^{*}

This paper proposes a cost reduction distribution policy for an integrated manufacturing system operating under quality assurance practice. We reexamine the problem studied by Chiu et al. [Numerical method for determination of the optimal lot size for a manufacturing system with discontinuous issuing policy and rework. International Journal for Numerical Methods in Biomedical Engineering, doi:10.1002/cnm.1369] and improve its replenishment lot-size solution in terms of lowering producer’s stock holding cost. Mathematical modeling and analysis is employed in this study and optimal replenishment lot size is derived. A numerical example is provided to show the practical usage of research result as well as to demonstrate significant savings in producer’s holding cost as compared to that in Chiu et al.

This paper proposes a cost reduction distribution policy to improve the economic lot size solution of a manufacturing system with discontinuous issuing policy and imperfect rework [

Another unrealistic assumption of classic EPQ model is the continuous inventory issuing policy for satisfying product demand. In real-life vendor-buyer integrated production-inventory system, multiple or periodic deliveries of finished products are commonly used. Schwarz [

On improving the replenishment lot size solution derived by [

Reexamine the specific EPQ model studied by Chiu et al. [_{1}, within the same cycle when regular production ends. A θ_{1} portion (where 0θ_{1} 1) of reworked items fails during the rework process and becomes scrap. Under the regular operating schedule, the constant production rate P is larger than the sum of demand rate λ and production rate of defective items d. That is: (P – d – λ) > 0; where d can be expressed as d = Px. Let d_{1} denote production rate of scrap items during the rework process, then d_{1} can be expressed as d_{1} = P_{1}θ_{1}. Under the proposed n + 1 delivery policy, an initial installment of finished (perfect quality) products is distributed to customer for satisfying the demand during producer’s production uptime and rework time. Then, at the end of rework, when the rest of the production lot is quality assured, fixed quantity n installments of finished items are delivered to customer at a fixed interval of time. Such an n + 1 delivery policy is intended to reduce supplier’s stock holding cost.

The on-hand inventory of perfect quality items of the proposed model is depicted in

The proposed system includes the following cost related parameters: setup cost K per production run, unit production cost C, unit holding cost h, unit rework cost C_{R}, disposal cost per scrap item C_{S}, holding cost h_{1} for each reworked item, fixed delivery cost K_{1} per shipment, and delivery cost C_{T} per item shipped to customers. Other notations used also include:

T = cycle lengtht = the production time needed for producing enough perfect items for satisfying product demand during the production uptime t_{1} and the rework time t_{2}t_{1} = the production uptime for the proposed EPQ modelt_{2} = time required for reworking of defective itemst_{3} = time required for delivering the remaining quality assured finished productsH = the level of on-hand inventory in units for satisfying product demand during manufacturer’s regular production time t_{1} and rework time t_{2}H_{1} = maximum level of on-hand inventory in units when regular production endsH_{2} = the maximum level of on-hand inventory in units when rework process finishesQ = production lot size to be determined for each cyclet_{n} = a fixed interval of time between each installment of products delivered during t_{3}n = number of fixed quantity installments of the rest of finished lot to be delivered during t_{3}I(t) = on-hand inventory of perfect quality items at time tI_{d}(t) = on-hand inventory of defective items at time tI_{s}(t) = on-hand inventory of scrap items at time tφ = overall scrap rate per cycle (sum of scrap rates in t_{1} and t_{2})TC(Q) = total production-inventory-delivery costs per cycle for the proposed modelE[TCU(Q)] = the long-run average costs per unit time for the proposed model.

From the assumption of the proposed model and

It is noted that the maximum level of defective items is dt_{1}. Among them a θ portion is scrap and the other (1 – θ) portion of defective items is considered to be re-workable. Time needed for reworking is shown in Equation (8). During the reworking, a portion θ_{1} of reworked items fail and becomes scrap. _{1} and t_{2}. One notes that maximum level of scrap items during a cycle is φxQ (Equation (9)).

Total production-inventory-delivery costs per cycle TC(Q) consists of the variable production cost, the setup cost, variable rework cost, disposal cost, (n + 1) fixed distribution costs and variable delivery cost, holding cost for perfect quality items during production uptime t_{1} and reworking time t_{2}, holding cost for defective items during t_{1}, variable holding cost for items reworked during t_{2}, and holding cost for finished goods during the delivery time t_{3} where n fixed-quantity installments of the finished

batch are delivered to customers at a fixed interval of time (one can refer to Appendix-2 of [

Taking into account of the randomness of defective rate x (which is assumed to be a random variable with a known probability density function), one can use the expected values of x in the related cost analysis. Substituting all related parameters from Equations (1) to (9) in TC(Q), one obtains the expected total E[TCU(Q)] as follows (see Appendix A for details).

The optimal production lot size can be obtained by minimizing E[TCU(Q)]. Differentiating E[TCU(Q)] with respect to Q, the first and the second derivatives of E[TCU(Q)] are

The second derivative of E[TCU(Q)] is resulting positive, because K, n, K_{1}, λ, Q, and (1 – φE[x]) are all positive. Hence, E[TCU(Q)] is convex, for all Q different from zero. Optimal lot size Q* can be obtained by setting the first derivative of E[TCU(Q)] equal to zero.

With further derivations, one obtains the optimal replenishment lot size as follows.

With the purpose of comparison of the proposed model and Chiu et al.’s model [_{1} = 2,200 units per year. A θ_{1} = 0.1 portion of reworked items fails and becomes scrap during rework. Other parameters include: K = $20,000; C = $100 per item; C_{R} = $60 per item reworked; C_{S} = $20 per scrap item; h = $20; h_{1} = $40 per item reworked; a fixed cost K_{1} = $4,350 per shipment; and C_{T} = $0.1 per item delivered.

In order to show practical usages of our research results, the following two different scenarios are demonstrated, respectively.

Scenario 1: Let total number of deliveries remain 4 (i.e. n = 4 as was used in [_{1}, for satisfying the product demand during producer’s production uptime and rework time. Then, at the end of rework, fixed quantity three other installments of finished items are delivered to customer at a fixed interval of time. Also, for the purpose of comparison, we use the lot-size solution Q = 3,553 (from [

Scenario 2: Let total number of deliveries remain 4 (that is (n + 1) = 4 in our model). By applying Equations (15) and (11), one obtains the optimal replenishment lot size Q* = 4,271 and the expected total costs E[TCU(Q*)] = $441,949, respectively. It is noted that overall reducetion in production-inventory-delivery costs amounts to $12397, or 12.16% of total other related costs.

Chiu et al. [

model [

The authors greatly appreciate the support of National Science Council (NSC) of Taiwan under grant number: NSC 99-2410-H-324-007-MY3.

Computation of Equation (11) is given below.

Recall Equation (10) as follows:

Substituting all related parameters from Equations (1) to (9) in Equation (10) one obtains

or

Because

and

Substituting Equations (A-2) and (A-3) in Equation (A-4) one obtains

With further rearrangements one has