Optimal inventory policy through dual sourcing

Abstract

Profit maximization in the retail and manufacturing industry is currently focused on offshore production to utilize the resulting low production costs. However, in the face of uncertain customer demand, it is difficult to determine an optimal order quantity which maximizes profit. We consider a risk-averse firm that utilizes dual-sourcing for perishable or seasonal goods with uncertain customer demand. Using real options theories, we provide two models aimed at determining optimal order quantities to maximize the firm’s expected profit. Furthermore, we can consider the demand to be an observable process correlated to a traded asset, which can be hedged to reduce profit uncertainty. A single offshore single local order period model provides a pseudo-analytical solution which can be easily solved to determine optimal offshore and local order quantities based on the manufacturers’ lead times, and a more realistic single offshore multiple local order period model which uses numerical methods to determine optimal order quantities. Finally, a method for matching distributions of expected demands based on managerial estimates can be applied to the two models, providing managers a tool for practical application.

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Notes

  1. 1.

    We note that the process, \(X_t\), can be modelled in the actual or risk-neutral probability space.

  2. 2.

    For a risk-neutral valuation, the drift can be adjusted accordingly.

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Appendices

Appendix

A Derivation of \(U^*\)

To derive \(U^*\), we rewrite \({\mathbb {E}}_{\tau }[G(X_T;M,U)]\) from Eq. 2 as

$$\begin{aligned} {\mathbb {E}}_{\tau }[G(X_T;M,U)]= & {} \int ^{\infty }_{0} [\underbrace{\min (x,M+U)P}_\text {A1} + \underbrace{P_{Salv}(M + U - x)^{+}}_\text {A2} \nonumber \\&- \underbrace{P_{Strat}(x - M - U)^{+}}_\text {A3}]f_{X_T|X_\tau }(x;y=X_\tau )dx - \underbrace{UC_{U}}_\text {A4}\nonumber \\ \end{aligned}$$
(18)

and maximize the expected profit by taking the derivative with respect to U and equating it to zero. Separating the equation into four parts \(A_{1}\), \(A_{2}\), \(A_{3}\) and \(A_{4}\) allows for easy expression management. Beginning with \(A_1\), the discontinuous nature of the equation makes it necessary to separate it into components where integration is possible as shown with,

$$\begin{aligned} \begin{aligned} A_{1} =&\int ^{\infty }_{0} (1_{x < M+U} \cdot x+1_{x > M+U} \cdot (M+U)) \cdot P \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx. \end{aligned} \end{aligned}$$
(19)

Next, taking the derivative of \(A_1\) with respect to U and integrating the function yields,

$$\begin{aligned} \begin{aligned} \dfrac{dA_{1}}{dU} =&\int ^{\infty }_{0} 1_{x > M+U} \cdot P \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\\ =&\int ^{\infty }_{M+U} P \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\\ =&P \cdot (1-F_{X_T|X_\tau }(M+U;y=X_\tau )). \end{aligned} \end{aligned}$$
(20)

The same process is applied for parts \(A_2 - A_4\);

$$\begin{aligned} A_{2}= & {} \int ^{\infty }_{0} P_{Salv} \cdot 1_{x < M+U} \cdot (M+U-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx, \end{aligned}$$
(21)
$$\begin{aligned} \dfrac{dA_{2}}{dU}= & {} \int ^{\infty }_{0} P_{Salv} \cdot 1_{x < M+U} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\nonumber \\= & {} \int _{0}^{M+U} P_{Salv} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\nonumber \\= & {} P_{Salv} \cdot F_{X_T|X_\tau }(M+U;y=X_\tau ), \end{aligned}$$
(22)
$$\begin{aligned} A_{3}= & {} \int ^{\infty }_{0} P_{Strat} \cdot 1_{x > M+U} \cdot (x-M-U)) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx, \end{aligned}$$
(23)
$$\begin{aligned} \dfrac{dA_{3}}{dU}= & {} -\int ^{\infty }_{0} P_{Strat} \cdot 1_{x > M+U} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\nonumber \\= & {} -\int ^{\infty }_{M+U} P_{Strat} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\nonumber \\= & {} -P_{Strat} \cdot (1-F_{X_T|X_\tau }(M+U;y=X_\tau )), \end{aligned}$$
(24)
$$\begin{aligned} A_{4}= & {} UC_{U}, \end{aligned}$$
(25)
$$\begin{aligned} \dfrac{dA_{4}}{dU}= & {} C_{U}. \end{aligned}$$
(26)

Substituting Eqs. 20, 22, 24, 26 into Eq. 18 and isolating for U, allows us to determine the optimal onshore order quantity \(U^*\), which maximizes the overall profit at time \(t=\tau \),

$$\begin{aligned} U^* = \underbrace{F^{-1}_{X_T|X_{\tau }}\bigg (\dfrac{P + P_{Strat} - C_{U}}{P+P_{Strat}-P_{Salv}};y\bigg )}_{\gamma (y)} - M \equiv \gamma (y)-M. \end{aligned}$$
(27)

