The combined effect of rework , postponement , multiple shipments , and overtime producing common-component on a multiproduct vendor-client incorporated system

Article history: Received November 6 2021 Received in Revised Format December 27 2021 Accepted February 26 2022 Available online February, 26 2022 The study examines the combined effect of rework, multiple shipments, postponement, and overtime producing common-component on a multiproduct vendor-client incorporated system. Clients’ product demand trend turns to diversity, quality, and rapid response in the current supplychain environment. Under such a stiff competitive environment, today’s manufacturers must effectively plan their multi-item fabrication to boost utilization and product quality, minimize total relevant costs, and meet short given order lead time. By considering the commonality of the finished goods, required quality, and completion lead time, this study presents an exact model featuring rework of defects, multiple shipments, postponement, overtime producing the mutual component, and satisfying the market needs. Through the techniques of explicitly modeling, formulating, and system cost minimization, this study simultaneously derives the optimal cycletime and shipping frequency for the studied problem. A numerical example helps show how our model works for any given parameter values and how the variation in single and multiple factors of the problem affects the crucial system performances (e.g., total uptimes, each relevant cost, utilization, total cost, etc.) A wide variety of today’s industries (e.g., automotive, household goods, etc.) and their related supply chains can utilize our decisional model to reveal in-depth managerial insights for planning their fabrication and shipments. © 2022 by the authors; licensee Growing Science, Canada


