A delayed differentiation multiproduct model with the outsourcing of common parts, overtime strategy for end products, and quality reassurance

Article history: Received December 1


Introduction
This study presents a two-stage delayed-differentiation multiproduct model that considers the outsourcing options for common parts, quality reassurance, and overtime strategy for end products to assist in making fabrication runtime decisions that are cost-effective. Production planners always seek a better multiproduct manufacturing scheme such as the postponement (or delayed-differentiation) policy to reduce operation uptime and total relevant expenses, especially when dealing with end products with common components. Van Hoek (1999) conducted interviews from various industries, including clothing, electronics, food, and automotive manufacturers in Germany, Belgium, and the Netherlands, to explore the challenge and advantage of the postponement/reconfiguration of the industrial supply chains. Their results could facilitate managerial decision-making. Blecker and Abdelkafi (2006) analyzed mass customization systems' complexity and variety to offer managerial insights to deal with potential problems cause by the complexity and variety in the mass customization 2. The proposed delayed differentiation multiproduct model

Assumption and description
The proposed delayed differentiation multiproduct model incorporates strategies of outsourcing common parts in stage one, overtime production of end products in stage two, and quality reassurance. This study builds a decision model to represent our studied problem. A Nomenclature (in Appendix A) defines the related symbols used in our study. The assumption and descriptions of the problem are as follows. A two-stage batch multiproduct manufacturing plan is designed to satisfy annual requirements λi of L distinct products (where i = 1, 2, … , L). Since the existence of commonality among these multiproduct, stage one fabricates all needed common parts, and stage two produces the finished products. Fig. 1 depicts the stock status of the proposed model featuring outsourcing in stage one, overtime in stage two, and quality reassurance. A constant common part's completion rate γ is assumed. Both the common part and end product's fabrication rates P1,0 and P1,i depend on γ. For example, if γ = 50%, then both P1,0 and P1,i are twice as much as their actual rates in a single-stage manufacturing system. This study employs the rotation cycle time discipline, so the cycle length is as follows: 1, 2, 3, where 0, 1, 2, ... , (1) Fig. 1. The inventory status in a replenishment cycle of the proposed model featuring the strategies of outsourcing in stage one, overtime in stage two, and quality reassurance compared to the same model without neither strategies To effectively reduce the uptime, we use an external contractor to supply a π0 portion of the common part's batch in stage one and employ an overtime strategy to produce L different products. We exhibit the following consequent changes in cost parameters and manufacturing/ reworking rates due to implementing outsourcing (see Eqs. (2) and (3)) and overtime strategies (Eqs. (4) to (8)): Production in both stages is assumed imperfect. Random defective rates x0 and xi exist in both stages. The scrap portions θ1,0 and θ1,i are associated with these faulty items. They are identified, and either reworked/repaired or removed. From Fig. 1, we notice that in stage one, when uptime completes the stock accumulates to H1,0, and at the end of rework, the stock arrives at H2,0. Then, the outsourced items are received as per the predetermined schedule, and the stock level reaches H3,0 before stage two. In the second stage, at the end of uptime of the product i (where i = 1, 2, …, L), its stock accumulates to H1,i, and when rework finishes, the stock arrives at H2,i. Then, its inventory starts to deplete under a continuous issuing plan (see Fig. 1). In this study, no shortages are permitted, so the following formulas must hold: in stage-1, (P1,0 -d1,0 ) > 0 and in stage-2, (PT1,i -dT1,i -λi) > 0. Meantime, in stage two (see Fig. 1), the common parts' level declines a quantity of Qi (from H3,0) to the level of Hi at the end of each product's uptime (see also Fig. 2).

Fig. 2.
The status of common parts in stage two of a replenishing cycle Fig. 3 and Fig. 4 exhibit the inventory status of defective and scrapped items in our proposed model. Fig. 3 shows that when the common part's uptime completes, the defective stock level reaches (d1,0 t1,0). Then, all scrapped items are removed, and the defective stock gradually depletes to zero during the rework time. The same status occurs for end products in stage two.   Fig. 4 shows that when stage one' uptime and rework times complete, the scrap stock level reaches its maximum level [d1,0(θ1,0)t1,0 + d2,0 t2,0]. For the end product i in stage two, the maximum level accumulates to [dT1,i (θ1,i)t1,i + dT2,i t2,i] at the end of uptime and rework times.

Formulations of stage two
We start with formulation of manufacturing L products in stage two. For i = 1, 2, …, L, we can directly observe the following formulas from Figs. 1 through 4:

Formulations of stage one
According to the lot size of each end product i (as shown in Eq. (9)), we notice the needed total common parts are as follows: For stage one, from Figs. 1 through 4, we also can directly observe the following formulas:

System cost function and problem solution
The total cost per replenishing cycle TC(TZ) includes the following expenses incurred in both stages: (a) for stage one, variable and setup costs for outsourcing and in-house fabrication, and the rework, disposal, and inventory holding costs; (b) for stage two: the summation of variable, setup, rework, removal, and stock holding costs for manufacturing of L different products.

