Optimal payment time and replenishment decisions for retailer’s inventory system under trade credit and carbon emission constraints

Abstract

This study presents a multi-item inventory and pricing model by considering marketing, service activities, trade credit, carbon emissions, and the restrictions of production cost and storage space. In the proposed model, shortages are allowed and demand rate is a power function of service and marketing costs, and selling price. The main objective of this study is to optimize retailer’s payments time, service and marketing expenditure, and replenishment decisions in order to maximize retailer’s total profit and minimize carbon emissions, simultaneously. Model is developed in a fuzzy environment under carbon tax regulation when the length of credit period provided by supplier is less than or equal to the length of time in which no shortage happens. To solve the proposed model, we first transform the original problem into a multi-objective Signomial Geometric Programming (SGP) problem using fuzzy and hybrid parameters, which minimizes both the mean value and the total dispersion value of the objective function. Then a global optimization problem method has been used to solve the SGP problem. Efficiency of this algorithm is tested and compared with multi-objective genetic algorithm, multi-objective genetic algorithm with varying population, and hybrid heuristic algorithm. At the end, several numerical examples and sensitivity analysis are performed to demonstrate the application of the proposed model and solution procedure to obtain managerial insights.

Appendices

Appendix 1: Hybrid numbers

Let $\underline{{\tilde{R}}} = \left( {\tilde{R},\underline{r} } \right)$ be a hybrid number, which the couple $\left( {\tilde{R},\underline{r} } \right)$ displays the addition to a fuzzy number with a random variable without changing characteristic of each one and without reducing the amount of variable information, where $\tilde{R}$ is a fuzzy number and $\underline{r} = \left( {\mu ,\sigma^{2} } \right)$ is the random variable with density function $f_{{\underline{r} }} \left( r \right)$ whose mean and variance are $\mu$ and $\sigma^{2}$ respectively. In this paper, we denote the couple $\left( {\tilde{R},\underline{r} } \right)$ by the symbol $\tilde{R}( + )^{\prime } \underline{r}$.

Appendix 2: Transforming SGP problems into a series of standard GP problems

As mentioned earlier, a global optimization method is applied for solving SGP problem proposed in Steps 1, 2, and 5. We first present a SGP problem, and then explain this approach in detail for transforming the SGP problem to a series of standard GP problems according to the type of the proposed problems.

2.1 A SGP program is equal to an optimization problem as follows

$$\begin{aligned} & \hbox{min} \,\,\xi_{0} (y) = \sum\limits_{k = 1}^{{n_{0} }} {\theta_{0k} c_{0k} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{oik} }} } \quad \,c_{0k} > 0, \, \theta_{0k} = \pm 1 \\ & {\text{s}} . {\text{t}} .\quad \xi_{j} (y) = \sum\limits_{k = 1}^{{n_{j} }} {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} \le 1} \quad c_{jk} > 0, \, \theta_{jk} = \pm 1, \, \alpha_{jik} \in R,j = 1,2, \ldots ,t \\ & \quad \quad y_{i} > 0,i = 1,2, \ldots ,m, \\ \end{aligned}$$

(43)

where $n_{j} \left( {j = 0,1,2, \ldots ,t} \right)$ shows the number of elements of the objective function and restrictions, and $\xi_{j} (y)$ is Signomial function.

2.2 Global optimization approach

This method defines all functions $\xi_{j}$ as:

$$\xi_{j} (y) = \xi_{j}^{ + } (y) - \xi_{j}^{ - } (y)\quad j = 0,1,2, \ldots ,t$$

(44)

where $\xi_{j}^{ - } (y)$ and $\xi_{j}^{ + } (y)$ are formulated as:

$$\xi_{j}^{ + } (y) = \sum\limits_{k = 1}^{{n_{j} }} {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} } \quad \theta_{jk} = + 1,\,j = 0,1,2, \ldots ,t$$

(45)

$$\xi_{j}^{ - } (y) = \sum\limits_{k = 1}^{n_{j} } {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} } \quad \theta_{jk} = - 1, \, j = 0,1,2, \ldots ,t$$

(46)

Next it defines a large number, $L > 0$, so that $\xi_{0}^{ + } (y) - \xi_{0}^{ - } (y) + L > 0$ and rewrites the model (43) as the following problem:

$$\hbox{min} \,\,\,\xi_{0} (y) = \xi_{0}^{ + } (y) - \xi_{0}^{ - } (y) + L$$

(47)

$${\text{s}} . {\text{t}} .\quad \xi_{j}^{ + } (y) - \xi_{j}^{ - } (y) \le 1$$

(48)

$$y_{i} > 0,\quad j = 1,2, \ldots ,t,\,\,i = 1,2, \ldots ,m$$

(49)

