Fuzzy Optimization for Multi-item Supply Chain with Trade Credit and Two-Level Price Discount Under Promotional Cost Sharing

Abstract

Here, a multi-item two-level supply chain is considered where supplier offers a cash discount and a credit period to its retailer to boost the demand of the items. Due to this facility, retailer also offers a cash discount to its customers to increase the base demand of the items. Retailer also introduces some promotional costs to boost the base demand of the items. It is established theoretically that if the supplier shares a part of the promotional cost then the channel profit as well as the individual profit increase. It is also established that the cash discount given to the customers has sufficient significance in a supply chain. The supply chain model is also considered in imprecise environment when different inventory costs are fuzzy in nature. In this case, individual profits as well as channel profit become fuzzy in nature. As optimization of fuzzy objective is not well defined, following credibility measure of fuzzy event, an approach is followed for comparison of fuzzy objectives and a particle swarm optimization algorithm is used to find marketing decisions. In another approach graded mean integration values of the fuzzy objectives are optimized to find marketing decisions. Models are illustrated with numerical examples.

Appendix

Proof

(Lemma 1) The Hessian matrix for $\Pi _\mathrm{R}$ is

$$\begin{aligned} D_{n+2}= \begin{bmatrix} \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}^{2}}&0&\cdots&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial f_\mathrm{R}} \\ 0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}^{2}}&\cdots&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial f_\mathrm{R}} \\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 0&0&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}^{2}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial f_\mathrm{R}} \\ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial \rho _{1}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial \rho _{2}}&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial \rho _{n}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T^{2}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial f_\mathrm{R}} \\ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{1}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{2}}&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{n}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}^{2}} \end{bmatrix} \end{aligned}$$

Since $\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}}<0$ for $i=1,2,\ldots ,n$, so $(-1)^{i}.|D_{i}|>0$ for $i=1,2,\ldots ,n$. If $(-\,1)^{i}.|D_{i}|>0$ for $i=n+1$ and $i=n+2$, then there must be a solution of the given set of equations to maximize $\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})$.

Multiplying each element in each row i, $(i=1,2,\ldots ,n)$ of $D_{n+2}$ by $-\,\left( \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T \partial \rho _{i}}\big /\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right)$ and adding it to the corresponding element in $(n+1)$th row, the above matrix becomes

$$\begin{aligned} D_{n+2}= \begin{bmatrix} \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}^{2}}&0&\cdots&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial f_\mathrm{R}} \\ 0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}^{2}}&\cdots&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial f_\mathrm{R}} \\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 0&0&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}^{2}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial f_\mathrm{R}} \\ 0&0&\cdots&0&V_\mathrm{R}&V_\mathrm{R}' \\ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{1}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{2}}&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{n}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial T}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}^{2}} \end{bmatrix} \end{aligned}$$

$$\begin{aligned} \text{ where } \,\, V_\mathrm{R}&= \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T^{2}} -\sum \limits _{i=1}^{n} \left[ \left( \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial T} \right) ^{2} \bigg / \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right] \\&= -\,\frac{2\left[ A_\mathrm{R}+\sum \nolimits _{i=1}^{n}a_{R,i} \right] }{T^3} - \sum \limits _{i=1}^{n} (1-f_\mathrm{S})w_{i}(I_{p}-I_{e})\frac{\rho _{i}'\xi _{i}}{T^{3}}t_\mathrm{S}^{2} \\&+\, \sum \limits _{i=1}^{n} \left[ \left\{ -\frac{\xi _{i}}{2}\{h_{R,i}+(1-f_\mathrm{S})w_{i}I_{p}\} + (1-f_\mathrm{S})w_{i}(I_{p}-I_{e})\frac{\xi _{i}t_\mathrm{S}^{2}}{2T^{2}}\right\} ^{2} \bigg /2K_{i}\xi _{i}^{\alpha _{i}} \right] \\ \text{ and } \,\, V_\mathrm{R}'&= \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial f_\mathrm{R}} -\sum \limits _{i=1}^{n} \left[ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial f_\mathrm{R}}. \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial T} \bigg / \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right] \\&= -\sum \limits _{i=1}^{n} \frac{\lambda \xi _{i}}{2} \left\{ h_{R,i}+(1-f_\mathrm{S})w_{i}I_{p}\right\} +\sum \limits _{i=1}^{n} (1-f_\mathrm{S})w_{i}(I_{p}-I_{e}) \frac{\lambda \xi _{i}t_\mathrm{S}^{2}}{2T^{2}} \\&+\sum \limits _{i=1}^{n} \left[ \left\{ \frac{r_{i}\xi _{i}^{2}}{2}\{h_{R,i}+(1-f_\mathrm{S})w_{i}I_{p}\} - (1-f_\mathrm{S})w_{i}(I_{p}-I_{e})\frac{r_{i}\xi _{i}^{2}t_\mathrm{S}^{2}}{2T^{2}} \right\} \bigg /2K_{i}\xi _{i}^{\alpha _{i}} \right] \\ \end{aligned}$$

