Capacity reservation and utilization for a manufacturer with uncertain capacity and demand

Abstract

We consider an original equipment manufacturer (OEM) who has outsourced the production activities to a contract manufacturer (CM). The CM produces for multiple OEMs on the same capacitated production line. The CM requires that all OEMs reserve capacity slots before ordering and responds to these reservations by acceptance or partial rejection, based on allocation rules that are unknown to the OEM. Therefore, the allocated capacity for the OEM is not known in advance, also because the OEM has no information about the reservations of the other OEMs. Based on a real-life situation, we study this problem from the OEM’s perspective who faces stochastic demand and stochastic capacity allocation from the contract manufacturer. We model this problem as a single-item, periodic review inventory system, and we assume linear inventory holding, backorder, and reservation costs. We develop a stochastic dynamic programming model, and we characterize the optimal policy. We conduct a numerical study where we also consider the case that the capacity allocation is dependent on the demand distribution. The results show that the optimal reservation policy is little sensitive to the uncertainty of capacity allocation. In that case, the optimal reservation quantities hardly increase, but the optimal policy suggests increasing the utilization of the allocated capacity. Further, in comparison with a static policy, we show that a dynamic reservation policy is particularly useful when backorder cost and uncertainty are low. Moreover, we show that for the contract manufacturer, to achieve the desired behavior, charging little reservation costs is sufficient.

Appendices

Appendix 1: Proof of theorem 1

Let

$$\begin{aligned} g_t \left( {x_t ,a_t } \right)= & {} min_{{\begin{array}{l} {r_t \ge 0} \\ {x_t \le y_t \le x_t a_t } \\ \end{array} }} \left\{ {sa_t +\mathcal{L}\left( {y_t } \right) +\alpha E_{D_t ,C_t } \left[ {g_{t+1} \left( {y_t -D_t ,A_t } \right) } \right] } \right\} \\&=min_{{\begin{array}{l} {r_t \ge 0} \\ {x_t \le y_t \le x_t a_t } \\ \end{array} }} \left\{ {sa_t +h_t \left( {y_t ,a_t } \right) } \right\} =sa_t +i_t \left( {x_t ,a_t } \right) \end{aligned}$$

• The functions $g_t \left( {x_t ,a_t } \right)$ and $h_t \left( {y_t ,a_t } \right)$ are jointly convex functions for any $t\in \left[ {1,T} \right]$. We prove this by induction. $g_{T+1} \left( \cdot \right) =0$ and is convex. Assume that $h_{t+1} \left( \cdot \right)$ is also convex. Then, the function $h_t \left( {y_t ,a_t } \right) =\mathcal{L}\left( {y_t } \right) +\alpha E_{D_t ,C_t } \left[ {g_{t+1} \left( {y_t -D_t ,A_t } \right) } \right]$ is also convex, because:

• $\mathcal{L}\left( {y_t } \right)$ is a convex function;

• $E\left[ {g_{t+1} \left( {y_t -D_t ,A_t } \right) } \right]$ is convex due to the convexity of the expected value operator. Rule: If $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ is convex, then the fction $g\left( x \right) =E_w \left\{ {f\left( {x+w} \right) } \right\}$ is also a convex function, where w is a random vector in ${\mathbb {R}}^{n}$., provided that the expected value is finite for every $x\in {\mathbb {R}}^{n}$. (Bertsekas 2005);

• the linear combination of two convex functions remains convex. Rule: Let K be a non-empty index set, X a convex set, and for each $k\in K$, let $f_k \left( \cdot \right)$ be a convex function on X and let $p_k \ge 0$. Then $\mathop \sum \limits _{k\in K} p_k f_k (x)$, $x\in X$ is a convex function on any convex subset of X, where the sum takes finite values (Heyman and Sobel 2004).

• $i_t \left( {x_t ,a_t } \right) =min_{{\begin{array}{l} {r_t \ge 0} \\ {x_t \le y_t \le x_t a_t } \\ \end{array} }} \left\{ {h_t \left( {y_t ,a_t } \right) } \right\}$ is also convex when $h_t \left( {y_t ,a_t } \right)$ is convex. Rule: Let X be a non-empty set with $A_x$ a non-empty set for each $x\in X$. Let $C=\left\{ {\left( {x,y} \right) :y\in A_x ,x\in X} \right\}$, let J be a real-valued function on C and define $f\left( x \right) =inf\left\{ {J\left( {x,y} \right) :y\in A_x } \right\} ,x\in X$. If C is a convex set and J is a convex function on C, then f is a convex function on any convex subset of $X^{*}=\left\{ {x:x\in X,f\left( x \right) >-\infty } \right\}$ (Heyman and Sobel 2004).

• The function $g_t \left( {x_t ,a_t } \right) =sa_t +i_t \left( {x_t ,a_t } \right)$ is then also convex.

Appendix 2: Conditional probability distribution

In part of the numerical studies, we consider the case where the capacity allocation $C_t$ by the contract manufacturer is positively or negatively dependent on OEM’s demand $D_t$. Therefore, we adapt the probability mass function of $C_t$ to a conditional probability mass function in which $C_t$ is conditioned on $D_t$: $P\left\{ {C_t =c|D_t =d} \right\}$. This function is basically the matrix G of size $m\times n$, where $m=d_{max}$ (the maximum demand) and $n=c_{max} -c_{min} +1$ (where $c_{min}$ and $c_{max}$ are the bounds of the capacity distribution).

In case of positive dependency, elements $G_{m,1} =G_{1,n}$ = 0 and $G_{1,1} =G_{m,n} =\left\{ \begin{array}{lll} \frac{2}{n} &{} if &{} n \ge 2 \\ 1 &{} if &{} n=1 \\ \end{array}\right.$

Then, the rows and columns are filled by a linear decrease/increase from 0 to $G_{1,1}$ or $G_{m,n}$. Finally, the probabilities are rescaled, such that distribution sums up to 1.

In case of negative dependency, the same procedure is applied, but now $G_1,1=G_{m,n}= 0$ and $G_{m,1} =\left\{ \begin{array}{lll} \frac{2}{n} &{} if &{} n \ge 2 \\ 1 &{} if &{} n=1 \\ \end{array}\right.$

$$\begin{aligned} G=\left[ \begin{array}{lll} 0 &{} \cdots &{} {G_{1,n} } \\ \vdots &{} \ddots &{} \vdots \\ {G_{m,1} }&{} \cdots &{} 0 \\ \end{array} \right] \end{aligned}$$

Appendix 3: Optimal policy parameters $\left( {z_t^*,y_t^*} \right)$ for different values of h,b, and s

Fig. 8

Order-up-to $(y_{t})$ and reservation-up-to $(z_{t})$ levels for different levels of demand uncertainty, backorder cost, and reservation cost