Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint

Abstract

Modeling of real-world problems requires data as input parameter which include information represented in the state of indeterminacy. To deal with such indeterminacy, use of uncertainty theory (Liu in Uncertainty theory, Springer, Berlin, 2007) has become an important tool for modeling real-life decision-making problems. This study presents a profit maximization and time minimization scheme which considers the existence of possible indeterminacy by designing an uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint (UMMFSTPwB) at each destination. Here, items are purchased at different source points with different prices and are accordingly transported to different destinations using different types of vehicles. The items are sold to the customers at different selling prices. In the proposed model, unit transportation costs, fixed charges, transportation times, supplies at origins, demands at destinations, conveyance capacities and budget at destinations are assumed to be uncertain variables. To model the proposed UMMFSTPwB, we have developed three different models: (1) expected value model, (2) chance-constrained model and (3) dependent chance-constrained model using uncertain programming techniques. These models are formulated under the framework of uncertainty theory. Subsequently, the equivalent deterministic transformations of these models are formulated and are solved using three different methods: (1) linear weighted method, (2) global criterion method and (3) fuzzy programming method. Finally, numerical examples are presented to illustrate the models.

Appendices

Appendix A

In this section, some theorems related to uncertain programming are revisited.

Theorem A.1

(Liu 2010) Let $\zeta _{1}$, $\zeta _{2}{,}~ .~ .~ .$ , $\zeta _{n}$ are independent uncertain variables with uncertainty distributions ${\Phi }_{\mathrm {1}}$, ${\Phi }_{\mathrm {2}}$,...,$~{\Phi }_{n}$, respectively, and $f_{1}\left( x \right)$, $f_{2}\left( x \right) {,}~ .~ .~ .{,}~ f_{n}\left( x \right)$, $\overline{f} (x)$ are real valued functions. Then,

$$\begin{aligned} \mathcal {M}\left\{ \sum \nolimits _{i=1}^n {{\zeta _{i}f}_{i}\left( x \right) } \le \overline{f} (x)~\right\} \ge {\alpha } \end{aligned}$$

holds if and only if,

$$\begin{aligned}&\sum \nolimits _{i=1}^n {{\Phi }_{i}^{-1}\left( {\alpha } \right) f_{i}^{+}\left( x \right) } -\sum \nolimits _{i=1}^n {{\Phi }_{i}^{-1}\left( 1-\alpha \right) f_{i}^{-}\left( x \right) } \le \overline{f} \left( x \right) , \end{aligned}$$

(A1)

where

$$\begin{aligned} f_{i}^{+}\left( x \right) =\left\{ {\begin{array}{rl} f_{i}\left( x \right) ~ ;&{}f_{i}\left( x \right) >0 \\ 0~ ;&{}f_{i}\left( x \right) \le 0 \\ \end{array}} \right. \end{aligned}$$

(A2)

and

$$\begin{aligned} f_{i}^{-}\left( x \right) =\left\{ {\begin{array}{rl} 0~ ;&{}f_{i}\left( x \right) \ge 0 \\ -f_{i}\left( x \right) ~ ;&{}f_{i}\left( x \right) <0 \\ \end{array}} \right. \hbox { for } i=1,2,\ldots ,n. \end{aligned}$$

(A3)

If $f_{1}\left( x \right) {,}~ f_{2}\left( x \right) {,}~ .~ .~ .{,}~ f_{n}\left( x \right)$ are all nonnegative, then $\mathrm {(A1)}$ becomes $\sum \nolimits _{i=1}^n {{\Phi }_{i}^{-1}\left( {\alpha } \right) f_{i}\left( x \right) } \le \overline{f} \left( x \right)$, and if $f_{1}\left( x \right)$, $f_{2}\left( x \right) {,}~ .~ .~ .{,}~ f_{n}\left( x \right)$ are all nonpositive, then $\mathrm {(A1)}$ becomes $\sum \nolimits _{i=1}^n {{\Phi }_{i}^{-1}\left( 1-\alpha \right) f_{i}\left( x \right) } \le \overline{f} \left( x \right) .$

Theorem A.2

(Liu 2010) Let $x_{1}{,}~ x_{2}{,}~\ldots , x_{n}$ are nonnegative decision variables and $\zeta _{1}$, $\zeta _{2}$, .  .  ., $\zeta _{n}$ are independent zigzag uncertain variables which are represented as $\mathcal {Z}\left( g_{1},h_{1},l_{1} \right)$, $\mathcal {Z}\left( g_{2},h_{2},l_{2} \right) {,}~ .~ .~ .{,}~ \mathcal {Z}\left( g_{n},h_{n},l_{n} \right)$, respectively. Then,

(A4)