Introduction
Clients' product demand trend turns to diversity, quality, and rapid response in the current supply-chain environment. Under such a stiff competitive environment, today's manufacturers must effectively plan their multi-item fabrication to boost utilization and product quality, minimize total relevant costs, and meet short given order lead time. In consideration of the needs of various goods with commonality and plant efficiency in terms of cost, utilization, and expediting order completion time, production planners often evaluate the alternatives of using a single piece of equipment to fabricate multiple goods in sequence, associating a postponement strategy. Wadhwa et al. (2006) examined the postponement strategies through knowledge innovation and redesigning business processes in automotive industries. By developing managers' innovative mindset and adequately managing the existing processes knowledge, they evaluate the impact of postponement on the product flexibility, service levels, and potential benefits. In addition, they demonstrated their results using various scenarios in simulated cases. Qrunfleh and Tarafdar (2013) respectively explored the effect of strategic postponement and the role of supplier partnership on the agile and lean supply chain. The researchers also investigated the relationship between supplychain approachability and corporation performance. They proposed a research model to study the problem and conducted a survey using data from two hundred and five managers and executives in the manufacturing and purchasing functions of business corporations in the USA. By applying the strategic-choice method and the firm resource-based supply-chain viewpoint and specific computer software, the researchers demonstrated how a proper supply-chain strategy could boost its responsiveness and performance. Finally, they found that a strategic supplier partnership can effectively enhance a lean supply chain's responsiveness; the postponement can partially increase the agile supply chain's responsiveness; both can boost the firm's performance. Mauri et al. (2021) proposed two hybrid-metaheuristics to explore a two-stage multi-item capacitated facility location system. The first stage of the studied system delivers goods from various plants to depots, and then the second stage transports goods from depots to clients. The aim is to meet demands and minimize overall operating costs under capacity constraints of plants and depots. Finally, the researchers applied the clustering search approaches and biased random-key algorithm to resolve their multiproduct system with various sets of experimental examples. Additional studies (Van Mieghem, 2004;Gualandris and Kalchschmidt, 2015;Ivanov et al., 2019) examined the impact of multi-item fabrication and different features of commonality and postponement strategies on multi-item production and supply-chain systems. In a multiproduct manufacturing plan with the postponement, an overtime arrangement helps expedite the completion uptime of making heavy quantities of mutual components. Ren et al. (2010) explored the multiple shifts vehicle routing problem with time windows and overtime. The study considered that variable demands in a finite time horizon are served by a limited set of vehicles with allowable overtime. The researcher presented a shift-dependent heuristic with overtime to construct routes for each shift and explore the tradeoff between regular operating costs (e.g., transportation, unmet demand, and diver costs) and overtime cost. The proposed heuristic can meet random demand within a tight time window and significantly save vehicle quantity and total costs. Madeira (2014) proposed a New Keynesian Phillips Curve (NKPC) model to evaluate the marginal overtime cost against the straight time. The research showed that the obtaining coefficient estimates were in line with the theory of hybrid-NKPC (which incorporates backward-looking price-setters), but not for the NKPC with only a forward-looking setting). In addition, the study exposed the characteristics of the hybrid model and the influence of employment frictions on the price changes/setting of the NK model. Keyvanshokooh et al. (2021) utilized a primal-dual approach to explore an online advance scheduling problem with overtime. Arrivals of jobs have dissimilar service times, and the services can be completed through a finite set of servers having regular or overtime capacities. Based on worst-case performance, the system selects a server, a corresponding service time, and competitive ratio (CR) for each arrival job. Then, the researchers developed an online primaldual method to assign server-date and potential overtime plans for each arriving job. Finally, they used actual health system appointment data to demonstrate that their online scheduling algorithm significantly outperformed existing scheduling rules, such as the first-come-first-served and the nested threshold rules. Additional works (Goldenhar et al., 2003;Lambooij et al., 2007;Conway and Sturges, 2014;Chiu et al., 2020;Soriano et al., 2020) studied the effect of diverse overtime aspects on different corporation planning, manufacturing operations, and supply-chain management.
Meeting order with a fixed-quantity multiple shipment policy helps reduce clients' inventory effectively. Hence, it is a commonly used delivery plan in vendor-client incorporated systems. Khan et al. (2014) explored the combined impact of learning in fabrication and quality inspection errors on an integrated supply chain system. The researchers presented a mathematical model with learning in fabrication at the vendor-side and quality inspection errors at the client-side to decide the optimal inventory for the integrated supply chain system. They demonstrated the model's applicability using numerical examples and discussed how it could benefit managerial decisions in investment, personnel training, and process and product design. Karakaya et al. (2021) analyzed the service systems' benefits and impact from preannounce delivery time and price to customers. The researchers used numerical studies to explore customers' expectations and reactions by knowing their anticipated services' price and delivery time. Additional works (Sahebi et al., 2019;Mabrouk, 2020;Sumrit, 2020;Tran et al., 2020;Farmand et al., 2021;Sazvar et al., 2021) studied the influence of diverse products' transportation plans on various supply-chain operations. Product quality is one of the crucial factors for current vendors to stay competitive in today's markets. However, real fabrication systems inevitably generate defeats. Repairing the defectives to meet clients' anticipation becomes vendors' essential task. Raviv (2013) presented an algorithm to maximize the profit of a serial fabrication line with screening stations and rework. The results help determine the optimal design of quality control stations and fabrication rates that bring the most profits. Ponte et al. (2021) applied control engineering approaches to a hybrid fabrication and re-fabrication system with product quality classification and returned goods and evaluated the supply chain's inventory performance and Bullwhip effect in a supply chain. Different re-fabrication lead times and operations are assigned to explore the potential benefits, operational costs, and savings from the quality grading. Additional works (Eskandari and Hosseinzadeh, 2014;Sahebi et al., 2019;Assia et al., 2020;Sztorc and Savenkovs, 2020;Son and Van Hop, 2021;Abukhader and Onbaşıoğlu, 2021) explored the influence of diverse fabrication quality situations on various manufacturing systems and supply chains. For few prior studies that examined the combined effect of rework, multiple shipments, overtime producing common-component, and postponement on a multiproduct vendor-client incorporated system, we wish to bridge the gap.

