To ease readability, in this study we use same notations as those used in mathematical modeling and formulation in Chiu et al.’s study (2013). Recall the problem description of their specific intra-supply chain system as follows. A product can be manufactured at an annual rate *P* by a single production unit, and an *x* portion of defective items may be randomly produced at a rate *d* during the production process. The unit manufacturing cost including the inspection cost is *C*. All defective items are reworked immediately after the regular production process ends in each cycle at a rate of *P*
_{1}, and there exists a rate of failure in rework *θ*
_{1}. To prevent shortages the production rate *P* must satisfy (*P* – *d* − *λ*) > *0*, where *λ* is the sum of the demands of all customers (i.e., the sum of *λ*
_{
i
}), and *d* can be expressed as *d* = *Px*. Cost parameters used in cost analysis include the following: unit holding cost *h*; set-up cost per production cycle *K*; unit cost *C*
_{R} and unit holding cost *h*
_{1} for each reworked item; unit disposal cost *C*
_{S} for failures in rework; fixed delivery cost *K*
_{1i
} per shipment delivered to regional sales office *i*; unit holding cost *h*
_{2i
} for items retained by regional sales office *i*; and unit shipping cost *C*
_{Ti} for items shipped to sales office *i*. Additional notations used in this study is listed in “Appendix A”.

Under the proposed delivery policy, an initial shipment of finished products is distributed to multiple sales locations to meet demand during the production unit’s uptime and rework time. After rework and once the remaining production lot goes through quality assurance, *n* fixed quantity installments of the finished products are transported to sales locations at a fixed time interval.

Figure 1 depicts the expected reduction in sales officers’ stock holding costs (yellow shaded area) of the proposed model (in blue) in comparison with that of Chiu et al.’s model (2013) (in black).

Apply the similar mathematical techniques used in the conventional economic production quantity (EPQ) model (Hillier and Lieberman 2001; Nahmias 2009) and in the specific EPQ model with discontinuous delivery policy (Chiu et al. 2014) to the proposed model, total delivery costs can be obtained as

$$\left( {n + 1} \right)\sum\limits_{i = 1}^{m} {K_{1i} } + \sum\limits_{i = 1}^{m} {C_{{{\text{T}}i}} \left[ {Q\left( {1 - \theta_{1} x} \right)} \right]}$$

(1)

The variable holding costs at the producer’s side during the delivery time *t*
_{3} where *n* fixed-quantity installments of the finished batch are delivered to retailers are

$$h\left( {\frac{n - 1}{2n}} \right)Ht_{3}$$

(2)

From Fig. 1, the total inventory holding costs for items stocked by retailers during the cycle are

$$\sum\limits_{i = 1}^{m} {h_{2i} } \left[ {\frac{{\lambda_{i} (t_{1} + t_{2} )^{2} }}{2} + n\left( {\frac{{D_{i} + 2I_{i} }}{2}} \right)t_{n} } \right]$$

(3)

Therefore, total production-inventory-delivery costs per cycle for the proposed *n* +1 delivery model, *TC*(*Q*, *n* + 1) consists of (1) the costs of variable manufacturing, setup, quality assurance, and inventory holding incurred in the production unit; (2) the costs for product distribution; and (3) the stock holding costs incurred in the sales offices, as follows:

$$\begin{gathered} TC\left( {Q,n{ + 1}} \right) = CQ + K + C_{R} \left[ {xQ} \right] + C_{S} \left[ {x\theta_{1} Q} \right] + \sum\limits_{i = 1}^{m} {C_{Ti} \left[ {Q\left( {1 - \theta_{1} x} \right)} \right]} + \left( {n + 1} \right)\sum\limits_{i = 1}^{m} {K_{1i} } \hfill \\ \, + h\left[ {\frac{{H_{1} }}{2}\left( t \right) + \frac{{H_{2} }}{2}\left( {t_{1} - t} \right) + \frac{{dt_{1} }}{2}\left( {t_{1} } \right) + \frac{{H_{2} + H}}{2}\left( {t_{2} } \right) + \left( {\frac{n - 1}{2n}} \right)Ht_{3} } \right] + h_{1} \cdot \frac{{dt_{1} }}{2} \cdot \left( {t_{2} } \right) \hfill \\ \, + \sum\limits_{i = 1}^{m} {h_{2i} } \left[ {\frac{{\lambda_{i} (t_{1} + t_{2} )^{2} }}{2} + n\left( {\frac{{D_{i} + 2I_{i} }}{2}} \right)t_{n} } \right] \hfill \\ \end{gathered}$$

