Multi-objective solid transportation-location problem with variable carbon emission in inventory management: a hybrid approach

Abstract

The most important strategic issue for several industries is where to find facilities so as to discover a transportation path for optimizing the objectives at the same time. This paper acquaints a streamlining model with incorporate the facility location problem, solid transportation problem, and inventory management under multi-objective environment. The aims of the stated formulation are multi-fold: (i) to seek the optimum locations for potential facilities in Euclidean plane; (ii) to find the amount of distributed commodities; and (iii) to reduce the overall transportation cost, transportation time, and inventory cost along with the carbon emission cost. Here, variable carbon emission cost is taken into consideration because of the variable locations of facilities and the amount of distributed products. After that, a new hybrid approach is introduced dependent on an alternating locate-allocate heuristic and the intuitionistic fuzzy programming to get the Pareto-optimal solution of the proposed formulation. In fact, the performances of our findings are discussed with two numerical examples. Sensitivity analysis is executed to check the resiliency of the parameters. Ultimately, managerial insights, conclusions and avenues of future studies are offered at the end of this study.

Appendix A

Iterative formula

• I. Herein, the iterations of

  $$\begin{aligned} Z_{1(x,y)}= \sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}\big (\alpha _{i}\phi _l(u_{i},v_{i};r_{k},s_{k})+\alpha ^\prime _{k}\phi _l(r_{k},s_{k};x_{j},y_{j})\big )w_{ikj}^{lB} \end{aligned}$$

  are derived iteratively in similar way of Das et al. (2020b) (see Appendix A) which are used at the ‘Solution methodology’ Sect. 3. Hence, the iterations of $(x_j,y_j)$ are as per the following:

  $$\begin{aligned}&x^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\alpha ^\prime _{k}\epsilon _lw_{ikj}^{lB}r_k}{\varphi (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\alpha ^\prime _{k}\epsilon _lw_{ikj}^{lB}}{\varphi (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \\&y^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\alpha ^\prime _{k}\epsilon _lw_{ikj}^{lB}s_k}{\varphi (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\alpha ^\prime _{k}\epsilon _lw_{ikj}^{lB}}{\varphi (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \end{aligned}$$

  where $\varphi (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})=[{(r_{k}-x_{j}^{k^{\prime }})^{2}+(s_{k}-y_{j}^{k^{\prime }})^{2}}+\delta _{kj}]^{1/2}$. The initial iteration of $(x_j,y_j)$ is the weighted mean coordinate:

  $$\begin{aligned}&x_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }_lw_{ikj}^{lB}r_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }w_{ikj}^{lB}}~~~~( j=1,2,\ldots , p), \\&y_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }w_{ikj}^{lB}s_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }w_{ikj}^{lB}}~~~~( j=1,2,\ldots , p). \end{aligned}$$

• II. Similarly, the iterations for

  $$\begin{aligned} Z_{2(x,y)} = \sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}\big (\beta _{i}\psi _l(u_{i},v_{i};r_{k},s_{k})+\beta ^\prime _{k}\psi _l(r_{k},s_{k};x_{j},y_{j})\big )\zeta \left( w_{ikj}^{lB}\right) \end{aligned}$$

  are

  $$\begin{aligned}&x_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) r_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) }~~~~( j=1,2,\ldots , p), \\&y_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) s_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) }~~~~( j=1,2,\ldots , p),\\&x^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) r_k}{\tau (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) }{\tau (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \\&y^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) s_k}{\tau (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\beta ^\prime _{k}\epsilon ^{\prime }_l\zeta \left( w_{ikj}^{lB}\right) }{\tau (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \end{aligned}$$

  where $\tau (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})=[{(r_{k}-x_{j}^{k^{\prime }})^{2}+(s_{k}-y_{j}^{k^{\prime }})^{2}}+t_{kj}+\delta _{kj}]^{1/2}$.

• III. Furthermore, the iterative formula for $Z_3$ are:

  The iterations for

  $$\begin{aligned} Z_{3(x,y)}= & {} \sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}\big (d_{2k}A_{ik}+d_{1j}G_{kj}\big )w_{ikj}^{lB}+\sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L} H_kw_{ikj}^{lB}\\&+\sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}D_kw_{ikj}^{lB}+\sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L} B_iw_{ikj}^{lB}+\sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}g_kw_{ikj}^{lB}\\&+\sum _{k \in K} f_k+\gamma \sum _{i\in I}\sum _{j \in J}\sum _{k \in K}\sum _{l \in L}\big (\rho _l(u_{i},v_{i};r_{k},s_{k})+\rho _l(r_{k},s_{k};x_{j},y_{j})\big )w_{ikj}^{lB} \end{aligned}$$

  are stated subsequently:

  $$\begin{aligned}&x_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\epsilon ^{\prime \prime }_lw_{ikj}^{lB}r_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\epsilon ^{\prime \prime }_lw_{ikj}^{lB}}~~~~( j=1,2,\ldots , p), \\&y_j^{0}=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\epsilon ^{\prime \prime }_lw_{ikj}^{lB}s_k}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\epsilon ^{\prime \prime }_lw_{ikj}^{lB}}~~~~( j=1,2,\ldots , p),\\&x^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\epsilon ^{\prime \prime }_lw_{ikj}^{lB}r_k}{\varrho (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\epsilon ^{\prime \prime }_lw_{ikj}^{lB}}{\varrho (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \\&y^{k^{\prime }+1}_j=\frac{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\epsilon ^{\prime \prime }_lw_{ikj}^{lB}s_k}{\varrho (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}{\sum _{i\in I}\sum _{k \in K}\sum _{l \in L}\frac{\epsilon ^{\prime \prime }_lw_{ikj}^{lB}}{\varrho (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})}}~~~~( j=1,2,\ldots ,p;~k^{\prime }\in {\mathbb {N}}), \end{aligned}$$

  where $\varrho (r_{k},s_{k};x_{j}^{k^{\prime }},y_{j}^{k^{\prime }})=[{(r_{k}-x_{j}^{k^{\prime }})^{2}+(s_{k}-y_{j}^{k^{\prime }})^{2}}+\delta _{kj}]^{1/2}$.