Incorporating Outsourcing Strategy and Quality Assurance into a Multiproduct Manufacturer–Retailer Coordination Replenishing Decision

: This study explores the multiproduct manufacturer-retailer coordination replenishing decision featuring outsourcing strategy and product quality assurance. Globalization has generated enormous opportunities. Consequently, transnational ﬁrms now face tough competition in global markets. To stay competitive, a ﬁrm should meet the client’s multi-item and quality requirements under capacity constraints and optimize the intra-supply chain system to allow the timely distribution of ﬁnished goods under minimum system cost. The outsourcing option is considered to release machine loadings and reduce cycle time e ﬀ ectively. All items fabricated are screened for quality, and reworkable and scrap items are separated. Any reworked items that fail the quality reassurance screening are discarded, whereas all outsourced products are quality-guaranteed by the provider. A ﬁxed-quantity multi-shipment plan is used when the whole ﬁnished lot is quality-ensured to help present-day transnational ﬁrms gain competitive advantage by making e ﬃ cient and cost-e ﬀ ective multiproduct manufacturing and delivering decisions. Mathematical modeling is built to portray the system’s characteristics, and conventional di ﬀ erential calculus is used to solve and derive the optimal operating policy for the proposed problem. Simultaneously, we ﬁnd the optimal delivery frequency and common cycle time for the problem mentioned above. A simulated numerical example and sensitivity analysis demonstrate the research result’s capability and applicability. Our precise analytical model can reveal / highlight the impact of deviations in quality- and outsourcing-related features on the optimal operating policy and several performance indicators that help managerial decision-making.


Introduction
This study explores the multiproduct manufacturer-retailer coordination replenishing decision incorporating an outsourcing strategy and product quality assurance. Due to the growing tendency of the market's multi-item requirements in past decades, studies on multi-item fabrication planning and controlling have been broadly carried out. Lee et al. [1] employed goal programming to study a multi-period multiproduct fabrication scheduling problem. A model including three distinct fabrication lines and one inspection facility was built to deal with this multi-machine multiple objectives scheduling problem. An example was used to demonstrate how the proposed goal programming can handle the repairs to them, or remove them from the quality-ensured finished lot. Dohi et al. [20] examined the economical manufacturing quantity (EMQ) models incorporating the Poisson machine failure rate and repairs. Models were constructed and formulated under two separate machine repairing policies to minimize the manufacturing operations' steady cost functions. Inderfurth et al. [21] explored a deterministic production planning problem wherein the regular fabrication and rework processes are scheduled on the same machine. The reworked items have a limited deterioration time while waiting to be repaired, and as waiting time gets longer the rework cost increases accordingly. The authors aimed to find the optimal regular lot sizes and amount of reworked items that keep total cost at a minimum. A proposed polynomial dynamic programming algorithm solved the problem. Extra studies [22][23][24][25] investigated different features of imperfect fabrication systems and their consequent actions.
Optimization of the intra-supply chain system (i.e., similar to the manufacturer-retailer integrated type of system) in current transnational enterprises will allow the timely distribution of their finished goods and minimize total system cost. Banerjee and Banerjee [26] examined a single product single-vendor multi-buyer inventory system that features order-less replenishment. They built a model to depict the problem's characteristics using a common cycle replenishing policy. It could also be computerized to enable real-time data interchange between trading parties. Viswanathan and Piplani [27] explored the benefit of a single product single-vendor multi-buyer coordinated supply-chain system under the common inventory replenishment periods. It was assumed that the vendor offered a price discount, so the vendor determines the common replenishing periods, and all buyers that follow these preset times to refill their stocks will receive the benefit through price discount. The objective was to jointly decide for the vendor the optimal replenishment times and the offering of a discounted price. Sancak and Salmann [28] studied a multiproduct dynamic lot-sizing problem featuring delayed delivery policy, wherein multiple items were purchased by a producer to meet its production needs, and the objective was to optimize the policies of ordering and inbound delivery that kept delivery and stock holding cost at a minimum. Regular delivery cost charge is assumed based on a full truckload. The authors explored using safety stocks to delay delivery to the following period with less than a full truckload. Real data from a transportation manufacturer was used to examine this delay delivery option's effect on service levels and system costs. As a result, the total delivery and stock holding cost were reduced without increasing the stock-out risk. Additional recent works [29][30][31][32] studied various natures of intra-supply chain or vendor-buyer integrated types of systems. The urgent need for a precise model to help managers of present-day transnational firms make efficient and cost-effective multiproduct manufacturing and delivering decisions. As few studies mainly focused on this specific area, this study aims to bridge the research gap by building a decision-support mathematical model to optimize the multi-item manufacturer-retailer integrated inventory system incorporating outsourcing and quality guarantee. This study's main contribution is that it can reveal/highlight the impact of deviations in quality-and outsourcing-related features on the optimal operating policy and several performance indicators that help managerial decision-making.
The rest of the paper includes the problem's description and mathematical modeling in Section 2 (containing notation, assumption, formulations, convexity, solution process, and prerequisite condition). Numerical example with sensitivity analyses in Section 3, and Conclusion in Section 4.

