Mathematical modeling for a multiproduct manufacturing system featuring postponement, external suppliers, overtime, and scrap

Manufacturing firms operating in today’s competitive global markets must continuously find the appropriate manufacturing scheme and strategies to effectively meet customer needs for various types of quality of merchandise under the constraints of short order lead-time and limited in-house capacity. Inspired by the offering of a decision-making model to aid smooth manufacturers’ operations, this study builds an analytical model to expose the influence of the outsourcing of common parts, postponement policies, overtime options, and random scrapped items on the optimal replenishment decision and various crucial system performance indices of the multiproduct problem. A two-stage fabrication scheme is presented to handle the products’ commonality and the uptime-reduced strategies to satisfy the short amount of time before the due dates of customers’ orders. A screening process helps identify and remove faulty items to ensure the finished lot’s anticipated quality. Mathematical derivation assists us in finding the manufacturing relevant total cost function. The differential calculus helps optimize the cost function and determine the optimal stock-replenishing rotation cycle policy. Lastly, a simulated numerical illustration helps validate our research result’s applicability and demonstrate the model’s capability to disclose the crucial managerial insights and facilitate manufacturing-relevant decision making.


Introduction
This study builds an analytical model to explore a multiproduct fabrication problem featuring postponement, external suppliers, overtime, and scrap. We aim to help manufacturers smoothen their fabrication planning and operations so that they could effectively meet today's customer needs for various types of quality merchandise under short-order lead-time and limited in-house capacity constraints. Multiproduct' commonality feature inspires production managers to design alternative manufacturing schemes such as the postponement option to gain potential savings in uptime or total fabrication-relevant costs. Aviv and Federgruen (2001) explored the benefits of a postponement model featuring dependent and unknown-distribution random demand. Accordingly, their model needed to continuously revise the estimation of demand-distribution-related parameters under a Bayesian framework to find the near-optimal ordering policies for various ordering costs. Ceryan et al. (2012) examined the admission control and fabrication decisions of a two-stage assembly-production system that must meet end-product' and intermediate parts' demands. The admission controls decisions related to acceptance or rejection of orders based on current stock-level and whether they are profitable. The researchers attempted to use a simple heuristic to test different example problems for exploring the optimal replenishment policy and its performance. Cerdá et al. (2020) studied the short-term production schedule for fabricating semifinished and final products using separate sequential batch processes in a make-to-order environment. The researchers built an efficient mixed-integer linear programming model based on the processes precedence relationship to decide the lot sizes and the synchronized fabrication schedules of the semifinished and final products. Recent works (Chiu et al., 2020;Nogueira, et al., 2020;Sharma and Rai, 2021) studied the influence of postponement strategies on diverse multiproduct manufacturing systems' planning, scheduling, and operations.
Meeting customer's anticipated merchandise quality is one of the primary competitive strategies of present-day manufacturers. The screening process helps manufacturers identify and remove inevitable faulty items produced to ensure the finished lot's quality. Kenne et al. (2003) explored the optimal fabrication control for a multiproduct multi-machine production system subject to machine failures. Their approach included a corrective maintenance plan on a failed machine to improve the system's productivity. In their model, both the fabrication and machine-repairing rates are the decision variables that affect the system's capacity and stock-level. The researchers used numerical methods and a computational algorithm to solve the optimal fabrication control problem that minimized the system's repair activities and total system costs. Finally, the researchers extended and discussed more a complex system. Sarkar and Moon (2011) studied a stochastic economic production quantity-based system featuring Weibull-distributed imperfect fabrication process, rework, shortages, and inflation. The researchers considered uniform-rectangular-distributed and general-distributed demands and developed a profit function for their model. By using computational experiments and graphical demonstration, they explored and discussed their probability model's optimality. Kundu et al. (2019) examined a seasonal multiproduct batch manufacturing system featuring time-sensitive demand and variable fabrication and defective rates. The remanufacturing of faulty items initiates with extra cost, right after these items are produced and identified. The researchers formulated a profit-maximization model with space and investment constraints to explore their proposed batch manufacturing problem. The researchers further incorporated fuzzy constraints into their model and used numerical experiments with sensitivity analyses to demonstrate the model's performance. Recent works (Alves et al., 2019;Belogusev and Egorov, 2019;Pinto et al., 2020;Salehi Amiri et al., 2020;Sarkar and Chung, 2020) explored the influence of imperfect manufacturing processes on diverse fabrication planning, controlling, and management.
Meeting the growing trend of the short amount of time before customer orders' due dates and coping with limited in-house capacity, present-day's fabrication planners always implement effective ways such as subcontracting or overtime shifts to reduce production uptime. Insinga and Werle (2000) examined the impact of aggressive outsourcing policies on diverse business operations. The researchers identified and discussed the outsourcing-relevant dependencies, their potential pitfalls, and corresponding coping actions to guide managerial decision making on linking outsourcing to corporate strategy. Boctor and Poulin (2007) proposed composite heuristics to explore a dynamic-demand economic lot-sizing and scheduling problem featuring multi-product, multi-stage, and limited capacity. The researchers first found each product's feasible solution in each period. Then, they used a recursive improving procedure to derive a more economical and capacity-feasible solution. The researchers further extended their work by considering the overtime options in the improving stage to increase capacity and save the total system costs. They offered numerical illustrations to validate their heuristics and solution procedures and show their performance. Westphal and Sohal (2013) proposed and evaluated outsourcing models' classification to help practitioners choose the appropriate outsourcing model for their corporation. The researchers also identified and discussed the limitations of outsourcing decision processes presented in the existing literature. Abdel-Aal (2019) utilized a mixed-integer linear programming approach to study a multi-period multi-product lot-sizing problem featuring uncertain demands/budgets, setup times, limited capacity, overtime and permitted backlogging. The research started with considering the model of deterministic demands and then extended to the stochastic demands ones. The author provided numerical experiments to validate their results. Recent works (Fontinha et al., 2019;Pasupa and Suzuki, 2019;Soriano et al., 2020;Abdul Halim et al., 2021;Chiu et al., 2021;Keyvanshokooh et al., 2021) investigated the impact of different subcontracting and overtime policies on different fabrication-inventory systems, supply chains, and business operating controls. Fewer past works specifically focused on investigating the impact of external suppliers, postponement, overtime, and scrap on the multiproduct fabrication problem; we aim to bridge this gap.

