the trend of increasing customer's needs of timely demands of multiproduct, the classic FPR model requires to be revised and expanded to meet the needs from present-day manufacturing firms.Past articles relating to diverse aspects of multiproduct inventory systems are surveyed below.Zahorik et al. (1984) considered production scheduling for multi-item multi-stage capacitated problems.For a single main facility, they proposed and examined two separate multi-item systems using linear related costs.One problem has a limitation on delivery capability, and the other one was assumed to be a bottleneck facility in the last stage of multi-stage process.Linear network programming technique was employed in their proposed rolling heuristic based on a three-period result.They provided discussion on conditions where their heuristic fails to locate optimal solutions.Rosenblatt and Rothblum (1990) examined a multi-item single resource inventory problem with the objective of determining the most economic capacity to the resource of the problem.Two different common cycle length solution procedures were proposed for deciding the optimal policy.They showed the applicability of their proposed procedures through numerical examples.Clausen and Ju (2006) proposed a hybrid algorithm to resolve the economic batch size and shipping schedule problem, wherein single producer fabricates and ships different parts in batches to a buyer.The purpose of their study was mainly to decide the cycle time that keeps the annual cost minimum.Additionally, in each cycle they also determined the best fabrication sequence for different types of parts.They employed two existing algorithms, an optimal one and a heuristic from prior studies and observed their computational performances.When the size of problem is large, in terms of numerous types of parts, running time becomes long for deriving the optimal solution.Hence, a hybrid algorithm was developed by them to shorten time to locate the optimal solution.Other studies (Song, 1998;Absi & Kedad-Sidhoum, 2009;Ma et al., 2010;Taleizadeh et al., 2013;Chiu et al., 2016a,b;Fergany, 2016;Jawla & Singh, 2016;Razmi et al., 2016;Zahedi et al., 2016;Vujosevic et al., 2017;Chiu et al., 2018a) also focused on diverse subjects of multiproduct stock refilling systems.
To reduce fabrication completion time a commonly used strategy is to expedite manufacturing rate.Villeda et al. (1988) considered a just-in-time (JIT) system that has kanbans of three assembly lines merging into a final station of the assembly.The times for operations are varied, and the impacts of variability can be decreased by increasing level of work-in-process (WIP) or by unbalancing three assembly lines via work assignment at each station.They analyzed different unbalancing approaches to find out that unbalanced stations have a consistent output rate improvement for the proposed JIT system and these rates outperformed that of the perfectly balanced stations.They also discussed the extent of output rate improvement and their relationship with the inter-stage buffer capacities of the system.Larsen (1997) reexamined a well known economic production lot-size problem and treated the manufacturing rate as a decision variable.His study pointed out an interesting finding that was in response to an increased demand rate, the mechanism of decreasing manufacturing rate can be optimal.Pellerin et al. (2009) studied fabrication rate control problem for stochastic remanufacturing and repair systems.They first formulated the system as a multi-level control problem, then, proposed a suboptimal control policy, which uses inventory thresholds to activate distinct control executing modes.Parameters of control policy are decided according to optimization of analytical cost equations.A real numerical case is applied to confirm the applicability of their controlling approach, which leads to a major savings in total cost as compared with a system without employing the control policy.Glock (2013) demonstrated that applying random shifts in machine fabrication rate, for instance, in the case of technical defects, may lead to a decline in cost, and hence, an increase in profit.Inspired by this idea, he used an existing inventory model to demonstrate how it works and traced it back to ordinary assumptions made in the literature.Extra studies (Wolisz, 1984;Gallego, 1993;Bylka & Rempala, 2001;Neidigh & Harrison, 2010;Muller & Clarkson, 2016;Rakyta et al., 2016;Liu et al., 2017;Kumar et al., 2017;Ameen et al., 2018;Chiu et al., 2018b,c) also addressed different issues and effects on systems with variable production rates.
Further, due to diverse unexpected/uncontrollable factors in manufacturing environments, fabrication of scrap items is inevitable.Rosenblatt and Lee (1986) investigated the impact of imperfect manufacturing process on the economic fabrication cycle time.The process is first assumed to be in-control, and it may randomly switch to outof-control state and causes fabrication of a fixed proportion of defective products.They proposed approximate solution procedures to locate optimal lot-size for the problem.Cheung and Hausman (1997) examined a fabrication system with stochastic equipment failures, with the objective of simultaneously determining best policies for preventive maintenance and safety stock.Cost-effectiveness of investment on these policies were explored and discussed.Both exponential and deterministic repair time distributions were assumed and analyzed.The resulting optimal conditions that minimize relating costs under either one or both policies are found respectively.Maddah et al. ( 2010) studied a production-inventory model with random supply.They assumed that the fabrication process may shift to out-of-control state under a constant probability, and begin to produce defective items.Two separate models were explored with different policies for handling the defects: (1) removal of defects from the inventory at no disposal cost, and (2) defective items are consolidated as batches and there is a disposal cost in removing/shipping them.Both models were studied to discover the optimal lot-size and expected system cost, respectively.Other research focused on diverse imperfect features of manufacturing systems can also be found (e.g.Makis, 1998;Eroglu & Ozdemir, 2007;Chakraborty et al., 2013;Kaylani et al., 2016;Majumder et al., 2016;Zhang et al., 2016;Khanna et al., 2017;Rezazadeh & Khiali-Miab, 2017;Shakoor et al., 2017;Al-Bahkali & Abbas, 2018;Pearce et al., 2018).Because few studies focuse on multiproduct fabrication system by considering an expedited rate and scraps, this research aims to fill the gap.