Discussion on the setup times condition
If the summation of the setup times Si is greater than the idle time in a cycle length (see Fig. 1), then, it is required to calculate Tmin (as pointed out in Nahmias (2009)) and select the maximum of (TZ*, Tmin) as the final answer for cycle length for the proposed problem.

Prerequisite condition of the proposed problem
The following prerequisite formulas must hold to ensure the equipment has sufficient capacity in both stages of this study for manufacturing and reworking common parts and L different products (Nahmias, 2009).

Example
A multiproduct batch plan needs to meet the annual requirements of five different products. This plan employs a two-stage postponement strategy to cope with the common part among these products. It also adopts an outsourcing alternative for common parts in stage-1, an overtime option for end products in stage-2, and quality reassurance in both stages. Table 1 shows the assumption of variables' values in stage one, while Tables 2 and 3 exhibit the relating variables' values in stage two. In contrast, the variables' values used for a single-stage fabrication plan for the same problem are displayed in Appendix C (see Tables C-1 and C-2).

Table 1
Assumption of variables' values in stage 1   The present study implements two different strategies to reduce the uptimes in both fabrication stages. Their effects are analyzed explicitly in terms of machine utilization and the cost they are paying as follows. Figure 6 depicts the impact of outsourcing (in stage one) on machine utilization. It reveals that when outsider suppliers provide 40% of the common parts, the utilization drops 23.58% (i.e., from 0.2500 to 0.1910, please refer to Table D-1 in Appendix D for details).   Fig. 7 illustrates that to reduce machine utilization by 23.58%, we are paying the price of 4.74% in system cost E[TCU(TZ * )] increase, i.e., the system cost goes up from $2,312,415 to $2,422,005 (see Table D-1 in Appendix D). Fig. 8 exhibits the influence of overtime (in stage two) on machine utilization. It reveals that when output rate increases 50% in the fabrication of end products, the utilization declines 21.2% (i.e., from 0.2423 to 0.1910, please refer to Table D-2 in Appendix D for details). Fig. 9 shows that for reducing machine utilization by 21.2%, we are paying the price of 13.26% in system cost E[TCU(TZ * )] increase, i.e., the system cost goes up from $2,138,414 to $2,422,005 (see Table D-2 in Appendix D). The collective impact of overtime factor α1,i and outsourcing ratio π0 on machine utilization are explored and depicted in Fig.  10. It indicates that as α1,i rises, the machine utilization decreases noticeably; as π0 goes up, utilization drops severely. Fig. 11 illustrates the analytical outcomes of the combined effect of overtime factor α1,i and outsourcing ratio π0 on system cost E[TCU(TZ * )]. It clearly illustrates that as both α1,i and π0 rise, the system cost increases noticeably. Our example shows that α1,i has more impact than π0 on the system cost E[TCU(TZ * )] increase. Fig. 12 explicitly analyzes the individual and collective impact of α1,i and π0 on system cost. Based on our parameter assumptions, this study suggests that it is more economical first to outsource 40% of the common parts and, in the meantime, gradually implement the overtime factor. As soon as we reach the point of α1,i = 0.5 and π0 = 0.4, to further reduce the machine utilization, it is more economical to stop increase the overtime factor (i.e., let α1,i remains at 0.5) and solely increase outsourcing percentage. Our proposed model can conduct a similar analysis for any given parameter assumptions and offers in-depth analytical information to facilitate managerial decisionmaking.

.3. Impact of common part's completion rate and its value on the proposed problem
The analytical outcomes of the influence of common part's completion rate and its value on the proposed problem are illustrated below.  Fig. 13 displays the influence of γ on the optimal fabrication uptime plus rework time t0 * . It reveals that as γ rises, more fabrication and rework time for the common part are needed. At our assumption where γ = 0.5, t0 * = 0.0513 (see Table D-1 in Appendix D). It also indicates that at γ = 0.5, by outsourcing 40% of common parts, t0 * drops 37.5% (i.e., it declines from 0.0821 to 0.0513). The impact of the linear (i.e., δ = γ 1 ) and nonlinear (e.g., δ = γ 1/3 or δ = γ 3 ) relationships between  and  on the optimal decision variable TZ* is explicitly explored and exhibited in Fig. 14. First, it reconfirms our example that when  = 0.50, TZ* = 0.5791. It also reveals the crucial information of TZ* variations concerning nonlinear relationships between δ and γ.

The influence of quality reassurances on the proposed problem
The collective impact of average defective and scrap rates on E[TCU(TZ * )] is illustrated in Fig. 15. It indicates that as  rises, E[TCU(TZ * )] surges noticeable; as x increases, the system cost upsurges significantly. The combined influence of overtime factor α1,i and average scrap rate on the sum of end products' uptime and rework time is depicted in Figure 16. It shows that as  rises, ti * changes slightly; as the overtime factor α1,i increases, ti * declines severely.

