Computational Optimization of Manufacturing Batch Size and Shipment for an Integrated Epq Model with Scrap

This paper employs mathematical modeling and algebraic approach to derive the optimal manufacturing batch size and number of shipment for a vendor-buyer integrated economic production quantity (EPQ) model with scrap. Unlike the conventional method by using differential calculus to determine replenishment lot size and optimal number of shipments for such an integrated system, this paper proposes a straightforward algebraic approach to replace the use of calculus on the total cost function for solving the optimal production-shipment policies. A simpler form for computing long-run average cost for such a vendor-buyer integrated EPQ problem is also provided.


Introduction
The economic production quantity (EPQ) model was first introduced by Taft [1] to assist practitioners in production and inventory control field to determine the economic replenishment batch size that minimizes total production-inventory costs.Classic economic production quantity model assumes a continuous inventory issuing policy for satisfying customer's demand.However, in real world vendor-buyer system, multiple or periodic deliveries of finished products are often adopted.Therefore, "how many shipments should a manufacturing lot be broken down to?" becomes another critical issue that practitioners must address in order to minimize overall production-inventory-delivery costs.
Studies related to various aspects of supply chain optimization have been extensively carried out (see for example [2][3][4][5][6][7][8][9]) in past decades.Goyal [2] examined an integrated single supplier-single customer problem.He proposed a method that is typically applicable to those inventory problems where a product is procured by a single customer from a single supplier.Example was provided to demonstrate his proposed model.Schwarz et al. [3] considered the system fill-rate of a one-warehouse N-identical retailer distribution system as a function of warehouse and retailer safety stock.They used an approximation model from a prior study to maximize system fill-rate subject to a constraint on system safety stock.As results, properties of fill-rate policy lines are suggested.They may be used to provide managerial insight into system optimization and as the basis for heuristics.Lu [4] studied a one-vendor multi-buyer integrated inventory model with the objective of minimizing vendor's total annual cost subject to the maximum costs that the buyers may be prepared to incur.Lu's model required to know buyer's annual demand and previous order frequency.As a result, an optimal solution for the onevendor one-buyer case was obtained and a heuristic approach for the one-vendor multi-buyer case was also provided.Sarker and Khan [5] considered a production system that procures raw materials from suppliers in a lot and processes them into finished products which are then delivered to outside buyers at fixed points in time.A general cost model was formulated considering both raw materials and finished products.Using this model, a simple procedure was developed to determine the optimal ordering policy for raw materials as well as the manufacturing batch size, so that the overall costs for such a supply chain system can be minimized.Chiu et al. [9] incorporated a multi-delivery policy and quality assurance into an imperfect economic production quantity (EPQ) model with scrap and rework.They assumed that the random defective items produced are partially repairable and are reworked in each cycle when regular production ends, and the finished items can only be delivered to customers if the whole lot is quality assured at the end of rework.Fixed quantity multiple installments of the finished batch are delivered to customers at a fixed interval of time.The expected integrated cost function per unit time was derived.A closed-form optimal batch size solution to the problem was obtained.
Imperfect quality items produced in real world manufacturing environments is another inevitable and important issue that practitioners in the production management filed must deal with.In the past decades, many studies have been carried out to address the issue of defective items in the production lines (see for example [10][11][12][13][14]).The nonconforming items sometimes can be repaired through rework, hence overall production costs can be significantly reduced [15][16][17][18][19][20].Yu and Bricker [15] presented an informative application of Markov Chain analysis to a multistage manufacturing problem.Jamal et al. [16] studied the optimal manufacturing batch size with rework process at a single-stage production system.Cases of rework being completed within the same production cycle, and rework being done after N cycles are examined.They developed mathematical models for each case and derived total system costs and optimal batch sizes accordingly.Chiu et al. [19] proposed a numerical method for expediting scrap-or-rework decision making in EPQ model with failure in repair.
Algebraic approach for determining economic order quantity (EOQ) model with backlogging was introduced by Grubbström and Erdem [21].They proposed algebraic derivations to solve the optimal order quantity without reference to the first-order or second-order differentiations.Various aspects of supply chain optimization studies have employed the same or similar methodologies [22,23].This paper uses mathematical modeling to derive the long-run average cost function for the proposed vendor-buyer integrated EPQ model with scrap; then employs such a straightforward algebraic derivation to determine the optimal production-shipment policies for the proposed EPQ model.