B Derivation of the Profit

Following a procedure similar to the derivation of \(U^*\), we begin with Eq. 5

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}[Profit(M,U^*)] \\&\quad = \int ^{\infty }_{0}[\underbrace{\min (X,M+(\gamma (y) - M)^+)P}_\text {B1} + \underbrace{P_{Salv} \cdot (M + (\gamma (y) - M)^+ - X)^{+}}_\text {B2}\\&\qquad - \underbrace{P_{Strat} \cdot (X - M - (\gamma (y) - M)^+)^{+}}_\text {B3}]f_{X_T|X_\tau }(x;y=X_\tau )dx \\&\qquad \underbrace{-MC_{M} - (\gamma (y) - M)^+C_{U}}_\text {B4}, \end{aligned} \end{aligned}$$
(28)

and separate it into four parts \(B_{1}\) to \(B_{4}\) and solve each individually,

$$\begin{aligned} \begin{aligned} B_{1} =&\int ^{\infty }_{0} (1_{x< M+U} \cdot x+1_{x>M+U} \cdot (M+U)) \cdot P \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx,\\ B_{2} =&\int ^{\infty }_{0} (1_{x< M+U} \cdot (M+U-x)) \cdot P_{Salv} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx,\\ B_{3} =&\int ^{\infty }_{0} (1_{x > M+U} \cdot (x-(M+U))) \cdot P_{Strat} \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx,\\ B_{4} =&{\left\{ \begin{array}{ll} -MC_{M} - (\gamma (y) - M)C_{U} &{} \text { if } \gamma (y) \ge M\\ -MC_{M} &{} \text { if } \gamma (y)<M. \end{array}\right. } \end{aligned} \end{aligned}$$
(29)

Since the proposed functions are discontinuous, we must determine appropriate integration boundaries to the functions in Eq. 29, as can be seen for \(B_1\) where

$$\begin{aligned} B_{1} =&\int ^{\infty }_{0} (1_{x < M+U} \cdot x+1_{x>M+U} \cdot (M+U)) \cdot P \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx. \end{aligned}$$

Substituting \(U^*\) for U transforms the first part of the discontinuous function \((1_{x < M+U} \cdot x)\) into

$$\begin{aligned}&1_{x<M+(\gamma (y)-M)^+} \rightarrow {\left\{ \begin{array}{ll} 1_{x<\gamma (y)} &{} \text { if } \gamma (y) \ge M\\ 1_{x<M} &{} \text { if } \gamma (y)<M\\ \end{array}\right. } = 1_{\gamma (y)>M} \cdot 1_{x<\gamma (y)} + 1_{\gamma (y)<M} \cdot 1_{x<M} \end{aligned}$$

which exists under the conditions that demand is between M and \(\gamma (y)\) when \(\gamma (y)\) is greater than M, or when demand is less than M when \(\gamma (y)\) is less than M, as shown in the following plot line

figurea

The second part of the discontinuous function \((1_{x>M+(\gamma (y)-M)^+})\) is handled in a similar manner, initially by substituting \(U^*\) for U, and then determining the integration boundaries:

$$\begin{aligned} 1_{x>M+(\gamma (y)-M)^+} = 1_{\gamma (y)>M} \cdot 1_{x>\gamma (y)} + 1_{\gamma (y)<M} \cdot 1_{x>M} \end{aligned}$$
figureb

As a result, the function \(B_1\) simplifies to:

$$\begin{aligned} B_{1} =&P \left\{ \begin{array}{ll} \int ^{\gamma (y)}_{0} x \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx + \int ^{\infty }_{\gamma (y)} \gamma (y)&{} \text { if } \gamma (y) \ge M\\ \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{}\\ \int ^{M}_{0} x \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx + \int ^{\infty }_{M} M &{} \text { if } \gamma (y)<M. \\ \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{}\\ \end{array} \right. \end{aligned}$$
(30)

Applying a similar process, functions \(B_2\) and \(B_3\) simplify to

$$\begin{aligned} B_{2} =&P_{Salv} \left\{ \begin{array}{ll} \int ^{\gamma (y)}_{0} (\gamma (y)-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y) \ge M\\ \int ^{M}_{0} (M-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y)<M \end{array}\right. \end{aligned}$$
(31)
$$\begin{aligned} B_{3} =&P_{Strat} \left\{ \begin{array}{ll} \int _{\gamma (y)}^{\infty } (x-\gamma (y)) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y) \ge M\\ \int _{M}^{\infty } (x-M) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y)<M. \end{array}\right. \end{aligned}$$
(32)

By defining \(G(y;a,b) \equiv \int ^{b}_{a}xf_{X_T|X_\tau }(x;y=X_\tau )dx\) we are able simplify the function \(\int ^{\gamma (y)}_{0} (\gamma (y)-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx\) such that

$$\begin{aligned} \begin{aligned} \int ^{\gamma (y)}_{0} (\gamma (y)-x) \cdot f_{X_T|X_\tau }(x;y=&X_\tau )dx \equiv \gamma (y) \cdot F_{X_T|X_\tau }(\gamma (y);y) \\&- G(y;0,\gamma (y)). \end{aligned} \end{aligned}$$
(33)