Assumptions and Notations
This study optimizes a multi-item manufacturer-retailer integrated inventory system with outsourcing and quality guarantee. We consider an inventory system having a multiproduct fabrication plan on a single facility, under a rotation cycle discipline along with a partial outsourcing policy. Specifically, in each cycle a π i proportion of batch size Q i for each product i is provided by the outside contractor (where i = 1, 2, . . . , L). As a part of the agreement, outsourced items must have perfect quality and be received right before product delivery time (see Figure 1). Accordingly, K πi and C πi denote the constant setup and unit purchase costs for outsourced items.
portion fails (where 0 ≤ θ2i ≤ 1). So, the production rate of scrap d2i is θ2iP2i, and the maximum level of scrap items in a cycle is φi xi [(1 − πi) Qi], where φi is the sum of scrap rates among in-house defective items in t1iπ and t2iπ (so, φi = [θ1i + θ2i(1 − θ1i)]). Figure 3 exhibits the on-hand inventory of scraps in the proposed multi-item system. At the end of reworking, outsourced products are received in time to bring the stock level to Hi. Then, a fixed amount of multiple shipments of the quality-assured batch is shipped to the retailer at a fixed time interval tniπ (Figures 1 and 4).
Extra notations used in the proposed multi-item manufacturer-retailer integrated system are listed in Table 1   The other (1 − π i ) portion of Q i for each product is manufactured in-house at P 1i products per year. However, the in-house processes are not perfect. A random x i portion of defective items are generated at the d 1i rate. Defective items are checked to separate a θ 1i portion of scrap from the other (1 − θ 1i ) re-workable (where 0 ≤ θ 1i ≤ 1). To avoid stock-out circumstances, the manufacturing rate P 1i has to satisfy (P 1i − d 1i − λ i ) > 0 (where λ i represents product i's demand rate and d 1i equals to x i P 1i ). In each cycle, when the regular processes end, the reworking of each end product i is performed at the rate P 2i (Figure 1) with extra unit rework cost C Ri . Figure 2 shows the on-hand inventory of defective products in the proposed multi-item manufacturer-retailer integrated system. In the rework, a θ 2i portion fails (where 0 ≤ θ 2i ≤ 1). So, the production rate of scrap d 2i is θ 2i P 2i , and the maximum level of scrap items in a cycle is where ϕ i is the sum of scrap rates among in-house defective items in t 1iπ and t 2iπ (so, Figure 3 exhibits the on-hand inventory of scraps in the proposed multi-item system. At the end of reworking, outsourced products are received in time to bring the stock level to H i . Then, a fixed amount of multiple shipments of the quality-assured batch is shipped to the retailer at a fixed time interval t niπ (Figures 1 and 4).         Extra notations used in the proposed multi-item manufacturer-retailer integrated system are listed in Table 1 below (where i = 1, 2, . . . , L): stock level of product i in the retailer's side at time t, t 1i uptime for the product i in the proposed system without outsourcing plan, t 2i rework time in a system without outsourcing, t 3i delivery time in a system without outsourcing, T rotation cycle time a system without outsourcing, the long-run average system cost per unit time,

