Inventory control in dual sourcing commodity procurement with price correlation

Abstract

In this paper, we focus on the role of inventory management as a means for operational hedging by dual sourcing of commodities using a multi-period option contract and spot market. We consider a manufacturing company in a make-to-stock environment with uncertain product demand. We replace the common i.i.d. price assumption that is typical in operations management studies by the mean reverting price model, a more realistic spot price model with inter-temporal price–price correlation. Additionally, we address the case where the spot price in one period is correlated with the demand in the previous period (demand–price correlation). The contribution of the paper is threefold. First, we reveal that price–price correlation has a considerable impact on the structural properties of optimal stock-keeping policies. Furthermore, we isolate two main effects of correlation in spot-price dynamics when selecting policy parameters: a variability effect, which increases the benefits from stock-keeping and lessens the usage of the option contract, and a counteracting correlation effect that exploits persistence of low/high spot price incidences. Finally, in a numerical study we show under which circumstances disregarding the correlation can result in large performance losses.

Appendices

Appendix 1: Proofs of Propositions

1.1 Proof of Proposition 1

Since the current price state p in $D_t(I,R,p)$ does not have an impact on the future cost $C_{t+1}(I+Q_L+Q_S-x,R,x)$ and since $C_{t+1}(.)$ is convex for each x, the solution structure of the optimization problem is exactly the same as in the case without correlation (see Inderfurth et al. 2013) so that the policy structure carries over. $\square$

1.2 Proof of Proposition 2

The starting point is given by the dynamic programming recursive equations for $t=1,\dots ,T$:

$$\begin{aligned} {D_t}(I,R,p)&=\underset{\begin{array}{c} R\ge Q_L\ge 0 \\ Q_S\ge 0 \end{array} }{\mathop {\min }} \Bigg \{ rR+ c Q_L+p Q_S+L(I+Q_L+Q_S)\nonumber \\&\quad +\alpha \int \limits _0^\infty C_{t+1}(I+Q_L+Q_S- x,R,p)f(x)dx\Bigg \} \end{aligned}$$

(12)

with

$$\begin{aligned} {C_t}(I,R,q)=\int _0^{\infty }{{D_t}(I,R,p){g(p|q)}dp} \quad \text { and }\quad {C_{T+1}}(I,R,q)\equiv 0. \end{aligned}$$

The major steps of the proof include

• proving the optimality of the $(S_L(p),S_S(p))$ policy by complete induction,

• proving that this policy holds for any t if $C_{t+1}(I,R,q)$ is convex in I and R,

• proving that $D_t(I,R,p)$ is convex in I and R if this policy is applied,

• proving that this holds for the final period $t=T$ or $t=T-1$, respectively, and

• proving that ${C_1}(\bar{I},R,\bar{p})$ is convex in R where $\bar{I}$ is a given initial inventory and $\bar{p}$ an initial price.

The optimization problem in period t can be reformulated as

$$\begin{aligned} {D_t}(I,R,p)&=\underset{R\ge {Q_L}\ge 0,{Q_S}\ge 0}{\mathop {\min }}\,\left\{ c{Q_L}+p{Q_S}+{H_t}(I+Q_L+Q_S,R,p) \right\} \end{aligned}$$

(13)

$$\begin{aligned} \text {with } {H_t}(I+Q_L+Q_S,R,p)&\equiv L(I+Q_L+Q_S)\\&\quad +\alpha \int _0^\infty {{C_{t+1}}(I+Q_L+Q_S-x,R,p)f(x)dx} \end{aligned}$$

By assumption ${C_{t+1}}(I,R,p)$ is convex in I and R for each p, thus ${H_t}(I,R,p)$ is also convex in I and R for each p due to well-known convexity of L(I). So, under the assumption that a $(S_L(p),S_S(p))$ policy holds, we can analyze the properties of minimum cost functions ${D_t}(I,R,p)$ and ${C_t}(I,R,q)$

• (i) in case of $p\le c$:

  $$\begin{aligned} {D_t}(I,R,p)&={\left\{ \begin{array}{ll} p\cdot ({S_S}(p)-I)+{H_t}(S_S(p),R,p) &{} \text {if }I\le S_S(p) \\ {H_t}(I,R,p) &{} \text {if }I\ge S_S(p) \\ \end{array}\right. } \end{aligned}$$

  (14)

• (ii) in case of $p>c$:

  $$\begin{aligned} D_t(I,R,p)={\left\{ \begin{array}{ll} c\cdot R+p\cdot (S_S(p)- I- R)+H_t(S_S(p),R,p) &{} \text {if }I\le S_S(p)-R \\ c\cdot R+{H_t}(I+R,R,p) &{} \text {if }S_S(p)- R\le I\le S_L(p)- R \\ c\cdot (S_L(p)- I)+{H_t}(S_L,\,R,p) &{} \text {if }S_L(p)- R\le I\le S_L(p) \\ H_t(I,R,p) &{} \text {if }I\ge S_L(p) \end{array}\right. } \end{aligned}$$

  (15)