The FPR-based multiproduct inventory system with expedited rate and scraps
A finite production rate (FPR)-based multiproduct inventory system featuring expedited rate and random scraps is investigated.Unlike conventional FPR model, the proposed system adopts an expedited rate to reduce cycle time of the batch fabrication, and also examines the effect of random defective (scrap) items on optimal lot-size decision.Let L be diverse products with demand rates λi per year (where i = 1, 2, …, L), which must be satisfied by a production equipment that has an expedited manufacturing rate P1iA under common fabrication cycle time principle (Fig. 1).Manufacturing processes are imperfect a xi portion of random scrap is produced during the processes (Fig. 2) at a rate of d1iA as follows: The following requirement must be satisfied to make sure that the production equipment has sufficient capacity to fabricate L products (Nahmias, 2009): where E[xi] denotes the expected random scrap rate of product i.Further, shortages are not permitted in the proposed system, so for each product i, the following equation must hold to avoid the unwanted stock-out occurrences in the proposed batch production planning: (3) Additional notations also include: Qi = batch size per cycle of product i, TA = common production cycle time with expedited rate -the decision variable, P1iA = expedited manufacturing rate of product i per unit time (year), P1i = standard manufacturing rate of product i per unit time, α1i = expedited proportion of manufacturing rate (where α1i > 0), CiA = unit cost of product i with expedited rate, Ci = standard unit cost of product i without expedited rate,, α3i = added cost (between CiA and Ci) due to expedited rate (where α3i > 0), KiA = setup cost of product i with expedited rate, Ki = standard setup cost of product i, α2i = relating factor between KiA and Ki (where α2i > 0), t1iA = uptime of product i with expedited rate, t2iA = downtime time of product i, hi = unit holding cost of product i, CSi = disposal charge per scrapped product i, Hi = level of finished item i at uptime completion, T = common production cycle time in the same system without expedited rate, t1i = uptime of product i in the same system without expedited rate, t2i = downtime of product i in the same system without expedited rate, d1i = production rate of scrapped product i in the same system without expedited rate, E[TA] = the expected common production cycle time, I(t)i = level of finished item i at time t, IS(t)i= level of scrapped item i at time t, TC(TA) = total system cost in a cycle, E[TCU(TA)] = expected system cost per unit time, Since expedited rate has been incorporated into the proposed multiproduct FPR-based system, certain distinct relationships between relevant system parameters are assumed as follows: The following formulas can also be undoubtedly observed from Figs. (1-2):

Cost analysis and optimization
Total system cost in a cycle -TC(TA) contains: (1) the sum of setup and variable manufacturing costs for L products; (2) the sum of disposal costs, and (3) and the sum of holding costs for L products as follows: Hence, TC(TA) is the following: .
We use E [xi] to handle the randomness of xi, substitute Eqs.(4-12) in Eq. ( 16), and with extra efforts on derivations, E[TCU(TA)] to find the following,