Applying this process to Eqs. 30, 31, and 32, we are able to reduce \(B_1\) to \(B_3\) to

$$\begin{aligned} B_{1}= & {} P \left\{ \begin{array}{ll} G(y;0,\gamma (y)) + \gamma (y) \cdot (1-F_{X_T|X_\tau }(\gamma (y);y)) &{} \text { if } \gamma (y) \ge M\\ G(y;0,M)+M \cdot (1-F_{X_T|X_\tau }(M;y)) &{} \text { if } \gamma (y)<M \end{array}\right. \end{aligned}$$
(34)
$$\begin{aligned} B_{2}= & {} P_{Salv} \left\{ \begin{array}{ll} \int ^{\gamma (y)}_{0} (\gamma (y)-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y) \ge M\\ \int ^{M}_{0} (M-x) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y)<M \end{array}\right. \nonumber \\= & {} P_{Salv} \left\{ \begin{array}{ll} \gamma (y) \cdot F_{X_T|X_\tau }(\gamma (y);y)-G(y;0,\gamma (y)) &{} \text { if } \gamma (y) \ge M\\ M \cdot F_{X_T|X_\tau }(M;y)-G(y;0,M) &{} \text { if } \gamma (y)<M \end{array}\right. \end{aligned}$$
(35)
$$\begin{aligned} B_{3}= & {} P_{Strat} \left\{ \begin{array}{ll} \int _{\gamma (y)}^{\infty } (x-\gamma (y)) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y) \ge M\\ \int _{M}^{\infty } (x-M) \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx &{} \text { if } \gamma (y)<M\\ \end{array}\right. \nonumber \\= & {} -P_{Strat} \left\{ \begin{array}{ll} \gamma (y)[1-F_{X_T|X_\tau }(\gamma (y);y)]-\int ^{\infty }_{\gamma (y)}(x) &{} \text { if }\gamma (y) \ge M\\ \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx&{}\\ M[1-F_{X_T|X_\tau }(M);y)]-\int ^{\infty }_{M}(x) &{} \text { if } \gamma (y) \ge M\\ \cdot f_{X_T|X_\tau }(x;y=X_\tau )dx&{}\\ \end{array}\right. \nonumber \\= & {} -P_{Strat} \left\{ \begin{array}{ll} \gamma (y)-\gamma (y) \cdot F_{X_T|X_\tau }(\gamma (y);y)-G(y;\gamma (y),\infty ) &{} \text { if } \gamma (y) \ge M\\ M-M \cdot F_{X_T|X_\tau }(M;y)-G(y;M,\infty ) &{} \text { if } \gamma (y)<M. \end{array}\right. \nonumber \\ \end{aligned}$$
(36)

Rearranging and cancelling terms in the above expression reduces the profit function to

$$\begin{aligned} Profit = \left\{ \begin{array}{ll} \gamma (y) \cdot (P+P_{Strat}-C_{U}) + \gamma (y) \cdot F(\gamma (y);y) &{}\text {if } \gamma (y) \ge M\\ \cdot (P_{Salv}-P-P_{Strat}) -P_{Strat} \cdot G(y;\gamma (y),\infty ) &{}\\ + (P-P_{Salv})\cdot G(y;0,\gamma (y))+ M \cdot (C_{U}-C_{M})&{}\\ &{}\\ M \cdot (P+P_{Strat}-C_{M})+M \cdot F_{X_T|X_\tau }(M;y) &{} \text { if } \gamma (y)<M.\\ \cdot (P_{Salv}-P-P_{Strat})+G(y;0,M) \cdot (P-P_{Salv})&{}\\ -P_{Strat} \cdot G(y;M,\infty )&{} \end{array}\right. \end{aligned}$$
(37)

After some algebraic manipulation, the expected profit at \(t=0\) can be determined as

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[Profit(y)]&= \int ^{\infty }_{y=\gamma ^{-1}(M)}\big [ \gamma (y) \cdot (P+P_{Strat}-C_{U}) + \gamma (y) \cdot F_{X_T|X_\tau }(\gamma (y);y) \\&\quad \cdot (P_{Salv}-P-P_{Strat}) -P_{Strat} \cdot G(y;\gamma (y),\infty ) + (P-P_{Salv}) \\&\quad \cdot G(y;0,\gamma (y)) + M \cdot (C_{U}-C_{M}) \big ]f_X(y)dy \\&\quad +\int ^{\gamma ^{-1}(M)}_{y=0}\big [ M \cdot (P+P_{Strat}-C_{M})+M \cdot F_{X_T|X_\tau }(M;y)\\&\cdot (P_{Salv}-P-P_{Strat}) +G(y;0,M) \cdot (P-P_{Salv})\\&\quad -P_{Strat} \cdot G(y;M,\infty ) \big ]f_X(y)dy, \end{aligned} \end{aligned}$$
(38)

where \(f_X(\cdot )\) is the density of X.

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Davison, M., Lawryshyn, Y. & Miklyukh, V. Optimal inventory policy through dual sourcing. Comput Manag Sci (2020). https://doi.org/10.1007/s10287-020-00371-8

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