The model (43) converts to the following optimization problem, by introducing an extra variable $y_{0}$ in order to express restrictions and objective function as quotient and linear form, respectively.

$$\hbox{min} \,\,\,y_{0}$$

(50)

$${\text{s}} . {\text{t}} .\quad \frac{{\xi_{0}^{ + } (y) + L}}{{\xi_{0}^{ - } (y) + y_{0} }} \le 1$$

(51)

$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{j}^{ - } (y) + 1}} \le 1\quad j \in j_{1}$$

(52)

$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{j}^{ - } (y) + 1}} \le 1\,\quad \,j \in j_{2}$$

(53)

$$y_{i} > 0,\quad j = 1,2, \ldots ,t,i = 1,2, \ldots ,m$$

(54)

where $j_{1} = \{ \left. j \right|\xi_{j}^{ - } (y) + 1\,$ are monomials $\}$ and $j_{2} = \left\{ {\left. j \right|j \notin j_{1} } \right\}$. In the above model the objective function (50) is a posynomial function, restriction (54) is a posynomial inequality that both equations are allowable in standard GP problem, but restrictions (51)–(53) are not permitted in a standard GP problem. So this method used from arithmetic–geometric mean approximation to approximate every denominator of restrictions (51)–(53) with monomial functions as follows:

$$f(y) \ge \hat{f}(y) = \prod\limits_{u} {\left( {\frac{{v_{u} (y)}}{{\omega_{u} (x)}}} \right)}^{{\omega_{u} (x)}}$$

(55)

where the parameters $\omega_{u} (x)$ can be computed as:

$$\omega_{u} (x) = \frac{{v_{u} (x)}}{f(x)}\quad \forall u$$

(56)

where $f(y) = \sum\nolimits_{u} {v_{u} (y)}$ is a posynomial function, $v_{u} (y)$ are monomial terms, and $x > 0$ is a fixed point. Using the proposed monomial approximation approach to every denominator of restrictions (51)–(53), finally we have:

$$\hbox{min} \,\,\,y_{0}$$

(57)

$${\text{s}} . {\text{t}} .\quad \frac{{\xi_{0}^{ + } (y) + L}}{{\xi_{0}^{ - } (y,y_{0} )}} \le 1$$

(58)

$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{1j}^{ - } (y)}} \le 1\quad j \in j_{1}$$

(59)

$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{2j}^{ - } (y)}} \le 1\quad \,j \in j_{2}$$

(60)

$$y_{i} > 0,\quad j = 1,2, \ldots ,t,\;\;i = 1,2, \ldots ,m$$

(61)

where $\xi_{0}^{ - } (y,y_{0} )$, $\xi_{1j}^{ - } (y)$, and $\xi_{2j}^{ - } (y)$ are the corresponding monomial functions approximated using Eq. (55). Now, the problem (57)–(61) is a standard geometric programming model than can be optimized efficiently using GGPLAB solver in MATLAB (Mutapcic et al. 2006). So, the proposed algorithm can be summarized as an iterative algorithm as follows:

Step 0 Select an initial solution for decision variables $y_{0}$ and $y$,$y_{0}^{(0)}$ and $y^{(0)}$ respectively. Consider a solution accuracy $\varepsilon > 0$ and put iteration counter $r = 0$.

Step 1 In iteration $r$, calculate the monomial components in the denominator posynomials of Eqs. (51)–(53) by the determined $y_{0}^{(r - 1)}$ and $y^{(r - 1)}$. Calculate their corresponding parameters $\omega_{u} (y_{0}^{(r - 1)} ,y^{(r - 1)} )$ using Eq. (56).

Step 2 Do the condensation on the denominator posynomials of Eqs. (51)–(53) using Eq. (55) by parameters $\omega_{u} (y_{0}^{(r - 1)} ,y^{(r - 1)} )$.

Step 3 Solve the standard GP (57)–(61) to obtain $\left( {y_{0}^{(r)} ,y^{(r)} } \right)$.

Step 4 If $\left\| {y^{(r)} - y^{(r - 1)} } \right\| \le \varepsilon$, then stop. Else $r = r + 1$ and return to Step 1.