Theorem A.3

(Liu 2010) Let $x_{1}{,}\, x_{2}{,}~\ldots , x_{n}$ are nonnegative decision variables and $\zeta _{1}$, $\zeta _{2}, \ldots , \zeta _{n}$ are independent normal uncertain variables which are denoted as $\mathcal {N}\left( \rho _{1},\sigma _{1} \right)$, $\mathcal {N}\left( \rho _{2},\sigma _{2} \right) {,}~\ldots , \mathcal {N}\left( \rho _{n},\sigma _{n} \right)$, respectively. Then,

$$\begin{aligned}&\mathcal {M}\left\{ \sum \nolimits _{i=1}^n {\zeta _{i}x_{i}} \le c~\right\} \nonumber \\&\quad =\left( 1+\exp \left( \frac{\pi \left( \sum \nolimits _{i=1}^n {\rho _{i}x_{i}-c} \right) }{\sqrt{3} \sum \nolimits _{i=1}^n {\sigma _{i}x_{i}}} \right) \right) ^{-1}. \end{aligned}$$

(A5)

Theorem A.4

(Liu 2010) Let $\zeta$ be an uncertain variable with continuous uncertainty distribution ${\Phi }$. Then, for any real number x, we have

$$\begin{aligned} \mathcal {M}\left\{ \zeta \le x \right\} ={\Phi }(x),\quad \mathcal {M}\left\{ \zeta \ge x \right\} =1-{\Phi }(x). \end{aligned}$$

(A6)

Appendix B

In this section, we state the relevant theorems to formulate the crisp equivalents of chance-constrained model (CCM) and dependent chance-constrained model (DCCM) of UMMFSTPwB.

Crisp equivalents of chance-constrained model (CCM)

Lemma B.1

If a and r are positive real numbers, $\xi$ is an independent uncertain variable with uncertainty distribution ${\Phi }$ and $\alpha$ is the chance level. Then, $\mathcal {M}\left\{ a-\xi \ge r \right\} \ge \alpha$ holds if and only if $a-{\Phi }^{-1}\left( \alpha \right) \ge r$.

Proof

$$\begin{aligned} \mathcal {M}\left\{ a-\xi \ge r \right\}\ge & {} \alpha \quad \Leftrightarrow \quad \mathcal {M}\left\{ \xi \le a-r \right\} \ge \alpha \quad \Leftrightarrow \\ a-r\ge & {} {\Phi }^{-1}\left( \alpha \right) \quad \Leftrightarrow \quad a-{\Phi }^{-1}\left( \alpha \right) \ge r. \end{aligned}$$

Theorem B.1

Let $\xi _{c_{ijk}^{p}}$, $\xi _{f_{ijk}^{p}}$, $\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}}$ and $\xi _{B_{j}}$ are the independent uncertain variables, respectively, associated with uncertainty distributions ${\Phi }_{\xi _{c_{ijk}^{p}}}{,}~ {\Phi }_{\xi _{f_{ijk}^{p}}}{,}~ {\Phi }_{\xi _{t_{ijk}^{p}}}{,}~ {\Phi }_{\xi _{a_{i}^{p}}}{,} {\Phi }_{\xi _{b_{j}^{p}}}{,}~ {\Phi }_{\xi _{e_{k}}}$ and ${\Phi }_{\xi _{B_{j}}}$ then the crisp equivalent of chance-constrained model ($\mathrm {CCM})$ in model (10) can be equivalently formulated as model (B1).

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \bar{Z}_{1}=\left[ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left\{ \left( s_{j}^{p}-v_{i}^{p}\right. \right. \right. \\ \quad \quad \quad \qquad \left. \left. \left. -{\Phi }_{\xi _{c_{ijk}^{p}}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {1}} \right) \right) x_{ijk}^{p}-{\Phi }_{\xi _{f_{ijk}^{p}}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {1}} \right) y_{ijk}^{p} \right\} \right] \\ \mathrm{Min}~ \bar{Z}_{2}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {{\Phi }_{\xi _{t_{ijk}^{p}}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {2}} \right) y_{ijk}^{p}} ~ \\ \mathrm{subject~ to} \\ \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {x_{ijk}^{p}-{\Phi }_{\xi _{a_{i}^{p}}}^{\mathrm {-1}}\left( {\mathrm {1-\upbeta }}_{{i}}^{p} \right) } \le 0,\\ \quad \quad \quad \qquad i=1,2,\ldots ,m,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\Phi _{\xi _{b_{j}^{p}}}^{-1}\left( {\upgamma }_{{j}}^{p} \right) {\ge 0}} ,\\ \quad \quad \quad \qquad j=1,2,\ldots ,n,~ p=1,2,\ldots ,r\\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m {\sum \nolimits _{j=1}^n x_{ijk}^{p} -{\Phi }_{\xi _{e_{k}}}^{\mathrm {-1}}\left( \mathrm {1-}\delta _{k} \right) } \le 0,\\ \quad \quad \quad \qquad k=1,2,\ldots ,K \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K \left\{ {\left( v_{i}^{p}+{\Phi }_{\xi _{c_{ijk}^{p}}}^{\mathrm {-1}}\left( \rho _{j} \right) \right) }x_{ijk}^{p}\right. \\ \quad \quad \quad \qquad \left. +\,{\Phi }_{\xi _{f_{ijk}^{p}}}^{\mathrm {-1}}\left( \rho _{j} \right) y_{ijk}^{p} \right\} ~- \Phi _{B_{j}}^{-1}\left( \mathrm {1-}\rho _{j} \right) \le 0, j=1,2,\ldots ,n \\ x_{ijk}^{p}\ge 0,~ y_{ijk}^{p}=\left\{ {\begin{array}{l} 1;~~{x}_{ijk}^{p}>0~ \\ 0;~~\mathrm{otherwise}~ \\ \end{array}} \right. ~ \forall ~ p,i,j,k.~ \\ \end{array}} \right. \end{aligned}$$

(B1)

Proof

Considering the CCM of UMMFSTPwB presented in model (10), the corresponding constraints can be written as follow.

