Fixed Charge Solid Transportation Problem Based on Carbon Emission with Budget Constraints in Uncertain Environment (UFSTPCEBC)

This section contains some concepts on uncertainty theory that have been used in the research work.

Definition 2.1 [13]:

Let $\mathcal{L}$ be a σ- algebra of collection of events Λ of a universal set Γ. A set function M is said to be an uncertain measure [UM]defined on the σ- algebra where M{Λ} indicates the belief degree with which we believe that the event will happen and satisfies the following four axioms:

1. For the universal set Γ. we have:

$\mathrm{M}\{\Gamma\}=1$               (1)

2. For any event Λ, we have:

$\mathrm{M}\{\Lambda\}+\mathrm{M}\left\{\Lambda^c\right\}=1$               (2)

3. For every countable sequence of events Λ₁­, Λ₂,... we have:

$\mathrm{M}\left\{\mathrm{U}_{j=1}^{\infty} \Lambda\right\} \leq \sum_{j=1}^{\infty} \mathrm{M}\left\{\Lambda_j\right\}$                  (3)

4. Let $\left(\Gamma_j, \mathcal{L}_j, \mathrm{M}_j\right)$ be uncertainty spaces for. The product UM measure is a UM holds:

$\mathrm{M}\left\{\prod_{j=1}^{\infty} \Lambda_j\right\}=\Lambda_{j=1}^{\infty} \mathrm{M}\left\{\Lambda_j\right\}$                 (4)

where, $\Lambda_j \in \mathcal{L}_j$ fo $j=1,2, \ldots, \infty r$.

Definition 2.2 [13]:

A function $\xi:(\Gamma, \mathcal{L}, \mathrm{M}) \rightarrow \mathrm{R}$ is known as an uncertain variable [UV] such that:

$\{\xi \in B\}=\{\gamma \in \Gamma / \xi(\gamma) \in B\}$                  (5)

is an event for any Borel set B of real numbers.

Definition 2.3 [13]:

The uncertainty distribution [UD] ρ(y) of a UV ξ for any real number y is defined by:

$\rho(y)=\mathrm{M}\{\xi \leq \mathrm{y}\}$                (6)

Definition 2.4:

An uncertainty distribution [UD] ρ(y) is said to be regular uncertainty distribution [RUD] if it is a strictly increasing and continuous function with respect to y at which 0<ρ(y)<1 and:

$\lim _{y \rightarrow-\infty} \varphi(y)=0$                  (7)

$\lim _{y \rightarrow \infty} \varphi(y)=1$                  (8)

Definition 2.5:

Let ρ(y) be the RUD of an uncertain variable ξ. Then ρ^-1(y) is called an inverse uncertainty distribution [IUD] of ξ and it exists on (0, 1).

Definition 2.6 [13]:

The UV $\xi_t(t=1,2, \ldots, T)$ are said to be independent if:

$\mathrm{M}\left\{\cap_{t=1}^T\left(\xi_t \in B_t\right)\right\}=\Lambda_{t=1}^T \mathrm{M}\left(\xi_t \in B_t\right)$                      (9)

where, B_t (t=1,2,..,T) are the Borel sets with real numbers.

Theorem 2.7 [13]:

The RUD of independent UV $\xi_t(t=1,2, \ldots, T)$ are $\rho_t(t=1,2, \ldots, T)$ respectively. If the function $h\left(y_1, y_2, \ldots, y_t\right)$ is strictly increasing and strictly decreasing with respect to $y_1, y_2, \ldots, y_s$ and $y_{s+1}, y_{s+2}, \ldots, y_t$ respectively then the uncertain variable $\xi=h\left(\xi_1, \xi_2, \ldots, \xi_s, \ldots, \xi_t\right)$ has an IUD:

$\rho^{-1}(\gamma)=h\left(\rho_1^{-1}(\gamma), \rho_2^{-1}(\gamma), \ldots, \rho_s^{-1}(\gamma), f\left(\rho_{s+1}^{-1}(1\right.\right. \left.-\gamma), \rho_{s+2}^{-1}(1-\gamma), \ldots, \rho_t^{-1}(1-\gamma)\right)$                     (10)

Definition 2.8 [13]:

The expected value of UV ξ is given by:

$E(\xi)=\int_0^{\infty} \mathrm{M}\{\xi \geq \mathrm{y}\} d y-\int_{-\infty}^0 \mathrm{M}\{\xi \leq \mathrm{y}\} d y$                    (11)

This is valid only if at least one of the integrals is finite.