Formulations
From Figures 1-3, we observe the following formulas: Figure 4 exhibits the on-hand inventory status in t 3iπ . Total inventories of product i are [33] as follows: 1 Figure 5 depicts the stock status at the retailer's side. Because n fixed quantity shipments are transported to the retailer at a fixed t niπ time period, the following formulas can be observed:  Figure 5 depicts the stock status at the retailer's side. Because n fixed quantity shipments are transported to the retailer at a fixed tniπ time period, the following formulas can be observed: TC(T π , n) for L distinct end products consists of the fixed and variable outsourcing and in-house fabrication costs, variable in-house rework and disposal costs, fixed and variable distribution costs, holding costs for reworked, finished, and defective items during T π , and holding costs in the retailer side. Let β 1i be the linking factor between K i and K πi , and K πi = K i (1 + β 1i ). Because the in-house setup cost K i is often much greater than the fixed delivery cost K πi , we assume that −1 < β 1i < 0. Also, let β 2i denote the linking factor between C i and C πi , and C πi = C i (1 + β 2i ), since unit outsourcing price is more significant than unit in-house manufacturing cost, so we assume where β 2i > 0.
Apply expected values E[x i ] to deal with the randomness of x i , substitute Equation (1) to (15) and the aforementioned linking parameters K πi and C πi in Equation (16), with extra derivations E[TCU(T π , n)] are obtained as follows:

Convexity and the Optimal Solution
Before deriving the optimal (T π *, n*) solutions, we first verify that if E[TCU(T π , n)] is convex. Applying the Hessian matrix equations (Rardin [34]), Equation (18) can be obtained (for details refer to Appendix B): Equation (18) yields a positive result, since K i , (1 + β 1i ), and T π are positive. Therefore, E[TCU(T π , n)] is strictly convex for all n and T π values other than zero, and a minimum for E[TCU(T π , n)] exists.
In order to simultaneously decide T π * and n*, we set the following first derivative of E[TCU(T π , n)] concerning T π and n both equal to zero, and then solve this linear system (i.e., Equations (19) and (20)).
The following optimal T π * and n* can be derived simultaneously with extra derivations: and

The Prerequisite Condition of the Fabrication
Sufficient capacity for the proposed multi-item fabrication and rework processes need to be guaranteed. Therefore, the following prerequisite formula must hold: If the summation of setup time S i becomes significant to T π , then Equation (24) also must hold: or, T π must be larger than T min as follows (refer to Appendix C for details): Therefore, when incorporating setup times into the proposed problem, one should select the maximum of T π * (i.e., Equation (20)) or T min (i.e., Equation (24)) (Nahmias [35]) as the optimal length.