Description, modelling, and assumption
The present work investigates the collective impact of postponement, overtime fabrication of common parts, multi-shipment, and rework on a multiproduct vendor-buyer coordinated problem. A two-stage manufacturing scheme is designed to explore the problem. All necessary common parts are made in the 1 st stage, and the customer's multiproduct requirements are produced in the 2 nd stage. We assume a constant mutual/common part's completion proportion γ and production rate P1,0. An overtime plan is utilized to raise common parts' output rate by α1,0 to PT1,0 to shorten its uptime. Eq. (1) to Eq. (3) express the relationship between overtime and regular output rates and relevant costs: where CT0, KT0, C0, K0, α3,0, and α2,0 denote the different overtime unit and setup costs, regular unit and setup costs, and connecting factors of overtime and regular rates. Constant demand rates λi are assumed for L multiproduct, where i = 1, 2, …, L. However, the manufacturing rates P1,i rely on the γ (e.g., when γ = 0.5, then P1,0 and P1,i both double their regular output rates comparing to that in a single-stage fabricating scheme. Nonconforming proportion x0 and xi are randomly fabricated in both stages, and these items are repairable by the use of PT2,0 and P2,i, respectively. An overtime rework-rate PT2,0 and unit cost CTR,0 have the following relationships with their corresponding the regular parameters P2,0 and CR,0: Fig. 1 depicts this particular postponement problem's stock level. It discloses that when uptime of stage one completes, the stock level upsurges to H1,0. It continues to rise to H2,0 at the end of stage one's rework time. Starting from the 2 nd stage, the common parts' inventory level depletes when the end products' production begins. Meantime, each finished product's stock climbs to H1,i when uptime t1,i completes, and it rises further to H2,i when the rework time t1,i ends (see Figure 1). Moreover, we must have (PT1,0 -dT1,0 -λ0 > 0) and (P1,i -d1,i -λi > 0) to ensure stock-out situations for both stages. It specifies that when uptimes t1,0, and t1,i complete, the nonconforming inventory level surge respectively to (dT1,0 t1,0) and (d1,i t1,i), where expressions for dT1,0 and d1,i a are shown in Eq. (6) and Eq. (7). As the rework process starts, the nonconforming inventory levels start to decline at the rates of PT2,0 and P2,i, and they deplete to zero when the rework times t2,0 and t2,i end.

Formulas of buyers' stocks
The end products' inventory levels surge to H2,i when their rework times complete (see Fig. 1). It follows that n equal-size multi-shipment plan initiates every tn,i period in t3,i (as shown in Fig. 3) and Eq. (8) gives the total stocks in t3,i: The buyer side's end product i's inventory level is illustrated in Fig. 4. Eq. (9) gives the total stocks at the buyer's side. (12)

Formulas in fabricating multiproduct
Based on the problem statement along with Fig. 1 and Fig. 2, the following formulas for i = 1, 2, …, L are observed: 1, 1, 1, 1, 1, Refer to Eq. (13), we know that to accomplish making all end products the required common parts are as follows:

Formulas in fabricating the mutual pasts
Due the postponement strategy, the 1 st manufacturing stage one must prepare adequate common parts H2,i for the 2 nd stage's finished products' production. From Fig. 1 and Fig. 2, the following expressions are observed: 2,0 1,0 2,0 T2,0

The optimal solution
By applying the Hessian Matrix Equations (Rardin, 1998) Solving the linear system of Eq. (34) and Eq. (35), the optimal production-shipment (TA*, n*) policy is determined as follows:

Numerical illustration
A simulated numerical example to utilized to exhibit the capability and applicability of our study's results. Consider a multiproduct vendor-buyer coordinated system with the postponement, overtime alternative, multi-shipment, and rework is employed to meet the needs of buyer-required five distinct items. The assumption of relevant system variables are exhibited in Tables 1 and 2. In comparison, Table B-1 (in Appendix B) displays the assumed values of corresponding parameters in a single-stage scheme.

Table 1
The assumed values of stage one's relevant variables Apply equations (36)

Combined effect of crucial system features
The combined impact of the overtime added-output factor α 1,0 and average defective proportion x i on the total rework cost is demonstrated in Fig.6. As the mean xi increases, the total rework cost drastically upsurges; and as α1,0 increases, the total rework cost rises slightly. Fig. 7 8 illustrates the joint effect of the overtime factor α1,0 and γ on t0*. As α1,0 rises, t0* noticeably declines. In contrast, as γ increases, t0* upsurges radically. It also exhibits that the changing γ value between 0.8 and 0.95 has caused t0* to have irregular changes. The collective impact of the overtime relevant factors of unit and setup costs α3,0 and α2,0 on E[TCU(TA*, n*)] is demonstrated in Fig. 9. As the overtime unit cost factor α3,0 rises, E[TCU(TA*, n*)] severely surges, and as the overtime setup cost factor α2,0 goes up, and E[TCU(TA*, n*)] slightly rises. That is α3,0 has more effect on E[TCU(TA*, n*)] than α2,0.

