A closed-form solution for the distribution free continuous review integrated inventory model

Abstract

In this paper, we develop an integrated inventory model in a single-vendor single-buyer supply chain under an unknown demand distribution at the buyer. It is assumed that each lot delivered to the buyer contains a random fraction of defective items, and lead time can be reduced at an extra crashing cost. We also consider that the unmet demand at the buyer during the stockout period is partially backordered. A model is formulated to minimize the total expected relevant costs of the system considering an exact expression of the service level constraint to ensure that a certain percentage of customer orders are filled by the buyer. Closed-form expressions are derived for the optimal order quantity and safety factor for given lead time and shipment frequency, and then an algorithm is proposed to find the global optimal solution. Finally, a numerical example is presented to illustrate the solution procedure and sensitivity analysis is carried out to analyze the proposed model.

Appendices

Appendix 1

1.1 Derivation of the expected average inventory at the vendor

The average inventory at the vendor is given by the difference between the accumulated inventory at the vendor and the accumulated inventory (including defective items) at the buyer per unit time (please see Fig. 1).

$$\begin{aligned} & {\text{Average}}\,{\text{inventory}}\,{\text{at}}\,{\text{the}}\,{\text{vendor}}, \bar{I}_{v} \\ & \quad = \frac{{{\text{The}}\,{\text{vendor's}}\,{\text{accumulated}}\,{\text{inventory}} - {\text{The}}\,{\text{buyer's}}\,{\text{accumulated}}\,{\text{inventory}}}}{{{\text{The}}\,{\text{expected}}\,{\text{production}}\,{\text{cycle}}\,{\text{length}}}}. \\ \end{aligned}$$

$$\bar{I}_{v} = \left[ {\left\{ {mQ\left( {\frac{Q}{P} + \frac{Q}{D}\left( {p_{1} + p_{2} + \ldots + p_{m - 1} } \right)} \right) - \frac{{m^{2} Q^{2} }}{2P}} \right\} - \left\{ {\left( {Q\frac{{Qp_{1} }}{D} + 2Q\frac{{Qp_{2} }}{D} + \ldots + \left( {m - 1} \right)Q\frac{{Qp_{m - 1} }}{D}} \right)} \right\}} \right]\frac{D}{m\mu Q}.$$

(29)

Putting $p_{1} + p_{2} + \ldots + p_{m - 1} = m\mu - p_{m}$ in the above expression and rearranging the terms, we have

$$\bar{I}_{v} = \left[ {\left\{ {mQ\left( {\frac{Q}{P} + \frac{m\mu Q}{D}} \right) - \frac{{m^{2} Q^{2} }}{2P}} \right\} - \left\{ {\frac{{Q^{2} }}{D}\left( {p_{1} + 2p_{2} + 3p_{3} + \ldots + \left( {m - 1} \right)p_{m - 1} + mp_{m} } \right)} \right\}} \right]\frac{D}{m\mu Q}.$$

(30)

Let $Z = p_{1} + 2p_{2} + 3p_{3} + \ldots + \left( {m - 1} \right)p_{m - 1} + mp_{m} ,$ then the expected average inventory at the vendor can be given by

$$E\left( {\bar{I}_{v} } \right) = \left[ {\left\{ {mQ\left( {\frac{Q}{P} + \frac{m\mu Q}{D}} \right) - \frac{{m^{2} Q^{2} }}{2P}} \right\} - \frac{{Q^{2} }}{D}E\left( Z \right)} \right]\frac{D}{m\mu Q}.$$

(31)

Now, $E\left( Z \right) = E\left( {\sum\nolimits_{i = 1}^{m} {ip_{i} } } \right) = \sum\nolimits_{i = 1}^{m} {E\left( {ip_{i} } \right)} = \sum\nolimits_{i = 1}^{m} {iE\left( {p_{i} } \right)} .$

Since $E\left( {p_{i} } \right) = \mu$, therefore $E\left( Z \right) = \mu \sum\nolimits_{i = 1}^{m} {i = \frac{{m\left( {m + 1} \right)}}{2}\mu } .$

Putting $E\left( Z \right) = m\left( {m + 1} \right)\mu /2$ into Eq. (31) and on simplification, we get

$$E\left( {\bar{I}_{v} } \right) = \frac{Q}{2}\left[ {m\left( {1 - \frac{D}{\mu P}} \right) - 1 + \frac{2D}{\mu P}} \right].$$

(32)