Numerical Example
This section offers a simulated numerical example and the sensitivity analyses to illustrate our results' applicability. As exhibited in Table 2, these assumed parameters' values are for demonstration purposes.  Using the assumed values of variables (as shown in Table 2) to calculate Equations (21), (22) and (17), we obtain n* = 3, T π * = 0.5982, E[TCU(T π *, n*)] = $2,390,389 (for π at 0.4; see Table A1 in Appendix D). It is noted that the cost for quality guarantee in the proposed system is $70,423, that is 2.95% of E[TCU(T π *, n*)] (see Table A1 in Appendix D).
The effect of variations in average unit cost linking parameter β 2 on total cost of each end product is analyzed, and its outcome is depicted in Figure 6. It indicates that as β 2 increases, each item's total cost goes up accordingly because unit outsourcing cost is higher than unit in-house manufacturing cost.   The effect of variations in rotation cycle time Tπ on different cost contributors of E[TCU(Tπ, n)] is explored and the outcomes are illustrated in Figure 8. It is noted that as cycle time Tπ increases, the cost for quality assurance goes up accordingly, and both holding costs at customer and producer sides increase significantly. Conversely, annual delivery costs, and both in-house and outsourcing setup costs decrease notably. In Figure 8, as the cycle length Tπ varies, the expected annual variable cost (λCi) changes slightly. Because (λCi) represents the annual variable cost, it is not directly/significantly affected by lot-size Qi or cycle-time Tπ.    The effect of variations in rotation cycle time Tπ on different cost contributors of E[TCU(Tπ, n)] is explored and the outcomes are illustrated in Figure 8. It is noted that as cycle time Tπ increases, the cost for quality assurance goes up accordingly, and both holding costs at customer and producer sides increase significantly. Conversely, annual delivery costs, and both in-house and outsourcing setup costs decrease notably. In Figure 8, as the cycle length Tπ varies, the expected annual variable cost (λCi) changes slightly. Because (λCi) represents the annual variable cost, it is not directly/significantly affected by lot-size Qi or cycle-time Tπ. The effect of variations in rotation cycle time T π on different cost contributors of E[TCU(T π , n)] is explored and the outcomes are illustrated in Figure 8. It is noted that as cycle time T π increases, the cost for quality assurance goes up accordingly, and both holding costs at customer and producer sides increase significantly. Conversely, annual delivery costs, and both in-house and outsourcing setup costs decrease notably. In Figure 8, as the cycle length T π varies, the expected annual variable cost (λC i ) changes slightly. Because (λC i ) represents the annual variable cost, it is not directly/significantly affected by lot-size Q i or cycle-time T π .  Figure 9 displays the impact of changes in average outsourcing percentage on each item's total cost. It reveals that as becomes higher, each item's total cost increases accordingly, for outsourcing is a more expensive stock-replenishment policy.
The impact of differences in average outsourcing percentage on overall machine utilization for multi-item manufacturing processes is studied, and the outcome is depicted in Figure 10. It shows that machine utilization declines significantly as increases; and at = 0.4 (in our example), machine utilization drops from 65.8% (refer to Table A2 in Appendix D) to 39.0%. This utilization drop is at the cost of 6.75% increase in E[TCU(Tπ*, n*)] (for the system cost goes up from $2,239,231 to $2,390,389, refer to Table A1). Moreover, Table A2 reveals real production uptime, rework time, and idle time.   Figure 9 displays the impact of changes in average outsourcing percentage π on each item's total cost. It reveals that as π becomes higher, each item's total cost increases accordingly, for outsourcing is a more expensive stock-replenishment policy.  Figure 9 displays the impact of changes in average outsourcing percentage on each item's total cost. It reveals that as becomes higher, each item's total cost increases accordingly, for outsourcing is a more expensive stock-replenishment policy.
The impact of differences in average outsourcing percentage on overall machine utilization for multi-item manufacturing processes is studied, and the outcome is depicted in Figure 10. It shows that machine utilization declines significantly as increases; and at = 0.4 (in our example), machine utilization drops from 65.8% (refer to Table A2 in Appendix D) to 39.0%. This utilization drop is at the cost of 6.75% increase in E[TCU(Tπ*, n*)] (for the system cost goes up from $2,239,231 to $2,390,389, refer to Table A1). Moreover, Table A2 reveals real production uptime, rework time, and idle time.  The impact of differences in average outsourcing percentage π on overall machine utilization for multi-item manufacturing processes is studied, and the outcome is depicted in Figure 10. It shows that machine utilization declines significantly as π increases; and at π = 0.4 (in our example), machine utilization drops from 65.8% (refer to Table A2 in Appendix D) to 39.0%. This utilization drop is at the cost of 6.75% increase in E[TCU(T π *, n*)] (for the system cost goes up from $2,239,231 to $2,390,389, refer to Table A1). Moreover, Table A2 reveals real production uptime, rework time, and idle time. Moreover, our proposed model can reveal the critical ratio of to support the make-or-buy decision (see Figure 11). It shows that as goes up to 0.702 or higher, a 100% outsourcing option is more cost-effective (for details, please refer to Table A2, in Appendix D). Furthermore, Figure 12 illustrates the impact of changes in average scrap rate on E[TCU(Tπ*, n*)]. It specifies that as rises, E[TCU(Tπ*, n*)] increases considerably, and at = 0.4 and ̅ = 0.3, if increases to 0.349 or higher, a 100% outsourcing plan (i.e., the 'buy' decision) are more economical. Additionally, another critical ratio = 0.550 (at = 0 and = 0.3) is revealed by the proposed model to support managerial decision making. That is if the average scrap rate is greater than 0.550, then 'buy' is recommended.  Moreover, our proposed model can reveal the critical ratio of π to support the make-or-buy decision (see Figure 11). It shows that as π goes up to 0.702 or higher, a 100% outsourcing option is more cost-effective (for details, please refer to Table A2, in Appendix D). Moreover, our proposed model can reveal the critical ratio of to support the make-or-buy decision (see Figure 11). It shows that as goes up to 0.702 or higher, a 100% outsourcing option is more cost-effective (for details, please refer to Table A2, in Appendix D).
Furthermore, Figure 12 illustrates the impact of changes in average scrap rate on E[TCU(Tπ*, n*)]. It specifies that as rises, E[TCU(Tπ*, n*)] increases considerably, and at = 0.4 and ̅ = 0.3, if increases to 0.349 or higher, a 100% outsourcing plan (i.e., the 'buy' decision) are more economical. Additionally, another critical ratio = 0.550 (at = 0 and = 0.3) is revealed by the proposed model to support managerial decision making. That is if the average scrap rate is greater than 0.550, then 'buy' is recommended. Figure 11. Effect of variations in on E[TCU(Tπ*, n*)] for managerial make-or-buy decision. Figure 11. Effect of variations in π on E[TCU(T π *, n*)] for managerial make-or-buy decision. Furthermore, Figure 12 illustrates the impact of changes in average scrap rate ϕ on E[TCU(T π *, n*)]. It specifies that as ϕ rises, E[TCU(T π *, n*)] increases considerably, and at π = 0.4 and x = 0.3, if ϕ increases to 0.349 or higher, a 100% outsourcing plan (i.e., the 'buy' decision) are more economical. Additionally, another critical ratio ϕ = 0.550 (at π = 0 and π = 0.3) is revealed by the proposed model to support managerial decision making. That is if the average scrap rate ϕ is greater than 0.550, then 'buy' is recommended.