(4)

By substituting all parameters in Eq. (4), applying the expected values of *x* to account for randomness of defective rate and with further derivations E[*TCU*(*Q*, *n* + 1)] is obtained as

$$\begin{gathered} E\left[ {TCU\left( {Q,n{ + 1}} \right)} \right] = C\lambda E_{3} + \frac{K\lambda }{Q}E_{3} + C_{R} \lambda E_{4} + C_{s} \theta_{1} \lambda E_{4} + \sum\limits_{i = 1}^{m} {C_{Ti} \lambda_{i} } + \frac{(n + 1)\lambda }{Q}E_{3} \sum\limits_{i = 1}^{m} {K_{1i} } \hfill \\ \, + \frac{hQ\lambda }{2}\left\{ \begin{aligned} \frac{{2\lambda^{2} }}{{P^{3} }}E_{0} + \frac{{4\lambda^{2} }}{{P^{2} P_{1} }}E_{1} + \frac{{2\lambda^{2} }}{{PP_{1}^{2} }}E_{2} - \frac{{\left( {1 - 2\theta_{1} E\left[ x \right]} \right)}}{P}E_{3} - \frac{\lambda }{{P^{2} }}E_{3} - \frac{2\lambda }{{PP_{1} }}E_{4} + \frac{1}{{E_{3} \lambda }} \hfill \\ - \frac{1}{{P_{1} }}\left[ {1 + \frac{\lambda }{{P_{1} }} - \theta_{1} } \right]E_{5} - \left( {\frac{1}{n}} \right)\left[ {\frac{1}{{E_{3} \lambda }} - \frac{2}{P} - \frac{2E\left[ x \right]}{{P_{1} }} + \frac{\lambda }{{P^{2} }}E_{3} + \frac{2\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{(P_{1} )^{2} }}E_{5} } \right] \hfill \\ \end{aligned} \right\} \hfill \\ \, + \frac{{h_{1} Q\lambda }}{{2P_{1} }}E_{5} + \sum\limits_{i = 1}^{m} {h_{2i} } Q\lambda_{i} \left\{ \begin{aligned} \frac{\lambda }{{2P^{2} }}E_{3} + \frac{\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{2P_{1}^{2} }}E_{5} + \frac{1}{{2n\lambda E_{3} }} - \frac{1}{nP} - \frac{E\left[ x \right]}{{nP_{1} }} + \frac{\lambda }{{P^{2} }}E_{6} + \frac{\lambda }{{PP_{1} }}E_{7} \hfill \\ + \frac{\lambda }{{2nP^{2} }}E_{3} + \frac{\lambda }{{nPP_{1} }}E_{4} - \frac{{\lambda^{2} }}{{P^{3} }}E_{0} + \frac{\lambda }{{2nP_{1}^{2} }}E_{5} - \frac{{\lambda^{2} }}{{PP_{1}^{2} }}E_{2} - \frac{{2\lambda^{2} }}{{P^{2} P_{1} }}E_{1} \hfill \\ \end{aligned} \right\} \hfill \\ \end{gathered}$$

(5)

where *E*
_{i} denotes the following:

$$\begin{gathered} E_{0} = E\left( {\frac{1}{1 - x}} \right)E_{3} ; \, E_{1} = E\left( {\frac{x}{1 - x}} \right)E_{3} \, ; \, E_{2} = E\left( {\frac{{x^{2} }}{1 - x}} \right)E_{3} \, ; \, E_{3} = \frac{1}{{1 - \theta_{1} E\left[ x \right]}} \hfill \\ E_{4} = E\left[ x \right]E_{3} \, ; \, E_{5} = \left( {E\left[ x \right]} \right)^{2} E_{3} \, ; \, E_{6} = E\left( {\frac{1}{1 - x}} \right) \, ; \, E_{7} = E\left( {\frac{x}{1 - x}} \right) \hfill \\ \end{gathered}$$