1.2 Derivation of the expected average inventory at the buyer

The expected shortages at the end of the buyer’s cycle is given by $E\left( {X - r} \right)^{ + }$. Since the demand during stockout period at the buyer is partially backordered with backorder ratio β, then the expected number of backorders per cycle and the expected lost sales per cycle can be given by $\beta E\left( {X - r} \right)^{ + }$ and $\left( {1 - \beta } \right)E\left( {X - r} \right)^{ + }$, respectively. To find the expression for the holding cost per unit time of the buyer, the average inventory at the buyer is established adapting the approaches of Montgomery et al. (1973) and Liu and Çetinkaya (2011). The buyer receives m shipments in each production cycle of the vendor and the number of non-defective items in each shipment is a random variable, and so the replenishment cycles of the buyer are not identical. Thus, the average inventory at the buyer is found based on the vendor’s production cycle rather than the buyer’s replenishment cycle. The expected net inventory level of non-defective items just before arrival of the ith shipment is $r - DL + \left( {1 - \beta } \right)E\left( {X - r} \right)^{ + }$ and the expected net inventory level of non-defective items immediately after arrival of the ith shipment is $Qp_{i} + r - DL + \left( {1 - \beta } \right)E\left( {X - r} \right)^{ + }$. Assuming a linear decrease of the buyer’s inventory level during each replenishment cycle, the expected average inventory at the buyer based on the vendor’s production cycle can be obtained by

$$E\left( {\bar{I}_{b} } \right) = \frac{{E\left( {\mathop \sum \nolimits_{i = 1}^{m} \frac{{Qp_{i} }}{D}\left( {\frac{{Qp_{i} }}{2} + r - DL + \left( {1 - \beta } \right)E\left( {X - r} \right)^{ + } } \right)} \right)}}{{{\text{The}}\,{\text{expected}}\,{\text{production}}\,{\text{cycle}}\,{\text{length}}}}.$$

(33)

Putting $r - DL = k\sigma \sqrt L$ from Assumption (2) into Eq. (33), we get

$$E\left( {\bar{I}_{b} } \right) = \frac{{\frac{{Q^{2} }}{2D}\mathop \sum \nolimits_{i = 1}^{m} E\left( {p_{i}^{2} } \right) + \frac{Q}{D}k\sigma \sqrt L \mathop \sum \nolimits_{i = 1}^{m} E\left( {p_{i} } \right) + \frac{Q}{D}\left( {1 - \beta } \right)E\left( {X - r} \right)^{ + } \mathop \sum \nolimits_{i = 1}^{m} E\left( {p_{i} } \right)}}{{\frac{m\mu Q}{D}}}.$$

(34)

Since $E\left( {p_{i} } \right) = \mu$ and we know that $Var\left( {p_{i} } \right) = E\left( {p_{i}^{2} } \right) - \left[ {E\left( {p_{i} } \right)} \right]^{2} , \;{\text{i}} . {\text{e}} .\; \delta^{2} = E\left( {p_{i}^{2} } \right) - \mu^{2}$. Substituting $E\left( {p_{i} } \right) = \mu$ and $E\left( {p_{i}^{2} } \right) = \mu^{2} + \delta^{2}$ into Eq. (34) and on simplification, we get the expected average inventory at the buyer as

$$E\left( {\bar{I}_{b} } \right) = \frac{Q}{2}\left( {\frac{{\mu^{2} + \delta^{2} }}{\mu }} \right) + k\sigma \sqrt L + \left( {1 - \beta } \right)E\left( {X - r} \right)^{ + } .$$

(35)

Appendix 2

We consider the situation when the buyer and the vendor are not willing to cooperate and they make their optimal decisions independently. The buyer determines the optimal order quantity, safety factor and lead time by minimizing the total expected cost in Eq. (9) and satisfying the SLC in Eq. (17). Based on the optimal decision of the buyer, the vendor decides the optimal number of shipments per production cycle for which the total expected cost of the vendor in Eq. (4) is minimum.

To solve the buyer’s problem, we temporarily ignore the SLC in Eq. (17) and minimize the total expected cost of the buyer in Eq. (9). Similar to the solution procedure as discussed in Sect. 5, it can be shown that for fixed Q and k, $TEC_{b} \left( {Q,k,L} \right)$ is a concave function in $L \in \left[ {L_{i} , L_{i - 1} } \right]$, and so the minimum value of $TEC_{b} \left( {Q,k,L} \right)$ occurs at the end points of the interval $L \in \left[ {L_{i} , L_{i - 1} } \right]$. Furthermore, we can show that for fixed $L \in \left[ {L_{i} , L_{i - 1} } \right]$, $TEC_{b} \left( {Q,k,L} \right)$ is a monotonically increasing function in $k \in \left[ {0,\infty } \right)$ and is a convex function in Q. Thus, for fixed $L \in \left[ {L_{i} , L_{i - 1} } \right]$, if the SLC is ignored, the minimum value of $TEC_{b} \left( {Q,k,L} \right)$ will occur at $k = 0$ and Q that satisfies the condition $\partial TEC_{b} \left( {Q,k,L} \right)/\partial Q = 0$, that is

$$Q = \left[ {\frac{{2D\left( {A + C\left( L \right)} \right)}}{{h_{b} \left( {\mu^{2} + \delta^{2} } \right)}}} \right]^{{\frac{1}{2}}} .$$