Nomenclature and problem description
This study employs a two-stage mathematical modeling approach to solve the rotation cycle length for a single-machine multiproduct fabrication problem featuring postponement, external suppliers for common parts, overtime option for producing end products, and random existence of scraps. The notation used in our model is displayed in Appendix A. The following is the description and assumption of our two-stage mathematical model: assuming the common intermediate part exists in a twostage multiproduct batch fabrication plan, where the annual demand λi of these L finished products (i = 1, 2, … , L) must be satisfied. We use a delayed product differentiation (postponement) strategy to gain the potential benefits of rapid order response time and total cost savings. That is, stage one of the batch plan manufactures all necessary common parts, and stage-2 fabricates all end products. This study assumes a constant completion rate γ for common part's as compared to its finished product, and both production rates P1,0 and P1,i depending on γ. For instance, if γ = 0.5, then both P1,0 and P1,i are twice as much as their ordinary rates in a single-stage model. Furthermore, to expedite the batch process, we consider a partial outsourcing portion π0 of common parts in stage-1. An overtime option with additional proportion α1,i for fabricating product i in stage two. The impact of these process accelerated strategies on the relevant cost and output-rate parameters as exhibited below (refer to Nomenclature for details): Furthermore, the inevitable random scrap rates x0 and xi exist in both production stages. The faulty items are screened and removed. Fig. 1 exhibits our proposed multiproduct fabrication model's stock status featuring postponement, external suppliers, overtime, and scrap. It specifies the common part's stocks stack to H1,0 when t1,0 ends; meantime, the receipt of outsourced items brings its level to H2,0 before the starting time of stage two. Then, in stage two, the stocks of end product i stack to H1,i, when the uptime t1,i completes, it follows that the stock level of product i depletes under a continuous issuing discipline (see Fig. 1).

Formulations, derivation, and optimization
Fig. 1 points out that starting stage two, the needed common parts for manufacturing each end product i start to deplete Qi from H2,0 to Hi. Fig. 3 exhibits a clear picture of common parts' level in the second stage, and Eqs. (6) to (8) display its details.

Fig. 3. Status of common parts of our multiproduct fabrication model
The following equations can be straightforwardly observed according to Fig. 1 to Fig. 3, and the problem statement: 1, 2, , 0, 1, 2, ... , According to the total required lot sizes of end product i (as displayed in equation (10)), the following are the total common parts required: From Fig. 1 to Fig. 3, we can also straightforwardly observe the following equations: TC(TZ) consists of the expenses incurred in stage one's variable and setup costs for outsourcing and in-house manufacturing, stockholding of scrap and safety items, and disposal costs; in stage two's sum of setup cost, variable cost, holding cost of scrap and safety items, and disposal cost. Therefore, we have the following TC(TZ): Apply the expected E[xi] (for i = 0, 1, 2, …, L) to deal with the random scrap, replace Eqs. (1) to (19) Let E00, E10, E0j, E0i, E1i, E2i, and E3i represent the following: ; ; Substitute Eqs. (21) and (22) in Eq. (20), we derive E[TCU(TZ)] as follows: To determine TZ*, we apply the first-and second-derivative to E[TCU(TZ)] and result as follows: In Eq. (26)