Discussion and comparisons
In addition to exposing the system characteristics mentioned earlier, the proposed model can analyze various previously inaccessible managerial information on the problem. For example, Fig. 18 illustrates the variations of (t1,i + t2,i) of each end product concerning α1,i. It reveals that as overtime factor α1,i increases, the sum of each end product's uptime and reworking time declines considerably.  Fig. 19 compares the proposed model's machine utilization with that in other similar models. It demonstrates that by implementing the dual strategies (i.e., outsourcing and overtime options) on reducing fabrication times, our proposed model has the lowest machine utilization (i.e., 0.1910), which is a 21.2% lower than a model solely using outsourcing in stage one (Chiu et al., 2021). It is a 36.6% and 39.2% less utilization than the model without applying any expediting strategy or without delayed differentiation (i.e., the single-stage manufacturing scheme). Further analysis discloses that we are paying the price of a 13.26%, 19.40%, or 14.00% increase in E[TCU(TZ * )], respectively, for the utilization reduction exhibited in Fig. 19. This crucial information was previously inaccessible to managers in supporting their production planning and management.

Conclusions
To simultaneously face the time and quality demands of various goods externally and meet limited capacity internally, production planners today must seek an efficient tool to make cost-effective fabrication decisions. Motivated by helping their decision-making, this study presents a two-stage delayed-differentiation multiproduct model that considers the outsourcing options for common parts, overtime strategy for end products, and quality reassurance. The researchers use the mathematical modeling, analysis, and optimization approach to decide the best rotation cycle length that minimizes the system's expenses. We use a numerical illustration to validate our result's applicability and exhibit the proposed model's capability.
The present work contributes to practical multiproduct-fabrication by (1) Deriving the optimal manufacturing policy for a delayed-differentiation multiproduct system with dual uptime reduction policies and quality reassurance (refer to Sections 2 and 3, and Figure 5); (2) Offering a decisional model that allows present-day production planners to explore the collective/separate effect of a quality-ensured and dual uptime reduction strategy on (i) the problem's operating policy (see Figures 6, 8, 10, 12, and 13) and (ii) crucial system performance indicators (refer to Figures 7,9,11,and 14 to 18) to help cost-effective decision-making; and (3) Comparing/demonstrating the model's performance with various closely related models (see Figure 19). Combining a discontinuous product-shipment plan into this problem's context will be a worthy study subject in the future.

TZ
= the replenishing cycle length (i.e., decision variable), λi = annual demand rate for L distinct end product i (where i = 1, 2, …, L), λ0 = annual demand rate of the common part, Qi = lot size for product i, Q0 = in-house lot size for common parts in stage 1, I(t)i = stock level at time t (where i = 0, 1, 2, …, L), H3,0 = stock level of common parts upon receipt of the outsourced items, H2,0 = common parts' stock level when rework process completes, H1,0 = common parts' stock level when fabrication process finishes, t1,0 = common parts' fabrication uptime, t2,0 = common parts' rework time, t3,0 = common parts' depletion time, t0 * = the sum of optimal uptime and rework time of the common parts (in stage one), t1,i = uptime for end product i, t2,i = rework time for end product i, t3,i = depletion time for end product i, ti * = the optimal total uptimes and rework times of the end products (in stage two), π0 = the outsourcing portion of the lot size of common parts (in stage 1), K0 = in-house common part's setup cost, Kπ0 = fixed cost of outsourcing, β1,0 = the connecting factor between Kπ0 and K0, C0 = in-house common part's unit cost, Cπ0 = common part's unit outsourcing cost, β2,0 = the connecting factor between Cπ0 and C0, P1,0 = common part's annual fabrication rate, x0 = common part's random defective rate, d1,0 = defective common parts' production rate (i.e., d1,0 = P1,0 x0), θ1,0 = scrap portion of defective common parts, h1,0 = common part's unit holding cost, h 4,0 = safety common part's unit holding cost, P2,0 = annual reworking rate for defective common part's, CR,0 = common part's unit rework cost, θ2,0 = scrap portion of the reworked common parts, d2,0 = scrap common parts' production rate in t2,0 (i.e., d2,0 = P2,0 θ2,0), CS,0 = unit disposal cost for scrapped common parts, φ0 = common part's total scrap portion, h2,0 = unit holding cost for reworked common parts, i0 = the relating ratio for holding cost (e.g., h1,0 = i0 C0), γ = the completion rate of common part compared with the end item, S0 = common part's setup time, Si = setup time for end product i, H1,i = stock level of end product i when its production process completes, Hi = inventory level of common parts when the production process of item i ends, P1,i = standard annual fabrication rate for end product i, PT1,i = overtime annual output rate for end product i, α1,i = the connecting factor between PT1,i and P1,i, KT,i = setup cost for end item i when overtime is used, Ki = standard setup cost for end product i, α2,i = the connecting factor between KT,i and Ki, CT,i = unit fabrication cost for end product i when overtime is implemented, Ci = standard unit fabrication cost for end product i, α3,i = the connecting factor between CT,i and Ci, h1,i = holding cost for end product i, h4,i = unit holding cost for the safety end product i, xi = random defective rate for end product i, dT1,i = fabrication rate of defective end product i (i.e., dT1,i = PT1,i xi),

Table D-2
The impact of changes in overtime factor α1,0 on various system parameters