Theorem 2.9 [38]:

Let $\rho_t(t=1,2, \ldots, T)$ be RUD of independent $\xi_t(t=1,2, \ldots, T)$ with respectively. If the function $h\left(y_1, y_2, \ldots, y_t\right)$ is strictly increasing and strictly decreasing w.r. to $y_1, y, \ldots, y_s$ and $y_{s+1}, y_{s+2}, \ldots, y_t$ respectively, then:

$E(\xi)=\int_0^1 h\left(\rho_1^{-1}(\gamma), \ldots, \rho_s^{-1}(\gamma), \rho_{s+1}^{-1}(1-\gamma), \ldots, \rho_t^{-1}(1-\gamma)\right) d \gamma$                   (12)

From this theorem, we have,

$E(\xi)=\int_0^1 \rho^{-1}(\gamma) d \gamma$                  (13)

here ξ is a UV with RUD ρ.

Definition 2.10 [13]:

The distribution function of a normal uncertain variable [NUV] is:

$\rho(x)=\left[1+\exp^{\left[\frac{\pi(\mu-x)}{\sigma{\surd{3}}}\right] \quad}\right]^{-1}, x \geq 0$                   (14)

and represented by $N(\mu, \sigma) ; \mu, \sigma \in R$ with $\sigma>0$. The IUD and the expected value of N(µ,σ) are given as follows:

$\rho^{-1}(\gamma)=\mu+\frac{\sigma \surd{3}}{\pi} \ln \frac{\gamma}{1-\gamma}$                         (15)

$E[\xi]=\mu$                (16)

The general model for UFSTPCEBC is presented below in (17).

Definition 4.1:

A feasible solution $Y^*=\left\{Y_{m n v}^*\right\} \in S$ is an efficient (no dominated) solution for UFSTPCEBC if there does not exist  another $Y=\left\{y_{m d w v g}\right\} \in S$ such that $Z_i(Y) \leq Z_i\left(Y^*\right), 1 \leq i \leq I$ and $Z_i(Y) \neq Z_i\left(Y^*\right)$ for some l, $1 \leq i \leq I$.

$\left\{\begin{array}{c}\operatorname{Max} \tilde{Z}_1=\sum_{m=1}^M \sum_{n=1}^N \sum_{v=1}^V\left(\widetilde{{S}_n}-\tilde{c}_{m n v}-\tilde{\alpha} \tilde{\gamma}_{m n v}-\tilde{P}_m\right) x_{m n v}-\tilde{f}_{m n v} Y\left(x_{m n v}\right) \\ \operatorname{Min} \tilde{Z}_2=\sum_{m=1}^M \sum_n^N \sum_{v=1}^V \tilde{d}_{m n v} x_{m n v} \\ \operatorname{Min} \tilde{Z}_3=\sum_{m=1}^M \sum_n^N \sum_{v=1}^V \tilde{t}_{m n v} Y\left(x_{m n v}\right) \\ \text { Subject to } \\ \sum_{n=1}^N \sum_{v=1}^V x_{m n v} \leq \tilde{a}_m, \quad m=1 \text { to } M \\ \sum_{m=1}^M \sum_{v=1}^V x_{m n v} \leq \tilde{b}_n, \quad \mathrm{n}=1 \text { to } \mathrm{N} \\ \sum_{m=1}^M \sum_{n=1}^N x_{m n v} \leq \tilde{e}_v, \quad v=1 \text { to } V \\ \sum_{m=1}^M \sum_{v=1}^V\left(\tilde{c}_{m n v}+\tilde{\alpha} \tilde{\gamma}_{m n v}+\tilde{P}_m\right) x_{m n v} \leq \tilde{B}_n, \mathrm{n}=1 \text { to } \mathrm{N} \\ \sum_{m=1}^M \sum_{n=1}^N \tilde{\gamma}_{m n v} x_{m n v} \leq \tilde{E}_v, v=1 \text { to } V, \\ x_{m n v} \geq 0, \forall m, n, v \\ Y\left(x_{m n v}\right)=\left\{\begin{array}{l}1 \quad \text { if } x_{m n v}>0 \\ 0 \quad \text { otherwise }\end{array}\right.\end{array}\right.$               (17)