(36)

Now, if the SLC in Eq. (17) is satisfied for k = 0 and Q as in Eq. (36), then k = 0 and Q in Eq. (36) is the optimal solution for fixed $L \in \left[ {L_{i} , L_{i - 1} } \right]$. Otherwise, the SLC becomes active, and the solution procedure similar to that in Sect. 5can be applied to derive the optimal solution for Q and k. Thus, for fixed $L \in \left[ {L_{i} , L_{i - 1} } \right]$, if the SLC is active, the optimal value of Q and k can be obtained using Eqs. (37) and (38), respectively.

$$Q = Min\left\{ {\frac{{\sigma \sqrt L \left[ {1 - \alpha \left( {1 - \beta } \right)} \right] }}{2\alpha \mu }, \left[ {\frac{{4\alpha D\left( {1 - \alpha \left( {1 - \beta } \right)} \right)\left( {A + C\left( L \right)} \right) + h_{b} \sigma^{2} L\left( {1 - \alpha \left( {1 - \beta } \right)} \right)^{2} }}{{2h_{b} \alpha \left( {1 - \alpha \left( {1 - \beta } \right)} \right)\left( {\mu^{2} + \delta^{2} } \right) - 4h_{b} \alpha^{2} \mu^{2} \beta }}} \right]^{1/2} } \right\}$$

(37)

and

$$k = \frac{{\sigma \sqrt L \left[ {1 - \alpha \left( {1 - \beta } \right)} \right]}}{4\alpha \mu Q} - \frac{\alpha \mu Q}{{\sigma \sqrt L \left[ {1 - \alpha \left( {1 - \beta } \right)} \right]}}.$$

(38)

Now, Steps 2–3 of the algorithm presented at the end of Sect. 5 is modified as below to find the global optimal decision of the buyer.

• Step 1: For each $L_{i} , i = 0, 1, 2, \ldots ,n$, perform (1.1) to (1.3).

  • (1.1) Calculate Q _i using Eq. (36).

  • (1.2) If Q _i calculated in (1.1) satisfies the SLC given by Eq. (17) for $k_{i} = 0$, then set $k_{i} = 0$ and go to (1.3). Otherwise, calculate Q _i using Eq. (37) and then k _i using Eq. (38).

  • (1.3) Calculate the corresponding $TEC_{b} \left( {Q_{i} ,k_{i} ,L_{i} } \right)$ using Eq. (9).

• Step 2: Find $\mathop {\hbox{min} }\nolimits_{i = 0,1, \ldots ,n} TEC_{b} \left( {Q_{i} ,k_{i} ,L_{i} } \right)$. Let $TEC_{b} \left( {Q_{b}^{*} ,k_{b}^{*} ,L_{b}^{*} } \right) = \mathop {\hbox{min} }\nolimits_{i = 0,1, \ldots ,n} TEC_{b} \left( {Q_{i} ,k_{i} ,L_{i} } \right)$, then $\left( {Q_{b}^{*} ,k_{b}^{*} ,L_{b}^{*} } \right)$ is the optimal solution for the buyer.

Next, the total expected cost of the vendor given by Eq. (4) is minimized for the given optimal order quantity $(Q_{b}^{*} )$ of the buyer. Now, for fixed order quantity of the buyer, $TEC_{v} \left( {Q,m} \right)$ is a convex function in m, because $\partial^{2} TEC_{v} \left( {Q_{b}^{*} ,m} \right)/\partial m^{2} = 2SD/\left( {m^{3} \mu Q_{b}^{*} } \right) > 0$. Since the optimal production lot size of the vendor is an integer multiple of the buyer’s order quantity, and so an integer positive value of $m = m_{v}^{*}$ that satisfies the following condition is selected as an optimal number of shipments in a production cycle.

$$m_{v}^{*} \left( {m_{v}^{*} - 1} \right) \le \frac{2SD}{{h_{v} \mu Q_{b}^{*} \left( {1 - \frac{D}{\mu P}} \right)}} \le m_{v}^{*} \left( {m_{v}^{*} + 1} \right).$$