The influence of crucial system features
Our proposed two-stage multiproduct postponement model with multi-shipment, rework, and overtime can explore the impact of its crucial features on the problem. First of all, Fig. 11 and Fig. 12 illustrate the impact of dissimilar relationships of common component's completion rates γ and its relating values δ on E[TCU(TA*, n*)] and TA*, respectively. For a linear relationship between δ and γ, the analytical outcome confirms that at γ = 0.5, E[TCU(TA*, n*)] = $2,305,879 (see Fig. 11) and TA* = 0.5379 (refer to Fig. 12). Our model can reveal the optimal operating policy (TA*, n*) and system cost E[TCU(TA*, n*)] for any given nonlinear relationships as may exist in real application systems. For instances, when δ = γ 3 and γ remains at 0.5, the analytical results show E[TCU(TA*, n*)] = $2,237,446 (refer to Fig. 11) and TA* = 0.6654 (see Fig. 12). That is for a 50% added output quantity due to the overtime alternative.    Our model can explicitly study additional system characteristics. Fig. 16 exposes the mean rework cost ratio (CR,i / Ci) effect on each end product's rework cost. As the mean (CR,i / Ci) ratio rises, each end product's rework cost significantly surges.  . It shows two main cost contributors: the variable expenses in stages 2 and 1; each contributes 45.10% and 29.49%. The overtime relating expense of 7.50% is the third large expense. It follows that the buyer's holding cost 4.36%, the end products' setup and delivery cost 3.83% and 3.45%. Then, the end products' holding and rework cost 3.05% and 1.90%. Finally, we compare our utilization with an existing model without overtime option (Chiu et al., 2016), as exhibited in Fig. 19. A further investigation, indicating that by reducing a 16.16% in utilization (declining from 0.2964 to 0.2485). However, the price paying is a 8.02% increase in E[TCU(TA*, n*)] (surging from $2,134,736 to $2,305,879, refer to Fig. 15).

Fig. 19.
Comparison of this study's utilization with an existing work without overtime

Conclusions
In the current stiff competitive supply-chain environment, customers' product demand trend turns to diversity, quality, and rapid response. By considering the existence of products' commonality, required quality, and short lead time, this study proposes a model featuring postponement, overtime fabricating the mutual parts, rework of defects, and multiple deliveries to assist today's producers in their multiproduct fabrication-distribution planning. Techniques of explicitly modeling, formulating, and system cost minimization were used to simultaneously derive the optimal cycle-time and shipping frequency for the studied problem. The obtaining results can be applied to a variety of today's industries (e.g., automotive, household goods, etc.) in revealing in-depth managerial insights for planning fabrication-shipment policy in their supply chains. As a demonstrating example, this model discloses the following important managerial information: (1) The optimal fabrication-transportation policy (see Fig. 5).
(2) Joint impact of various system factors (such as overtime add-up rate, defect portion, mean rework-cost ratio, mutual part's completion rate, and overtime set-up cost) on the rework cost, total system cost, and mutual parts' uptime plus rework time. Refer to subsection 4.1., Fig. 6 to Fig. 10. (3) The impact of dissimilar relationships of mutual component's completion rate and its corresponding values on total system cost and cycle-time (see Fig. 11 to Fig. 12). (4) The effect of overtime-related ratios on the sum of optimal uptime and rework time, utilization, and each cost contributor ( Fig. 13 to Fig. 15). (5) The influence of variations in average rework-cost ratio and delivery frequency on the rework cost of each end product, and total system cost and separate cost contributor (see Figs. 16-17). (6) The breakup of detailed cost contributors and a comparison of this model's utilization against a closely related previous model (refer to Fig. 18 to Fig. 19).
In summary, the significant contributions of this study are: (1) it presents a multiproduct postponement model to analyze a real fabrication-distribution problem featuring overtime and rework explicitly; and (2) it discloses diverse, in-depth system information to facilitate managerial decision makings. In addition, considering variable end products' demand rates for the studied problem is worth investigating in the future study.