The Charnes clan cooper [45] proposed the GPT to find a workable solution when there were multiple objectives. Numerous authors, including Chang [46] and others, have researched and developed the GPT further. Mohammed [47] proposed the Fuzzy GP method for solving MOTP, and Zangiabadi et al. [48, 49] used it later to solve MOTP using both linear and nonlinear membership functions. The goal of GP is to minimizing the distance between $Z=\left(Z_1, Z_2, Z_3, \ldots \ldots Z_i\right)$, and aspiration (or) target level $\bar{Z}=\left(\bar{Z}_1, \bar{Z}_2, \bar{Z}_3, \ldots \ldots \bar{Z}_i\right)$, which are set by the DM. To apply GP in this proposed model, we introduce the negative and positive deviational variables.

$\begin{gathered}P_i^{+}=\max \left(0, Z_i-\bar{Z}_i\right) \\ N_i^{-}=\max \left(0, \bar{Z}_i-Z_i\right)\end{gathered}$

We must reduce either $\mathrm{P}_{\mathrm{i}}^{+}, \mathrm{N}_{\mathrm{i}}^{-}$or $\mathrm{P}_{\mathrm{i}}^{+}+\mathrm{N}_{\mathrm{i}}^{-}$ in order to reduce the distance between $\mathrm{Z}_{\mathrm{i}}$ and $\overline{\mathrm{Z}}_{\mathrm{i}}$. If $\mathrm{Z}_{\mathrm{i}}$ must be maximized, then $\mathrm{h}_{\mathrm{i}}\left(\mathrm{P}_{\mathrm{i}}^{+}, \mathrm{N}_{\mathrm{i}}^{-}\right)=\mathrm{N}_{\mathrm{i}}^{-}$. When minimizing $\mathrm{Z}_{\mathrm{i}}$, however, $\mathrm{h}_{\mathrm{i}}\left(\mathrm{P}_{\mathrm{i}}^{+},  \mathrm{N}_{\mathrm{i}}^{-}\right)=\mathrm{P}_{\mathrm{i}}^{+}$. When we require $\mathrm{Z}_{\mathrm{i}}=\overline{\mathrm{Z}}_{\mathrm{i}}$,  $\mathrm{h}_{\mathrm{i}}\left(\mathrm{P}_{\mathrm{i}}^{+}, \mathrm{N}_{\mathrm{i}}^{-}\right)=\mathrm{P}_{\mathrm{i}}^{+}+\mathrm{N}_{\mathrm{i}}^{-}$.

In addition to the solution, the membership functions are described below in order to convey the decision-maker’s satisfaction.

$\mu_i\left(Z_i\right)=\left\{\begin{array}{cl}1 & \text { if } Z_i \leq L_{Z_i} \\ 1-\frac{Z_i-L_{Z_i}}{U_{Z_i}-L_{Z_i}} & \text { if } L_{Z_i}<Z_i<U_{Z_i} \\ 0 & \text { if } Z_i \geq U_{Z_i}\end{array}\right.$                      (20)

The DM's satisfaction is represented by $\mu_n\left(Z_n\right)$. As a result, it needs to be maximized i.e., max $\left(\mu_1\left(Z_1(x)\right), \mu_2\left(Z_2(x)\right), \mu_3\left(Z_3(x)\right), \ldots \ldots \ldots \ldots \mu_i\left(Z_i(x)\right)\right)$ The highest acceptable and desired levels of performance for the $\widetilde{Z}_i,(i=1,2,3)$ objective function are shown here as $\mathrm{U}_{\mathrm{Z}_{\mathrm{i}}}$ and $\mathrm{L}_{\mathrm{Z}_{\mathrm{i}}}$.