We can easily show that ${D_t}(I,R,p)$ is twice continuously differentiable in I and R for each p. Due to convexity of ${H_t}(I,R,p)$ for each p we have:

$$\begin{aligned}&\frac{\partial ^2 H_t(I,R,p)}{\partial I^2}\ge 0,\quad \frac{\partial ^2 H_t(I,R,p)}{\partial R^2}\ge 0,\quad \text {and} \quad \frac{\partial ^2 H_t(I,R,p)}{\partial I^2}\cdot \frac{\partial ^2H_t(I,R,p)}{\partial R^2}\nonumber \\&\quad -\frac{\partial ^2 H_t (I,R,p)}{\partial I\partial R}\cdot \frac{\partial ^2 H_t(I,R,p)}{\partial R\partial I}\ge 0. \end{aligned}$$

(16)

So the Hessian of ${D_t}(I,R,p)$ is positive-semidefinite for each p, ${D_t}(I,R,p)$ is convex in I and R for each p, and ${C_t}(I,R,q)=\int _0^{\infty }{{D_t}(I,R,p){g(p|q)}dp}$ is convex in I and R due to ${g(p|q)}\ge 0$.

Steps of induction:

• For $t=T$ (start of induction) we have: ${H_T}(I,R,p)=L(I)$ which is independent of R and p, thus ${H_T}(I,R,p)$ is convex in I and a $(S_L,S_S(p))$ policy is optimal.

• For $t=T-1$ we have: ${H_T}(I,R,p)=L(I)$ we find: Function ${C_{T}}(I,R,p)$ depends on R and p and is convex in I and R for each p. From that it follows that also ${H_T-1}(I,R,p)$ depends on R and p and is convex in I so that a $(S_L(p),S_S(p))$ policy is optimal.

• For each $t < T-1$ the following holds: From convexity of ${C_{t+1}}(I,R,p)$ it follows that also ${C_t}(I,R,p)$ is convex in I and R for each p, so ${H_{t-1}}(I,R,p)$ is also convex in I and R, and consequently for each R and p a $(S_L(p),S_S(p))$ policy is optimal also for $t-1$.

Conclusions on policy structure:

For each R a $(S_L(p),S_S(p))$ policy is optimal for $1\le t\le T$.

• Policy parameter ${S_{L,t}}(p)$ is calculated from $\frac{\partial {H_t}(S,R,p)}{\partial S}+c=0$ for each R and p.

• Policy parameter ${S_{S,t}}(p)$ is calculated from $\frac{\partial {H_t}(S,R,p)}{\partial S}+p=0$ for each R and p.

• Policy parameter R is calculated from: $\frac{\partial {C_1}(\bar{I},R,\bar{p})}{\partial R}=0$ for given initial inventory $\bar{I}$ and price $\bar{p}$.

• Functions ${C_1}(\bar{I},R,\bar{p})$ and ${H_t}(I,R,p)$ are convex.

From unconstrained optimization under the optimal R-level we get as optimal inventory levels

• after $Q_L$-optimization: ${S_{L,t}}(p_t,R)$ from $\frac{\partial {H_t}(S,R,p_t)}{\partial S}+c=0$

• after $Q_S$-optimization: ${S_{S,t}}(p_t,R)$ from $\frac{\partial {H_t}(S,R,p_t)}{\partial S}+p_t=0$

Due to $\frac{\partial }{\partial {Q_L}}{H_t}(I+{Q_L}+{Q_S},R,p_t)=\frac{\partial }{\partial {Q_S}}{H_t}(I+{Q_L}+{Q_S},R,p_t)$, and due to restrictions $0\le {Q_L}\le R$ and $\text{0 }\le {Q_S}$ we get the policy structure described in Proposition 2(a) and find the relevant cost functions to be convex.

From convexity of ${H_t}(S,R,p)$ and respective optimality conditions for the order-up-to levels it immediately follows that

$$\begin{aligned} {S_{S,t}}(p_t)\text { } {\left\{ \begin{array}{ll}>S_{L,t}(p_t) &{} \text {if } p_t<c \\ ={S_{L,t}}(p_t) &{} \text {if } p_t=c \\ <{S_{L,t}}(p_t) &{} \text {if } p_t>c \\ \end{array}\right. }. \end{aligned}$$

(17)

Next we show monotonicity properties of $S_{L,t}(p)$ and $S_{S,t}(p)$. Starting point are first-order conditions for policy parameters $S_{L,t}(p)$ and $S_{S,t}(p)$ from which we define functions $F_{L,t}(.)$ and $F_{S,t}(.)$ as follows

$$\begin{aligned} F_{L,t}(S,R,p)&:=\frac{\partial {H_t}(S,R,p)}{\partial S}+c=0\ \ \text {and} \nonumber \\ F_{S,t}(S,R,p)&:=\frac{\partial {H_t}(S,R,p)}{\partial S}+p=0. \end{aligned}$$

(18)