Now, multiplying each element in each column i, $(i=1,2,\ldots ,n)$ of $D_{n+2}$ by $-\left( \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial T} \big /\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right)$ and adding it to the corresponding element in $(n+1)$th column, the matrix reduces to

$$\begin{aligned} D_{n+2}= \begin{bmatrix} \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}^{2}}&0&\cdots&0&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial f_\mathrm{R}} \\ 0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}^{2}}&\cdots&0&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial f_\mathrm{R}} \\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 0&0&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}^{2}}&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial f_\mathrm{R}} \\ 0&0&\cdots&0&V_\mathrm{R}&V_\mathrm{R}' \\ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{1}}&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{2}}&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{n}}&W_\mathrm{R}'&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}^{2}} \end{bmatrix} \end{aligned}$$

$$\begin{aligned} \text{ where } \,\,W_\mathrm{R}'&= \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial T\partial f_\mathrm{R}} -\sum \limits _{i=1}^{n} \left[ \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial f_\mathrm{R}}. \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial T} \bigg / \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right] \\&= V_\mathrm{R}'. \end{aligned}$$

Now, multiplying each element in each row i, $(i=1,2,\ldots ,n)$ of $D_{n+2}$ by $-\left( \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}\partial \rho _{i}} \big / \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right)$ and in $(n+1)th$ row by $-(W_\mathrm{R}'/V_\mathrm{R})$ and adding it to the corresponding element in $(n+2)$th row, the following matrix is obtained

$$\begin{aligned} D_{n+2}= \begin{bmatrix} \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}^{2}}&0&\cdots&0&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{1}\partial f_\mathrm{R}} \\ 0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}^{2}}&\cdots&0&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{2}\partial f_\mathrm{R}} \\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 0&0&\cdots&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}^{2}}&0&\frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{n}\partial f_\mathrm{R}} \\ 0&0&\cdots&0&V_\mathrm{R}&V_\mathrm{R}' \\ 0&0&\cdots&0&0&W_\mathrm{R} \end{bmatrix} \end{aligned}$$

$$\begin{aligned} \text{ where } \,\,W_\mathrm{R}&= \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial f_\mathrm{R}^{2}} -\sum \limits _{i=1}^{n} \left[ \left( \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}\partial f_\mathrm{R}} \right) ^{2} \bigg / \frac{\partial ^{2}\Pi _\mathrm{R}(\rho _{i},T,f_\mathrm{R})}{\partial \rho _{i}^{2}} \right] -\frac{V_\mathrm{R}'\times W_\mathrm{R}'}{V_\mathrm{R}} \\&= -\sum \limits _{i=1}^{n} 2\lambda r_{i}\xi _{i} +\sum \limits _{i=1}^{n} \left[ r_{i}^{2}\xi _{i}^{2} \bigg /2K_{i}\xi _{i}^{\alpha _{i}} \right] - \frac{V_\mathrm{R}'\times W_\mathrm{R}'}{V_\mathrm{R}}. \end{aligned}$$

Hence, the required condition is proved. $\square$

Proof

(Proposition 1)