We reduce its negative deviation from 1 to bring them as close to 1 as possible in order to maximize any of the membership functions since the maximum value of the membership function cannot be more than one. The LPP can be written down as follows:

$\min \left(\max \left(h_i\left(P_i^{+}, N_i^{-}\right)\right)\right)$

i.e, Min Q.

Subject to:

$\begin{array}{} \frac{U_{Z_i}-Z_i}{U_{Z_i}-L_{Z_i}}+N_i^{-}-P_i^{+}=1, \\  Q \geq N_i^{-}, \text {where } i=1,2,3 . \\  P_i^{+} \cdot N_i^{-}=0 .\end{array}$                 (21)

$N_i^{-}, P_i^{+} \geq 0,0 \leq Q \leq 1$ and with given constraints. Here, we have considered UFSTPCEBC type of problem; GPT will be the most suitable methodology for getting the most acceptable compromise solution.

To illustrate the effectiveness and efficiency of the proposed UFSTPWCE under the budget constraints model, a numerical example is presented in this section, whose parameters are NUV. Two different customers (destinations), sources (origins) and conveyances each are considered in this model. i.e m=n=v=2.

Selling price $=\left(\tilde{S}_n:\left\{\tilde{S}_1=(50,4) ;  \tilde{S}_2=(60,4)\right\}\right.$; Purchasing cost $=\left(\tilde{P}_m\right):\left\{\tilde{P}_1=(5,1) ;  \tilde{P}_2=(6,2)\right\}$; Carbon tax $=(\tilde{\alpha}):\{\alpha=(2,0.5)\}$; Source $=\left(\tilde{a}_m\right):\left\{\tilde{a}_1=(230,5) ; \tilde{a}_2=(240,10)\right\} ;$ Demand $=\left(\tilde{b}_n\right)$ : $\left\{\tilde{b}_1=(90,5) ; \tilde{b}_2=(220,10)\right\} ;$ Conveyance $=\left(\tilde{\mathrm{e}}_{\mathrm{v}}\right):\left\{\tilde{\mathrm{e}}_1=(270,5)\right.$; $\tilde{\mathrm{e}}_2=(290,10\} ;$ Budget $=\left(\tilde{b}_n\right):\left\{\tilde{b}_1=(3900,100) ; \tilde{b}_2=(3500,100)\right\}$; Carbon capacity $=\left(\mathrm{E}_{\mathrm{v}}\right):\left\{\mathrm{E}_1=(360,40) ; \mathrm{E}_2=(420,40)\right\}$.

Table 2 contains the transportation cost per unit and fixed cost of the products.

Table 2. Transportation cost per unit and fixed charge

Table 3 contains deterioration cost per unit.

Table 3. Deterioration cost d_mnv

Table 4 contains the transportation time of the product.

Table 4. The transportation time

Average carbon emission is presented in Table 5.

Table 5. Average carbon emission per unit

The steps involved in using the aforementioned algorithm to solve the problem are as follows:

Step 1: For the above data, the deterministic problem of the considered UFSTPWCE model is obtained using the ECCM as of (19) and solved.

Step 2: Solving the considered objectives separately, we have $Z_1=14976.72, Z_2=897.98$ and $Z_3=36$.

By using these solutions, the value of each objective function is found as follows:

$\begin{gathered}Z_1\left(X_1\right)=14976.72, Z_1\left(X_2\right)=13890.77 \text { and } Z_1\left(X_3\right)=13902.7 \\ Z_2\left(X_1\right)=964.93, Z_2\left(X_2\right)=897.05 \text { and } Z_2\left(X_3\right)=924.9 \\ Z_3\left(X_1\right)=27, Z_3\left(X_2\right)=34 \text { and } Z_3\left(X_3\right)=36\end{gathered}$

The boundary values of $Z_1, Z_2$ and $Z_3$ are given as follows;

$\begin{gathered}U_{z_1}=14976.72, L_{z_1}=13890.77 \\ U_{z_2}=964.93, L_{z_2}=897.05 \\ U_{z_3}=36, U_{z_3}=27\end{gathered}$

Step 3: The GP ECCM for the proposed model is defined as follows using the GPT.