Thus, we find the following first-order derivatives

$$\begin{aligned} \frac{dS_{L,t}}{dp}=-\frac{\partial F_{L,t}(S,R,p) / \partial p}{\partial F_{L,t}(S,R,p) / \partial S}\quad \text {and}\quad \frac{dS_{S,t}}{dp}=-\frac{\partial F_{S,t}(S,R,p) / \partial p}{\partial F_{S,t}(S,R,p) / \partial S}. \end{aligned}$$

(19)

From convexity of $H_t(S,R,p)$ in S it is evident that

$$\begin{aligned} \frac{\partial F_{L,t}(S,R,p)}{\partial S}=\frac{\partial F_{S,t}(S,R,p)}{\partial S}=\frac{\partial ^2 H_t(S,R,p)}{(\partial S)^2}\ge 0. \end{aligned}$$

(20)

Furthermore

$$\begin{aligned} \frac{\partial F_{L,t}(S,R,p)}{\partial p}=\frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}\quad \text {and}\quad \frac{\partial F_{S,t}(S,R,p)}{\partial p}=\frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}+1. \end{aligned}$$

(21)

It can be shown by complete induction (see below) that

$$\begin{aligned} -1\le \frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}\le 0, \end{aligned}$$

(22)

so that

$$\begin{aligned} \frac{dS_{L,t}}{dp}\ge 0\quad \text {and}\quad \frac{dS_{S,t}}{dp}\le 0. \end{aligned}$$

(23)

This is just the relationship described in Proposition 2(b).

Steps of induction

• $t=T$: $H_T(S,R,p)=L(S)$ so that $\frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}=0$.

• $t=T-1$: For the minimum cost function $D_T(S,R,p)$ in (14) and (15) under a $(S_L(p),S_S(p))$ policy we find due to $H_T(S,R,p)=L(S)$

  $$\begin{aligned} -p\le \frac{\partial D_T(S,R,p)}{\partial S}\le \frac{dL(S)}{dS}. \end{aligned}$$

  (24)

  In a next step we find that

  $$\begin{aligned} -1\le \frac{\partial ^2 D_T(S,R,p)}{\partial S\cdot \partial p}\le 0. \end{aligned}$$

  (25)

  Thus, it follows from $C_T(S,R,q)=\int _0^\infty D_T(S,R,p)\cdot {g(p|q)}\cdot dp$

  $$\begin{aligned} -1\le \frac{\partial ^2 C_T(S,R,p)}{\partial S\cdot \partial p}=\int _0^\infty \frac{\partial ^2 D_T(S,R,p)}{\partial S\cdot \partial p}\cdot {g(p|q)}\cdot dp\le 0. \end{aligned}$$

  (26)

  Further, we find from $H_{T-1}(S,R,q)=L(S)+\alpha \int _0^\infty C_T(S-x,R,p)\cdot f(x)\cdot dx$

  $$\begin{aligned} -1\le \frac{\partial ^2 H_{T-1}(S,R,p)}{\partial S\cdot \partial p}=\alpha \int _0^\infty \frac{\partial ^2 C_T(S-x,R,p)}{\partial S\cdot \partial p}\cdot f(x)\cdot dx\le 0. \end{aligned}$$

  (27)

• $t<T-1$: Due to

  $$\begin{aligned} -p\le \frac{\partial D_t(S,R,p)}{\partial S}\le \frac{\partial H_t(S,R,p)}{\partial S} \end{aligned}$$

  (28)

  under an $(S_L(p),S_S(p))$ policy we find for $-1\le \frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}\le 0$ that

  $$\begin{aligned} -1\le \frac{\partial ^2 D_t(S,R,p)}{\partial S\cdot \partial p}\le 0. \end{aligned}$$

  (29)

  Thus, the property $-1\le \frac{\partial ^2 H_t(S,R,p)}{\partial S\cdot \partial p}\le 0$ carries over for each t.

The stationarity property of the policy for infinite horizon problems for problems without discounting can be shown in just the same way as in the case without correlation (see Inderfurth et al. 2013). $\square$

Appendix 2: Additional figures and box plots

See Figs. 8 and 9.

Fig. 8

Impact of coefficient of correlation $\rho _{\text {PP}}$ on order-up-to levels $S_L(p)$ and $S_S(p)$

Fig. 9

Impact of model parameters capacity reservation price r, holding cost h, backorder cost v, demand standard deviation ${\sigma _{\text {X}}}$, and average spot-market price ${\mu _{\text {P}}}$ on expected cost error of misspecification (restricted to $\sigma =3$/${\rho _{\text {PP}}}=0.8$ instances)

Appendix 3: Impact of misspecification of policy structure in base case

We assume a simplified policy structure in which the order-up-to level for long-term supplier procurement is constant, i.e. $S_L(p)=S_L$. Table 4 provides the cost error for different $S_L$ levels in the base case scenario (optimal cost $C=95.79$) where all other policy parameters are taken from the optimal solution. Fixing the order-up-to level $S_L$ at $S_S^*(c)$ only yields small error which further reduces when re-optimizing over $S_L$. The remaining error could (possibly) be further reduced when appropriately adapting all other policy parameters, too.

Table 4 Impact of constant order-up-to level $S_L$ on cost performance in base case scenario