Numerical Example
Assume the following parameters' values (as displayed in Tables 1 and 2) are for a batch manufacturing plan of a multiproduct replenishing problem with postponement, random scrap, overtime strategy for end products manufacturing, and external suppliers for common parts. In contrast, the values of parameters assumed for the same problem with a single-stage scheme are exhibited in Table B-1 in Appendix B.  Applying Eqs. (27) and (24), we gain TZ* = 0.5516 years and E[TCU(TZ*)] = $2,516,667 for our proposed multiproduct fabrication problem featuring postponement, external suppliers, overtime, and scrap. Figure 4 demonstrates the convexity of E[TCU(TZ)] relating to TZ. It discovers that E[TCU(TZ)] surges both sides as TZ deviates from the optimal TZ*. Fig. 5. Changes of each end-product's uptime t1,i relating to α1,i 3.1 Individual impact of overtime and external supplier on the problem With our proposed model's help, we can reveal diverse crucial system characteristics such as the impact of the overtime for fabricating end products and the external supply of the common parts on the problem. For instance, Fig. 5 demonstrates the analytical results from variations in α1,i on each end-product's uptime t1,i. It discloses that as α1,i surges, end-products' uptimes knowingly declines. Fig. 6 illustrates the impact of the overtime for producing end products on utilization. It discovers that when the overtime increases the end products' output rate by 50%, the machine utilization declines by 20.7% (i.e., it decreases from 0.2434 to 0.1930, see Table C-1 in Appendix C).  Table C-1). Fig. 8 shows the analytical outcomes of the common parts' outsourcing portion π0 on facility utilization. It exposes that when external providers supply 40% common parts, the system's utilization decreases from 0.2545 to 0.1930 or a decline of 24.2% (see Table C-2, Appendix C).  Fig. 9 exhibits that by reducing 24.2% of machine utilization, the proposed system pays a price of 4.67% in E[TCU(TZ * )] increase, i.e., E[TCU(TZ * )] rises from $$2,404,296 to $2,516,667 (see Table C-2 in Appendix C). Fig. 10 explicitly studies the individual/combined influence of and π0 and α1,i on E[TCU(TZ * )]. According to this study's parameter assumptions, the analytical outcome indicates it is more economical to subcontract 40% common parts and gradually increase the overtime factor. Once α1,i reaches 0.5 and π0 remains at 0.4, to further decrease utilization, the economical way is to stop increasing α1,i (i.e., let overtime factor remain at 0.5) and only increase π0. Our model is capable of performing the investigation for any given variable assumptions and exposes crucial information to facilitate managerial decision-making. For instance, π0 = 0.5 is justified as a better starting value than π0 = 0.4, if the managers are looking for a more economical π0-value to cut down the utilization. The collective effect of α1,i and π0 on utilization are investigated and demonstrated in Fig. 11. It shows as α1,i surges, the utilization considerably declines; as π 0 increases, the utilization harshly drops. Fig. 12 illustrates the combined influence of π 0 and α1,i on E[TCU(TZ * )]. It noticeably reveals as both π0 and α1,i go up, E[TCU(TZ * )] rises considerably. It further exposes that α1,i has more impact than π0 on E[TCU(TZ * )] increase.  Fig. 13 displays the impact of γ on the optimal uptime t0 * . It discovers as γ increases, more manufacturing uptime t0 * is required. As we assume γ = 0.5, t0 * is 0.0509 years (see Table C-2 in Appendix C). It also shows that at γ = 0.5, by subcontracting 40% common parts, t0 * drops from 0.0815 to 0.0509 or a drop of 37.55%. Fig. 14 depicts the analytical results of TZ* changes concerning the nonlinear and linear relationships between δ and γ. First, it verifies that in our example, when γ = 0.50, TZ* is 0.5516 years. It explicitly exposes crucial information of TZ* difference concerning two separate examples of nonlinear relationships between γ and δ. Fig. 15 depicts the detailed investigative results of cost contributors to E[TCU(TZ * )]. It indicates total quality costs (due to scraps) add up to 5.63% (i.e., 0.27% for fabricating the common parts and 5.36% in producing end products; for details, see Tables C-1 and C-2). The overtime and outsourcing relevant costs contribute 11.51% and 16.40% of E[TCU(TZ * )]. The variable cost (in-house) for end products and common parts contribute 44.77% and 17.65%. Fig. 16 demonstrates the behavior of cost contributors to E[TCU(TZ * )] relating to the average scrap rate x. It discloses as x increases, the scrap related cost rises noticeably; other cost contributors slightly change.

Conclusions
This work built a two-stage postponement model to investigate a multiproduct fabrication problem featuring products' commonality, scrap, external supplier, and overtime. It aimed to help manufacturing firms smoothen their fabrication planning and operations so that they could effectively meet today's client needs of various types of quality merchandise under the constraints of short order lead-time and limited in-house capacity. Section two, problem formulation, mathematical derivation, and differential calculus, helped us analyze and optimize the problem. Section three, a simulated numerical illustration, helped us validate the research result's applicability and demonstrate our model's capability to expose the crucial managerial insights and facilitate fabrication-relevant decision-making. These included: (1) Total cost function's convexity and cost-minimized rotation cycle decision (refer to Fig. 4); (2) Individual impact of overtime and external supplier on the problem (see Fig. 5 to Fig. 9); (3) Combine impact of overtime and outsourcing strategies on the problem (refer to Figures 10 to 12); (4) Common part's completion rate and quality cost effect on the problem (see Fig. 13 to Fig. 16). An interesting and worthwhile topic for future work is combining a multi-delivery strategy for this multiproduct fabrication problem.

Funding
The researchers appreciate Ministry of Science and Technology of Taiwan for supporting this project (#: MOST 109-2221-E-324-015).