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Sequential unreliable newsboy ordering policies

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Abstract

In this paper, we show how ordering time can be used as a mechanism to mitigate the supply risks of the unreliable newsboy. We consider the situation where a firm has an option of placing a series of sequential orders such that a new order is placed only after the yield from the previous order is known and derive an optimal order quantity for each ordering stage. The effectiveness of our approach is assessed by comparing its expected cost with the expected cost of the conventional reliable newsboy problem. It is demonstrated that the adverse effect of supply side risks can be almost completely negated by adopting our approach when the combined purchasing and holding costs for a low cost supplier at an earlier stage are equal to those of a high cost supplier at the next stage. Computational experiments suggest that the sequential ordering policy performs better than a simultaneous ordering strategy when customer service level norms become stringent and inventory holding costs decrease.

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Correspondence to Rahul Patil.

Appendix

Appendix

Proof of Theorem 4

Assuming the opening inventory is zero and that the demand is uniformly distributed between 0 and \(a\).

Using Eqs. (3), (4) and (5), and also assuming opening inventory at stage 2 to be zero \(I_{2}\) = 0

$$\begin{aligned} q_2^*=K_2 \hbox { Assuming }c_2 +h=c_1 \quad \therefore K_2 =K_1 =F^{-1}\left[ {\frac{p-c_1 }{p+v}} \right] \end{aligned}$$

Also for uniformly distributed demand, \(f(x)=\frac{1}{a}\hbox { and } F(x)=\frac{x}{a},\;K_1 =a\left[ {\frac{p-c_1 }{p+v}} \right] ,\quad \alpha =\frac{a}{2}\).

Substituting in the expected cost expression,

$$\begin{aligned} EC_{1A}&= \theta _1 \left\{ {\left( {v+p} \right) \frac{K_1 ^{2}}{2a}+p\left( {\alpha -K_1 } \right) } \right\} +(1-\theta _1 )\left\{ {\left( {v+p} \right) \frac{K_1 ^{2}}{2a}+p\left( {\alpha -K_1 } \right) } \right\} \nonumber \\&= \frac{\left( {p-c_1 } \right) K_1 }{2}+p\left( {\alpha -K_1 } \right) \nonumber \\ EC_{1B}&= \theta _1 \left\{ {c_1 K_1 +\left( {v+p} \right) \frac{K_1 ^{2}}{2a}+p\left( {\alpha -K_1 } \right) } \right\} +(1-\theta _1 )p\alpha \nonumber \\&= p\alpha -\frac{\theta _1 \left( {p-c_1 } \right) K_1 }{2} \nonumber \\ \therefore EC_2^*&= \theta _2 c_1 K_1 +\frac{\theta _2 \left( {p-c_1 } \right) K_1 }{2}+p\theta _2 \left( {\alpha -K_1 } \right) \nonumber \\&\,+(1-\theta _2 )p\alpha -\theta _1 (1-\theta _2 )\frac{\left( {p-c_1 } \right) K_1 }{2} \nonumber \\ \hbox {or }EC_2^*&= \frac{pa}{2}-\frac{\theta _1 \left( {p-c_1 } \right) K_1 }{2}-\frac{\theta _2 \left( {1-\theta _1 } \right) \left( {p-c_1 } \right) K_1 }{2} \end{aligned}$$
(17)

Expected cost for newsboy problem

$$\begin{aligned} EC_N =c_1 q_1 +\left( {v+p} \right) \left[ {q_1 F(q_1 )-\int \limits _0^{q_1 } {xf(x)dx} } \right] +p\left( {\alpha -q_1 } \right) \end{aligned}$$

Also optimal \(q_1 =K_1 \hbox { where }K_1 =F^{-1}\left[ {\left( {p-c_1 } \right) /\left( {p+v} \right) } \right] \).

As demand is uniformly distributed between 0 to \(a, \therefore \quad f(x)=\frac{1}{a} ,F(x)=\frac{x}{a}\).

$$\begin{aligned} \therefore EC_N^*&= c_1 K_1 +\frac{\left( {p+v} \right) K_1 ^{2}}{2a}+p\left( {\frac{a}{2}-K_1 } \right) \nonumber \\ \hbox {or }EC_N^*&= pa/2-(p-c_1 )K_1/2 \end{aligned}$$
(18)

Using (17) and (18)

$$\begin{aligned} EC_2^*-EC_N^*=(1-\theta _1 )(1-\theta _2 )\frac{(p-c_1 )K_1 }{2} \end{aligned}$$