(6)

### Proof of convexity and the optimal operating policy

To derive the optimal production-shipment policy for the proposed model, one must first prove that E[*TCU*(*Q*, *n* + 1)] is convex. The Hessian matrix equations (Rardin, 1998) are used to prove its convexity since the following equations are true (see “Appendix B” for details):

$$\left[ {\begin{array}{*{20}c} Q & n \\ \end{array} } \right] \cdot \left( {\begin{array}{*{20}c} {\frac{{\partial^{2} E\left[ {TCU\left( {Q,n{ + 1}} \right)} \right]}}{{\partial Q^{2} }}} & {\frac{{\partial^{2} E\left[ {TCU\left( {Q,n{ + 1}} \right)} \right]}}{\partial Q\partial n}} \\ {\frac{{\partial^{2} E\left[ {TCU\left( {Q,n{ + 1}} \right)} \right]}}{\partial Q\partial n}} & {\frac{{\partial^{2} E\left[ {TCU\left( {Q,n{ + 1}} \right)} \right]}}{{\partial n^{2} }}} \\ \end{array} } \right) \cdot \left[ {\begin{array}{*{20}c} Q \\ n \\ \end{array} } \right] = \frac{2\lambda }{Q}\frac{1}{{1 - \theta_{1} E\left[ x \right]}}\left( {K + \sum\limits_{i = 1}^{m} {K_{1i} } } \right) \, {\mathbf{ > }}{ 0}$$

(7)

The results of Eq. (7) are positive, because *λ*, *Q* (1 − *θ*
_{1}
*E*[*x*]), *K,* and *K*
_{1i
} are all positive. Hence, E[*TCU*(*Q, n* + 1)] is a strictly convex function for all *Q* and *n* different from zero. So, the minimum of E[*TCU*(*Q, n* + 1)] exists. In order to derive the optimal lot size *Q** and number of shipments *n**, one can differentiate E[*TCU*(*Q, n* + 1)] with respect to *Q* and with respect to *n*, respectively, and solve the linear system of these equations [i.e., equations (B-1) and (B-3) in “Appendix B”] by setting these partial derivatives to zero. With further derivations, one obtains

$$Q^{*} = \sqrt { \, \frac{{2\left[ {K + \left( {n + 1} \right)\sum\limits_{i = 1}^{m} {K_{1i} } } \right]\lambda E_{3} }}{\begin{aligned} h\lambda \left\{ \begin{aligned} \frac{{2\lambda^{2} }}{{P^{3} }}E_{0} + \frac{{4\lambda^{2} }}{{P^{2} P_{1} }}E_{1} + \frac{{2\lambda^{2} }}{{P(P_{1} )^{2} }}E_{2} - \frac{{\left[ {1 - 2\theta_{1} E\left[ x \right]} \right]}}{P}E_{3} - \frac{\lambda }{{P^{2} }}E_{3} - \frac{2\lambda }{{PP_{1} }}E_{4} \hfill \\ - \frac{1}{{P_{1} }}\left[ {1 + \frac{\lambda }{{P_{1} }} - \theta_{1} } \right]E_{5} + \frac{1}{{E_{3} \lambda }} - \left( {\frac{1}{n}} \right)\left[ {\frac{1}{{E_{3} \lambda }} - \frac{2}{P} - \frac{2E\left[ x \right]}{{P_{1} }} + \frac{\lambda }{{P^{2} }}E_{3} + \frac{2\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{P_{1}^{2} }}E_{5} } \right] \hfill \\ \end{aligned} \right\} \hfill \\ + \frac{{h_{1} \lambda }}{{P_{1} }}E_{5} + \sum\limits_{i = 1}^{m} {h_{2i} } \lambda_{i} \left\{ \begin{aligned} \frac{\lambda }{{P^{2} }}E_{3} + \frac{2\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{P_{1}^{2} }}E_{5} + \frac{1}{{n\lambda E_{3} }} - \frac{2}{nP} - \frac{2E\left[ x \right]}{{nP_{1} }} + \frac{\lambda }{{P^{2} }}E_{6} + \frac{2\lambda }{{PP_{1} }}E_{7} \hfill \\ + \frac{\lambda }{{nP^{2} }}E_{3} + \frac{2\lambda }{{nPP_{1} }}E_{4} - \frac{{2\lambda^{2} }}{{P^{3} }}E_{0} + \frac{\lambda }{{nP_{1}^{2} }}E_{5} - \frac{{2\lambda^{2} }}{{PP_{1}^{2} }}E_{2} - \frac{{2\lambda^{2} }}{{P^{2} P_{1} }}E_{1} \hfill \\ \end{aligned} \right\} \hfill \\ \end{aligned} }}$$