Min Q

Subject to

$\boldsymbol{E}\left(\sum_{m=1}^M \sum_{n=1}^N \sum_{v=1}^V \left(\left(\widetilde{S_n} \right)-\tilde{c}_{m n v}-\widetilde{\alpha} \tilde{\gamma}_{m n v}-\widetilde{P}_m\right) x_{m n v}-\tilde{f}_{m n v} Y\left(x_{m n v}\right)\right)+1085.95\left(N_1^{-}-P_1^{+}\right)=14976.72$

$E\left(\sum_{m=1}^M \sum_{n=1}^N \sum_{v=1}^V \tilde{d}_{m n v} x_{m n v}\right)=897.05+67.88 *\left(N_2^{-}-P_2^{+}\right)$

$\mathrm{E}\left(\sum_{\mathrm{m}=1}^{\mathrm{M}} \sum_{\mathrm{n}=1}^{\mathrm{N}} \sum_{\mathrm{v}=1}^{\mathrm{V}} \tilde{\mathrm{t}}_{\mathrm{mnv}} \mathrm{Y}\left(\mathrm{x}_{\mathrm{mnv}}\right)\right)=27-9\left(\mathrm{~N}_3^{-}-\mathrm{P}_3^{+}\right)$

where, $Y\left(x_{m n v}\right)=\left\{\begin{array}{lc}1 & \text { if } x_{m n v}>0 \\ 0 & \text { otherwise }\end{array}\right.$.

$\begin{array}{cc}\sum_{n={1}}^N \sum_{v=1}^V x_{m n v} \leq e_m+\frac{\sigma_m}{\pi} * \sqrt{3} \quad \log \frac{1-\delta_m}{\delta_m}, m=1 \text { to } M \\ \sum_{m=1}^M \sum_{v=1}^V x_{m n v} \geq e_n+\frac{\sigma_n}{\pi} * \sqrt{3} \quad \log \frac{\sigma_n}{1-\sigma_n}, \mathrm{n}=1 \text { to } \mathrm{N} \\ \sum_{m=1}^M \sum_{n=1}^N x_{m n v} \leq e_v+\frac{\sigma_v}{\pi} * \sqrt{3} \quad \log \frac{1-\tau_v}{\tau_v}, v=1 \text { to } V \\ \sum_{m=1}^M \sum_{v=1}^V\left(\theta_{m n v}^{-1}\left(\alpha_1\right)+\sigma^{-1}\left(\alpha_2\right) \eta_{m n v}^{-1}\left(\alpha_3\right)+\mu_m^{-1}\left(\left(\alpha_4\right)\right) x_{m n v} \leq e_n+\frac{\sigma_n}{\pi} * \sqrt{3} \quad \log \frac{1-\alpha_5}{\alpha_5}, n=1 \text { to } N\right. \\ \sum_{m=1}^M \sum_{n=1}^N \tilde{\gamma}_{m n v} x_{m n v} \leq e_v+\frac{\sigma_v}{\pi} * \sqrt{3} \quad \log \frac{1-\varepsilon_v}{\epsilon_v}, v=1 \text { to } V \\ x_{m n v} \geq 0, \forall m, n, v, Q \geq N_i^{-},\end{array}$

where, n=1,2,3.

$\begin{gathered}P_i^{+} \cdot N_i^{-}=0 . \\ N_i^{-}, P_i^{+} \geq 0,0 \leq Q \leq 1 .\end{gathered}$                  (23)

Step 4: With the use of the GRG technique ( LINGO-18.0 Suite Solver), we obtain the efficient value of Q = 1 and the associated transportation plan is $P_1^{+}=0$, $N_1^{-}=1$, $P_2^{+}=0$, $N_2^{-}=0.27$, $P_3^{+}=0$, $N_3^{-}=1$, $\operatorname{Max} Z_1=13876.99$, $\operatorname{Min} Z_2=942.45$, $\operatorname{Min} Z_3=36 x_{111}=17.75$, $x_{122}=95.01$, $x_{211}=27.7$, $x_{212}=50.5$, $x_{221}=110.3$, $x_{222}=26.79$, $y_{111}=y_{121}=y_{211}=y_{221}=1$ and the other decision variables have values of 0. We can see that DM's goals have been met to a satisfactory extent.