Deciding optimal common production cycle time
The first-and second-derivative of E[TCU(TA)] are as follows: For (1 + α2i ), TA, and Ki are all positive and we confirm that the second-derivative of E[TCU(TA)] (Eq.( 19)) is positive.Hence, E[TCU(TA)] is convex for all TA other than zero.To find the optimal TA*, one can set the firstderivative of E[TCU(TA)] = 0 and solve the following: The following optimal TA* can be found after extra derivations: Finally, it should be noted that total setup times of L products may affect the aforementioned optimal cycle time if it cannot be accommodated in idle time of the proposed system (see Fig. 1 for idle time).If this is the case, then one should calculate the following Tmin (Nahmias, 2009) and choose max(TA*, Tmin) as the cycle length TA in order to assure that cycle length can contain the sum of setup times:

Numerical example
Suppose demands of 5 products need to be satisfied by a FPR-based inventory system with expedited manufacturing rate and random scrap rate.Assumptions of system's variables are listed in Table 1.=0.
Table 2 exhibits analytical effects of changes in average expedited proportion of manufacturing rate 1  on major parameters of the proposed based system.From Table 2, the quality cost (due to the existence of random scrap rate in manufacturing processes) is $208,724, which is about 8.00% of system cost E[TCU(TA*)].Moreover, the exploratory result on effect of changes in average random scrap rate x on total disposal cost is illustrated in Fig. 3.It shows that total disposal cost raises notably, as x increases; and at x = 0.15 (as assumed in this example), total disposal cost is $49,734.and its different cost factors Fig. 6.Impact of variations in ratio of A C / C on variable manufacturing cost for each product Fig. 6 shows the impact of variations in ratio of average expedited unit cost over average standard unit cost A C / C on variable manufacturing cost for each product.It is noted that variable manufacturing cost for each product notably increases, as the ratio of A C / C rises.Influences of changes in ratio of 1A P / 1 P on the sum of machine utilization are illustrated in Fig. 7.It points out that sum of utilization drops significantly, as 1A P / 1 P ratio rises; and the sum of utilization declines to 0.2047 (from 0.3070, for details refer to Table 2) when 1A P / 1 P = 1.5 (as in our example).

Fig. 7. Impact of changes in 1A
P / 1 P ratio on sum of utilization  on TA*.It indicates that TA* slightly decreases, as x raises; and conversely, TA* increases radically, as 1  goes up.Investigative result on distinctive cost elements in the proposed FPR system is presented in Fig. 9.It shows that quality cost (due to random scrap) contributes 8.00% to E[TCU(TA*)], and sum of variable manufacturing cost is 83.95% of E[TCU(TA*)] due to the added cost from the expedited manufacturing policy used (an increase of 25% as compared to when 1  = 0, see Table 2).

Conclusions
This paper has aimed to decide the most economic cycle time for a multiproduct FPR-based inventory system featuring the expedited rate and scraps.We cautiously constructed a mathematical model to examine and resolve the problem, and employed a numerical illustration to exhibits applicability investigation outcome.Other than obtaining the most economic cycle length for the problem, the core contribution of this research has also included revealing the individual and combined impact(s) of key factors to the problem (refer to section four), which have never been exposed until now.Examining the effect of random demands of multiproduct on the same system shall be an interesting area for future research.

Fig. 1 .
Fig. 1.Level of finished item i in the proposed FPR-based multiproduct inventory system with expedited rate and scraps (in red) as compared to the same system with standard rate (in grey)

Fig. 3 .
Fig. 3. Effect of changes in average random scrap rate x on total disposal cost Fig. 4. Impact of differences in 1A P / 1 P on the holding cost for each product

Fig. 5 .
Fig. 5. Influences of variations in TA on E[TCU(TA)] and its different cost factors Fig. 6.Impact of variations in ratio of A C / C on variable manufacturing cost for each product

Fig. 8 .
Fig. 8. Joint effects of differences in x and 1 on TA* Fig.8depicts the exploratory outcomes on combined effects of differences in x and average expedited proportion of manufacturing rate 1  on TA*.It indicates that TA* slightly decreases, as x raises; and conversely, TA* increases

Fig. 9 .
Fig. 9. Investigative result on distinctive cost elements in the proposed FPR-based system Fig. 10.Combined impacts of variations in x and 3  on E[TCU(TA*)].

Fig. 10
Fig. 10 depicts the analytical results on combined impacts of variations in average random scrap rate x and average added expedited unit cost 3 on the optimal system cost E[TCU(TA*)].It is noted that E[TCU(TA*)] raises drastically, as both x and 3  increase.

Table 1
Assumptions of system's variables

Table 2
Analytical effects of changes in 1  on major parameters of the proposed system