Trade credit policy of an inventory model with imprecise variable demand: an ABC-GA approach

Abstract

In this research work, an inventory model with fuzzy promotional effort induced dynamic demand under two level partial trade credit policy has been developed in an imprecise planning horizon. Here, it is assumed that in the planning horizon a retailer completed a finite number full cycles. In each of the retailer’s cycle, a wholesaler offers a credit period to the retailer on the full purchased amount and in turn the retailer offers a credit period to its customers on a part of his/her purchased amount. The imprecise marketing demand is influenced by the retailer’s fuzzy promotional effort, customers’ credit period, customers’ credit amount and retail selling price. Goal of this study is to find the optimal business strategy for the retailer with respect to his/her total fuzzy financial gain from the system. Due to imprecise nature of the demand, the problem is mathematically represented following fuzzy differential equation and fuzzy Riemann integration and alpha-cut of the entire fuzzy gain from the system is derived. Its graded mean integration representation is computed and optimized with respect to customer’s credit amount credit period, and retailer’s order quantity for most appropriate marketing decision. Hence, the problem reduced to a multivariate crisp optimization problem and a heuristic, multichoice artificial bee genetic algorithm (MCABGA) has been proposed for it. The efficiency of MCABGA is tested against some other existing artificial bee colony variants using a list of benchmark test functions available in the literature. The model is illustrated with some hypothetical test problems and some managerial insights are outlined. Finally, a conclusion is drawn and some future research directions are proposed.

Appendices

Appendix A

The GMIR value of the $\alpha$-cut of a fuzzy number $\widetilde{Z}$ and is given by Eq. (45).

(45)

$\hbox {HOC}$, IP, IE and $\hbox {PRC}$, respectively, represent the GMIR value of $\widetilde{\hbox {HOC}}[\alpha ]$, $\widetilde{IP}[\alpha ]$, $\widetilde{IE}[\alpha ]$, and $\widetilde{\hbox {PRC}}[\alpha ]$, where

$$\begin{aligned} \hbox {HOC}= & {} \frac{hQ^2}{2}\Big [\frac{1}{{(d_3-d_2)}^2}\Big \{d_3 \log \Big |\frac{d_3}{d_2}\Big |-(d_3-d_2)\Big \}\nonumber \\&+\frac{1}{{(d_2-d_1)}^2}\Big \{(d_2-d_1)-\nonumber \\&d_1 \log \Big |\frac{d_2}{d_1}\Big |\Big \}\Big ] \end{aligned}$$

(46)

$$\begin{aligned} IP= & {} \frac{pI_p}{12}\Bigg [\beta N^2(d_1+4d_2+d_3)\nonumber \\&+2\beta N\Big [3Q\big [\frac{1}{{(d_3-d_2)}^3}\big \{2d_2d_3(d_3-d_1)\log \Big |\frac{d_3}{d_2}\Big |\nonumber \\&-d_1{d_2}^2-3d_2{d_3}^2+d_1{d_3}^2+4{d_2}^2d_3-{d_2}^3\big \}\nonumber \\&+\frac{1}{{(d_2-d_1)}^3}\Big \{{d_2}^2d_3-{d_1}^2d_3\nonumber \\&-2d_1d_2(d_3-d_1)\log \Big |\frac{d_2}{d_1}\Big |\nonumber \\&+{d_2}^3+3{d_1}^2d_2-4d_1{d_2}^2\big \}\big ]-M(d_1+4d_2+d_3)\Big ]\nonumber \\&+6Q^2\big [\frac{1}{{(d_3-d_2)}^2}\big \{d_3 \log \Big |\frac{d_3}{d_2}\Big |-(d_3-d_2)\big \}\nonumber \\&+\frac{1}{{(d_2-d_1)}^2}\big \{(d_2-d_1)-\nonumber \\&d_1 \log \Big |\frac{d_2}{d_1}\Big |\big \}\big ]-12QM+M^2(d_1+4d_2+d_3)\Bigg ]\nonumber \\ \end{aligned}$$

(47)

$$\begin{aligned} IE= & {} \frac{pI_e}{12}(d_1+4d_2+d_3)\big \{(1-\beta )M^2+\beta {(M-N)}^2\big \}\nonumber \\ \end{aligned}$$

(48)

$$\begin{aligned} \hbox {PRC}= & {} \frac{P_c}{12}\Big [6\big \{{(\rho _1-1)}^2+{(\rho _3-1)}^2\big \}\nonumber \\&+3\big \{{(\rho _3-\rho _2)}^2+{(\rho _2-\rho _1)}^2\big \}+ \nonumber \\&8\big \{(\rho _1-1)(\rho _2-\rho _1)-(\rho _3-1)(\rho _3-\rho _2)\big \}\Big ]\nonumber \\ \end{aligned}$$

(49)

Appendix B

For any crisp number Q and the triangular fuzzy number $\widetilde{D}=(d_1,d_2,d_3)$, by Eq. (15) the retailer’s cycle length became a fuzzy number $\widetilde{T}$ and hence $n\,\widetilde{T}$ is also a fuzzy number. The membership function of the fuzzy number $\widetilde{nT}$ is presented by Eq. (50).

$$\begin{aligned} \mu _{\widetilde{nT}}(x)= & {} \left\{ \begin{array}{rl} \frac{d_3x-Qn}{(d_3-d_2)x} &{}\quad \hbox {for}\,\frac{Qn}{d_3}\le x \le \frac{Qn}{d_2}\\ \frac{Qn-d_1x}{(d_2-d_1)x} &{}\quad \hbox {for}\, \frac{Qn}{d_2}\le x \le \frac{Qn}{d_1}\\ 0 &{}\quad \mathbf{}\hbox {otherwise.} \end{array}\right. \end{aligned}$$

(50)

According to the definition of possibility and necessity measure (Zadeh 1978; Dubois and Prade 1980), for any two fuzzy numbers $\widetilde{nT}$ and $\widetilde{H}$, $\hbox {Pos}(\widetilde{NT}\le \widetilde{H})$ and $\hbox {Nes}(\widetilde{NT}\le \widetilde{H})$ are, respectively, given by Eqs. (51) and (52).

(51)

(52)

where $\eta _1$ (cf. Fig.-1) and $\eta _2$ are, respectively, given by Eqs. (53) and (54).

$$\begin{aligned}&\eta _1\nonumber \\&\quad =\frac{(d_3H_2-d_2H_3) +\sqrt{{(d_3H_2-d_2H_3)}^2+4Qn(d_3-d_2)(H_3-H_2)}}{2(d_3-d_2)}\nonumber \\ \end{aligned}$$

(53)

$$\begin{aligned}&\eta _2\nonumber \\&\quad =\frac{(d_2H_1-d_1H_2) +\sqrt{{(d_2H_1-d_1H_2)}^2+4Qn(d_2-d_1)(H_2-H_1)}}{2(d_2-d_1)}\nonumber \\ \end{aligned}$$

(54)

Fig. 1

Pictorial representation of possibility measure

Using Eqs. (51) and (52), according to Liu (2002), the credibility measure of the fuzzy event $\widetilde{NT}\le \widetilde{H}$ is given by Eq. (55).

$$\begin{aligned} \hbox {Cr}(\widetilde{NT}\le \widetilde{H})=\frac{1}{2}[\hbox {Pos}(\widetilde{NT}\le \widetilde{H})+\hbox {Nes}(\widetilde{NT}\le \widetilde{H})] \end{aligned}$$

(55)