(8)

and

$$n^{*} = \sqrt { \, \frac{{\left( {\sum\limits_{i = 1}^{m} {K_{1i} } + K} \right)\frac{1}{2}\left( {h\lambda - \sum\limits_{i = 1}^{m} {h_{2i} \lambda_{i} } } \right)\left[ {\frac{1}{{\lambda E_{3} }} - \frac{2}{P} - \frac{2E\left[ x \right]}{{P_{1} }} + \frac{\lambda }{{P^{2} }}E_{3} + \frac{2\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{P_{1}^{2} }}E_{5} } \right]}}{{\left( { - \sum\limits_{i = 1}^{m} {K_{i} } } \right)\left\{ \begin{aligned} \frac{h\lambda }{2}\left\{ \begin{aligned} \frac{{2\lambda^{2} }}{{P^{3} }}E_{0} + \frac{{4\lambda^{2} }}{{P^{2} P_{1} }}E_{1} + \frac{{2\lambda^{2} }}{{PP_{1}^{2} }}E_{2} - \frac{{\left[ {1 - 2\theta_{1} E\left[ x \right]} \right]}}{P}E_{3} \hfill \\ - \frac{\lambda }{{P^{2} }}E_{3} - \frac{2\lambda }{{PP_{1} }}E_{4} - \frac{1}{{P_{1} }}\left[ {1 + \frac{\lambda }{{P_{1} }} - \theta_{1} } \right]E_{5} + \frac{1}{{E_{3} \lambda }} \hfill \\ \end{aligned} \right\} \hfill \\ + \frac{{h_{1} \lambda }}{{2P_{1} }}E_{5} + \sum\limits_{i = 1}^{m} {h_{2i} } \lambda_{i} \left\{ \begin{aligned} \frac{\lambda }{{2P^{2} }}E_{3} + \frac{\lambda }{{PP_{1} }}E_{4} + \frac{\lambda }{{2P_{1}^{2} }}E_{5} + \frac{\lambda }{{P^{2} }}E_{6} \hfill \\ + \frac{\lambda }{{PP_{1} }}E_{7} - \frac{{\lambda^{2} }}{{P^{3} }}E_{0} - \frac{{\lambda^{2} }}{{PP_{1}^{2} }}E_{2} - \frac{{2\lambda^{2} }}{{P^{2} P_{1} }}E_{1} \hfill \\ \end{aligned} \right\} \hfill \\ \end{aligned} \right\}}}}$$

(9)

The computational result of Eq. (9) does not necessarily have to be an integer number. However, the number of deliveries in real supply chain situations can only take on an integer value. In order to determine the integer value of *n** that minimizes E[*TCU*(*Q, n* + 1)], two adjacent integers to *n* must be examined respectively. Let *n*
^{+} denote the smallest integer greater than or equal to *n* [derived from Eq. (9)] and *n*
^{−} denote the largest integer less than or equal to *n*. Substitute *n*
^{+} and *n*
^{−} respectively in Eq. (8) and then apply the results in Eq. (5), respectively. Choose the one that gives the minimum long-run average cost as the optimal replenishment- distribution policy.