Joint Impacts from Combined System Factors
Looking into the quality guarantee matter in manufacturing processes, joint impacts of variations in average scrap rate and average defective rate ̅ on E[TCU(Tπ*, n*)] are investigated. The results are presented in Figure 13. This specifies that E[TCU(Tπ*, n*)] raises drastically, as both ̅ and increase. Figure 14 shows

Joint Impacts from Combined System Factors
Looking into the quality guarantee matter in manufacturing processes, joint impacts of variations in average scrap rate ϕ and average defective rate x on E[TCU(T π *, n*)] are investigated. The results are presented in Figure 13. This specifies that E[TCU(T π *, n*)] raises drastically, as both x and ϕ increase.

Joint Impacts from Combined System Factors
Looking into the quality guarantee matter in manufacturing processes, joint impacts of variations in average scrap rate and average defective rate ̅ on E[TCU(Tπ*, n*)] are investigated. The results are presented in Figure 13. This specifies that E[TCU(Tπ*, n*)] raises drastically, as both ̅ and increase. Figure 14 shows    Figure 14 shows the analytical result of the joint effects of changes in rotation cycle time T π and average unit cost linking parameter β 2 on E[TCU(T π , n)]. It indicates that E[TCU(T π , n)] raises considerably, as β 2 goes up; and when T π deviates from its optimal value 0.5982, E[TCU(T π , n)] increases significantly. Furthermore, the joint impacts of changes in π and ϕ on E[TCU(T π *, n*)] is analyzed, and the outcome is presented in Figure 15. It can be seen that (1) when π is smaller than 0.5, E[TCU(T π *, n*)] boosts up considerably, as ϕ increases and; (2) quite the opposite, when ϕ > 0.65, E[TCU(T π *, n*)] declines notably, as π increases. However, (3) when ϕ < 0.4, as π goes up, E[TCU(T π *, n*)] increases accordingly.  The reasons for the situations mentioned above are as follows: (1) if the amount of outsourced items is less than that of in-house manufactured items, the impact of on E[TCU(Tπ*, n*)] is significant; (2) in contrast, although is high, when the outsourced amount is much larger than in-house made amount, E[TCU(Tπ*, n*)] decreases notably, as goes up; and (3) the impact from (in terms of the in-house quality cost) does not exceed that from (in terms of outsourcing added cost), hence, as increases, E[TCU(Tπ*, n*)] still raises accordingly.