Similarly it can be shown that for 3-stage sequential ordering:

$$\begin{aligned} EC_3^*-EC_N^*=(1-\theta _1 )(1-\theta _2 )(1-\theta _3 )\frac{(p-c_1 )K_1 }{2} \end{aligned}$$

And for n stages,

$$\begin{aligned} EC_n^*-EC_N^*=\prod _{i=1}^n {(1-\theta _i )} \frac{(p-c_1 )K_1 }{2} \end{aligned}$$

Proof of Lemma 1

For two simultaneous orders, the expected cost can be written as :

$$\begin{aligned} EC&= c_1 \theta _1 q_1 +c_2 \theta _2 q_2 +\left( {1-\theta _1 } \right) \left( {1-\theta _2 } \right) p\int _0^a xf\left( x \right) dx \\&+\,\,\theta _1 \theta _2 \left\{ v\int _0^{q_1 +q_2 } \left( {q_1 +q_2 -x} \right) f\left( x \right) dx\right. \\&+\,\left. p\int _{q_1 +q_2 }^a \left( {x-q_1 -q_2 } \right) f\left( x \right) dx \right\} \\&+\,\,\theta _1 \left( {1-\theta _2 } \right) \left\{ {v \int _0^{q_1 } \left( {q_1 -x} \right) f\left( x \right) dx+p\int _{q_1 }^a \left( {x-q_1 } \right) f\left( x \right) dx} \right\} \\&+\,\left( {1-\theta _1 } \right) \theta _2 \left\{ {v \int _0^{q_2 } \left( {q_2 -x} \right) f\left( x \right) dx+p\int _{q_2 }^a \left( {x-q_2 } \right) f\left( x \right) dx} \right\} \end{aligned}$$

Algebraic processing of the above equation yields following expected cost expression

$$\begin{aligned} EC=\frac{pa}{2}-(p-c_1 )\theta _1 q_1 -(p-c_2 )\theta _2 q_2 +\frac{(p+v)}{2a}\left[ {\theta _1 q_1^2 +\theta _2 q_2^2 +2\theta _1 \theta _2 q_1 q_2 } \right] \end{aligned}$$

Equating the differentiations of the equation with respect to \(q_1\) and \(q_2\) to zero gives

$$\begin{aligned} \frac{\partial EC}{\partial q_1 }&= \theta _1 \left\{ {c_1 -p+\frac{\left( {p+v} \right) \left( {q_1 +\theta _2 q_2 } \right) }{a}} \right\} =0\\ \frac{\partial EC}{\partial q_2 }&= \theta _2 \left\{ {c_2 -p+\frac{\left( {p+v} \right) \left( {q_2 +\theta _1 q_1 } \right) }{a}} \right\} =0 \end{aligned}$$

Processing of the above equation gives

$$\begin{aligned} q_1^*=\frac{a\left( {p-c} \right) }{\left( {p+v} \right) }-\theta _2 q_2^*. \end{aligned}$$

Similarly,

$$\begin{aligned} q_2^*=\frac{a\left( {p-c} \right) }{\left( {p+v} \right) }-\theta _1 q_1^*\end{aligned}$$

Substituting \(q_2^*\, in \,q_1^*\) gives

$$\begin{aligned} q_1^*=\frac{a\left( {p-c_1 } \right) -\theta _2 a\left( {p-c_2 } \right) }{\left( {1-\theta _1 \theta _2 } \right) \left( {p+v} \right) } \end{aligned}$$

Similarly,

$$\begin{aligned} q_2^*=\frac{a\left( {p-c_2 } \right) -\theta _1 a\left( {p-c_1 } \right) }{\left( {1-\theta _1 \theta _2 } \right) \left( {p+v} \right) } \end{aligned}$$

Let \(p_1 =\left( {p-c_1 } \right) \) and \(p_2 =\left( {p-c_2 } \right) \) and \(\frac{1}{X}=\frac{a}{\left( {p+v} \right) }\).

Initially substituting with these terms in the expected cost expression and algebraic processing yields the following optimal expected cost expression

$$\begin{aligned} EC^{*}=\frac{pa}{2}-\frac{a\left[ {\theta _1 \left( {p-c_1 } \right) ^{2}+\theta _2 \left( {p-c_2 } \right) ^{2}-2\theta _1 \theta _2 \left( {p-c_1 } \right) \left( {p-c{ }_2} \right) } \right] }{2\left( {1-\theta _1 \theta _2 } \right) \left( {p+v} \right) } \end{aligned}$$

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Tiwari, D., Patil, R. & Shah, J. Sequential unreliable newsboy ordering policies. Ann Oper Res 233, 449–463 (2015). https://doi.org/10.1007/s10479-014-1705-4

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