The Proposed Model and Mathematical Modeling
The proposed economic production quantity model as-sumes there is an x portion of defective items produced randomly at a production rate d during regular production time.All produced items are screened and inspection cost per item is included in the unit production cost C.All nonconforming items are assumed to be scrap and will be discarded at the end of production.Under regular supply (not allowing shortages), the constant production rate P must be larger than the sum of demand rate λ and production rate of scrap items d.That is: The production rate of scrap items d can be expressed as d = Px.
A multi-delivery policy is considered in this study and it is also assumed that the finished items can only be delivered to customers if the whole lot is quality assured at the end of production process.Fixed-quantity n installments of finished batch are delivered to customers at a fixed interval of time during the production downtime t 2 (see Figure 1).Additional notation is listed in Nomenclature in Appendix.
TC(Q, n), the total production-inventory-delivery costs per cycle consists of 1) setup cost; 2) variable production costs; 3) variable scrap disposal costs; 4) fixed delivery cost; 5) variable delivery costs; 6) variable holding costs at the supplier side for all items produced (defective and perfect quality items) in t 1 and all items waiting to be delivered in t 2 ; and 7) holding cost for finished goods stocked at customer's end.Therefore, Figure 2 shows supplier's inventory holding during delivery time t 2 .The variable holding costs for finished products kept by the supplier in delivery time t 2 are 1) When n = 1, total holding cost in delivery time = 0. 2) When n = 2, total holding costs in delivery time become (see Figure 2) 3) When n = 3, total holding costs in delivery time are 4) When n = 4, total holding costs in delivery time become Therefore, the following general term for total holding costs during t 2 can be obtained (as shown in Equation ( 1) above): Taking randomness of scrap rate into consideration and employing the expected values of it, and with further derivations, the long-run average costs per unit time for the proposed EPQ model, E[TCU(Q, n)] can be derived as follows (refer to a similar derivation procedure in [9]):
It is noted that if the following square terms (Equations (17) and ( 18)) equal zero, then Equation (15) will be minimized:

Demonstrative Example
  Consider a product can be produced at an annual rate of 60,000 units and this item has experienced a flat demand rate of 3,400 units per year.Assume that during production process a random scrap rate which follows a uniform distribution over the interval [0, 0.3].In additions, the following values of related variables are considered:  9) and (10) in Equation ( 18), the optimal replenishment lot size Q * can be obtains: K = $20,000 per production run, C T = $0.1 per item delivered.
Substituting Equations ( 11), (12), and (20) in Equation ( 19), the optimal number of shipments is From Equations ( 21), one obtains the optimal number of delivery n * = 3.By plugging n * back into Equation ( 7) and resolving the algebraic solution for Q * one finds the optimal production batch size Q * = 2652.Calculating Equation ( 22) one obtains the long-run average cost E × [TCU(Q * , n * )] = $512,047.It is noted that n * should practically be an integer number, but Equation ( 21) gives a real number.In order to obtain the optimal integer value of n, one should compute the E[TCU(Q, n)] for both integers that are adjacent to real number n * respectively (for instance, in this example Equation (21) gives n * = 3.1733, so both n = 3 and n = 4 must be plugged in E[TCU(Q, n)]), and select the one with minimum cost as our optimal n * .(21) One notes that Equation ( 21) is identical to what was obtained by using the conventional differential calculus method on E[TCU(Q, n)] [24].Further, from Equation ( 7) the optimal cost function E[TCU(Q * , n * )] is

Conclusions
This paper derives the optimal manufacturing batch size and number of shipment for a vendor-buyer integrated EPQ model with scrap using mathematical modeling and algebraic approach.It is confirmed the research results from the proposed algebraic derivations are identical to what were derived by the use of conventional differential calculus.In additions, this study also reveals a simpler computation formula (i.e.Equation ( 22)) for the long-run average cost function for such a vendor-buyer integrated EPQ problem.This straightforward algebraic approach enables practitioners or students who with little or no knowledge of calculus to learn or handle with ease the real-life EPQ model.

Figure 1 .
Figure 1.On-hand inventory of perfect quality items in the proposed EPQ model with scrap and a multiple shipment policy.

Figure 2 .
Figure 2. On-hand inventory of the finished items kept by supplier during t 2 in the proposed EPQ model.

Figure 3
shows the convexity of the long-run integrated cost function E[TCU(Q, n * = 3)].