Conclusions
To meet the client's multi-item and quality requirements under capacity constraints and to satisfy the timely distribution of finished goods under minimum system cost, a multi-item manufacturer-retailer integrated type of system incorporating outsourcing and quality guarantee is explored. All in-house fabricated/reworked products are screened to make sure of the desired  The reasons for the situations mentioned above are as follows: (1) if the amount of outsourced items is less than that of in-house manufactured items, the impact of on E[TCU(Tπ*, n*)] is significant; (2) in contrast, although is high, when the outsourced amount is much larger than in-house made amount, E[TCU(Tπ*, n*)] decreases notably, as goes up; and (3) the impact from (in terms of the in-house quality cost) does not exceed that from (in terms of outsourcing added cost), hence, as increases, E[TCU(Tπ*, n*)] still raises accordingly.

Conclusions
To meet the client's multi-item and quality requirements under capacity constraints and to satisfy the timely distribution of finished goods under minimum system cost, a multi-item manufacturer-retailer integrated type of system incorporating outsourcing and quality guarantee is explored. All in-house fabricated/reworked products are screened to make sure of the desired The reasons for the situations mentioned above are as follows: (1) if the amount of outsourced items is less than that of in-house manufactured items, the impact of ϕ on E[TCU(T π *, n*)] is significant; (2) in contrast, although ϕ is high, when the outsourced amount is much larger than in-house made amount, E[TCU(T π *, n*)] decreases notably, as π goes up; and (3) the impact from ϕ (in terms of the in-house quality cost) does not exceed that from π (in terms of outsourcing added cost), hence, as π increases, E[TCU(T π *, n*)] still raises accordingly.

Conclusions
To meet the client's multi-item and quality requirements under capacity constraints and to satisfy the timely distribution of finished goods under minimum system cost, a multi-item manufacturer-retailer integrated type of system incorporating outsourcing and quality guarantee is explored. All in-house fabricated/reworked products are screened to make sure of the desired quality, whereas the external provider guarantees the quality of outsourced items. A fixed-quantity multi-shipment plan starts when the entire lot is quality-ensured. Accordingly, we build a precise model to portray the system characteristics and use mathematical derivation and optimization approach to obtain the total system cost and optimal policy (in terms of common cycle time and frequency of deliveries).
This study's main contribution is that we developed a decision support model (please refer to Section 2) to enable production managers to explore such a specific multiproduct manufacturer-retailer coordination problem featuring outsourcing strategy and product quality assurance. Using the proposed optimization techniques, managers can simultaneously find the optimal delivery frequency and common cycle time for the problem (please see Section 2.3). This helps the managers in making efficient and cost-effective multiproduct manufacturing and delivering decisions. By taking advantage of our results, the diverse individual and collective impact of variations in a system's features on the proposed problem can now be revealed to facilitate managerial decision-making. For instance: (1) The effect of variations in outsourcing proportion, setup cost, or unit outsourcing add-up expense on the optimal operating policy, individual cost of each end product, utilization, and the total system cost (see Figures 6,7,9 and 10). (2) The impact of changes in the optimal cycle time on each cost contributors and the total system cost (refer to Figure 8). (3) The make-or-buy decision relating information based on outsourcing proportion or total in-house scrap rate (refer to Figures 11 and 12). (4) The collective influence of differences in system features on the total system cost (see . The limitations of this study concerning fabrication capacity and setup times for producing multiproduct are shown in Equations (23) and (25) (refer to Section 2.4). Future studies may examine the influence of random demand or another uptime-reduction strategy, such as the expedited fabrication rate on the system's optimal operating policy. The results obtained can also be compared/verified with the results from artificial neural networks.

Appendix C
Detailed derivations of T min in Equation (25) are presented as follows. From Equation (24) we have: (5)), so Equation (A10) becomes: or, the fabrication cycle length T must be greater than T min , as shown in Equation (25) or (A13) as follows: Appendix D Table A1. Effects of differences in π on distinct cost categories in the proposed system.