• (a) From

  $$\begin{aligned}&\Pi _\mathrm{R}^{F}\left( \rho _{i}^{l},T^{l},f_\mathrm{R}^{l}\right) -\Pi _\mathrm{R} \ge 0 \\\Rightarrow & {} - \frac{A_\mathrm{R}}{T^{l}} + \sum \limits _{i=1}^{n} \left[ \left\{ (1-f_\mathrm{R})r_{i}-(1-f_\mathrm{S})w_{i}\right\} \rho _{i}''\xi _{i} - \frac{a_{R,i}}{T^{l}} -(1-F)K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}}\right. \nonumber \\&\left. -\frac{\rho _{i}''\xi _{i}T^{l}}{2}h_{R,i} -(1-f_\mathrm{S})w_{i}I_{p}\frac{\rho _{i}''\xi _{i}}{2T^{l}}(T^{l}-t_\mathrm{S})^{2} +(1-f_\mathrm{S})w_{i}I_{e}\frac{\rho _{i}''\xi _{i}}{2T^{l}}t_\mathrm{S}^{2} \right] \ge \Pi _\mathrm{R} \\&\text{ where }, \rho _{i}''=\rho _{i}^{l}+\lambda f_\mathrm{R}^{l}. \\\Rightarrow & {} F \ge \left\{ \Pi _\mathrm{R} + \frac{A_\mathrm{R}}{T^{l}} - \sum \limits _{i=1}^{n} \left[ \left\{ (1-f_\mathrm{R})r_{i}-(1-f_\mathrm{S})w_{i}\right\} \rho _{i}''\xi _{i} -\frac{a_{R,i}}{T^{l}} - \frac{\rho _{i}''\xi _{i}T^{l}}{2}h_{R,i}\right. \right. \nonumber \\&-\left. \left. K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} -(1-f_\mathrm{S})w_{i}I_{p}\frac{\rho _{i}''\xi _{i}}{2T^{l}}(T^{l}-t_\mathrm{S})^{2} +(1-f_\mathrm{S})w_{i}I_{e}\frac{\rho _{i}''\xi _{i}}{2T^{l}}t_\mathrm{S}^{2} \right] \right\} \nonumber \\&\bigg / \sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \\ \end{aligned}$$

  Therefore, $F_\mathrm{min}$ is obtained. Also from $\Pi _\mathrm{S}^{F}(\rho _{i}^{l},T^{l},f_\mathrm{R}^{l})-\Pi _\mathrm{S} \ge 0$, $F_\mathrm{max}$ can be obtain.

• (b) The following relations are found from (a):

  $$\begin{aligned} F_\mathrm{max}-F&= \left[ \Pi _\mathrm{S}^{F}(\rho _{i}^{l},T^{l},f_\mathrm{R}^{l})-\Pi _\mathrm{S} \right] \bigg /\sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \\ \Rightarrow \bigtriangleup \Pi _\mathrm{S}^{F}&= (F_\mathrm{max}-F)\sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \\ \text{ Similarly, } \,\, \bigtriangleup \Pi _\mathrm{R}^{F}&= (F-F_\mathrm{min})\sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \end{aligned}$$

  Now, $\bigtriangleup \Pi _\mathrm{S}^{F}(F) \times \bigtriangleup \Pi _\mathrm{R}^{F}(F)$

  $$\begin{aligned}&= \left[ (F_\mathrm{max}-F)\sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \right] \times \left[ (F-F_\mathrm{min})\sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \right] \\&= (F_\mathrm{max}-F)(F-F_\mathrm{min}) \left\{ \sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \right\} ^{2}\\&= (F_\mathrm{max}\,\cdot \,F-F_\mathrm{max}\,\cdot \, F_\mathrm{min}-F^2+F\,\cdot \,F_\mathrm{min})\times \left\{ \sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \right\} ^{2} \\&= \left[ -\left( F-\frac{F_\mathrm{max}+F_\mathrm{min}}{2} \right) ^2+\frac{(F_\mathrm{max}-F_\mathrm{min})^2}{4} \right] \times \left\{ \sum \limits _{i=1}^{n}K_{i}(\rho _{i}^{l}-1)^2\xi _{i}^{\alpha _{i}} \right\} ^{2}. \end{aligned}$$

  Thus, the appropriate fraction of the promotional cost sharing is obtained as $F=(F_\mathrm{max}+F_\mathrm{min})/2$.

$\square$