• (i) The constraint $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K\right. \left. \left[ \left( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \right) x_{ijk}^{p}-\left( \xi _{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \ge \bar{Z}_{1} \right\} \ge {\alpha }_{\mathrm {1}}$ can be rewritten as $\mathcal {M}\left\{ Z_{1}\ge \bar{Z}_{1} \right\} \ge {\alpha }_{\mathrm {1}}$, since $\xi _{c_{ijk}^{p}}$ and $\xi _{f_{ijk}^{p}}$ are the independent uncertain variables with regular uncertainty distributions ${\Phi }_{\xi _{c_{ijk}^{p}}}$ and ${\Phi }_{\xi _{f_{ijk}^{p}}}$, respectively. Then according to Theorem 2.1 provided in Section 2 and the Lemma B.1, $\mathcal {M}\left\{ Z_{1}\ge \bar{Z}_{1} \right\} \ge {\alpha }_{\mathrm {1}}$ can be reformulated as

  $$\begin{aligned}&\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}\right. \right. \\&\quad \left. \left. -{\Phi }_{c_{ijk}^{p}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {1}} \right) \right) x_{ijk}^{p}-{\Phi }_{f_{ijk}^{p}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {1}} \right) y_{ijk}^{p} \right] \ge \bar{Z}_{1}. \end{aligned}$$

  In similar way, constraint $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m\right. \left. \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \xi _{t_{ijk}^{p}}y_{ijk}^{p} \right] \le \bar{Z}_{2} \right\} \ge {\alpha }_{\mathrm {2}}$ can be restructured as

  $$\begin{aligned} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {{\Phi }_{t_{ijk}^{p}}^{\mathrm {-1}}\left( {\alpha }_{\mathrm {2}} \right) y_{ijk}^{p}} \le \bar{Z}_{2}. \end{aligned}$$

• (ii) Constraint $\mathcal {M}\left\{ \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\xi _{a_{i}^{p}}} \le 0 \right\} \ge \beta _{i}^{p}$$\Leftrightarrow ~ \mathcal {M}\left\{ \xi _{a_{i}^{p}}\ge \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} \right\} \ge \beta _{i}^{p}$. Since, ${\Phi }_{\xi _{a_{i}^{p}}}$ is the uncertainty distribution of $\xi _{a_{i}^{p}}$ then from Theorem A.4 (cf. “Appendix A”),

  $$\begin{aligned}&\mathcal {M}\left\{ \xi _{a_{i}^{p}}\ge \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} \right\} \\&\quad \ge \beta _{i}^{p}\Leftrightarrow \mathrm {1-}{\Phi }_{\xi _{a_{i}^{p}}}\left( \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} \right) \ge \beta _{i}^{p}\\&\quad \Leftrightarrow \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} -{\Phi }_{a_{i}^{p}}^{\mathrm {-1}}\left( {{1-\upbeta }}_{{i}}^{\mathrm {p}} \right) \le 0 . \end{aligned}$$

• (iii) Constraint $\mathcal {M}\left\{ \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\xi _{b_{j}^{p}}} \ge 0 \right\} \ge \gamma _{j}^{p}$$\Leftrightarrow ~ \mathcal {M}\left\{ \xi _{b_{j}^{p}}\le \sum \nolimits _{i=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} \right\} \ge \gamma _{j}^{p}$.

  Since, ${\Phi }_{\xi _{b_{j}^{p}}}$ is the uncertainty distribution of $\xi _{b_{j}^{p}}$ then from Theorem A.4,

  $$\begin{aligned}&\mathcal {M}\left\{ \xi _{b_{j}^{p}}\le \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K x_{ijk}^{p} \right\} \nonumber \\&\quad \ge \gamma _{j}^{p}\Leftrightarrow {\Phi }_{\xi _{b_{j}^{p}}}\left( \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} \right) \\&\quad \ge \gamma _{j}^{p}\Leftrightarrow \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K x_{ijk}^{p} -{\Phi }_{\xi _{b_{j}^{p}}}^{\mathrm {-1}}\left( \gamma _{j}^{p} \right) \ge 0. \end{aligned}$$

  Similarly, constraint $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n x_{ijk}^{p} \right. \left. -\xi _{e_{k}} \le 0 \right\} \ge \delta _{k}$ can be equivalently transformed into $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m {\sum \nolimits _{j=1}^n x_{ijk}^{p} -{\Phi }_{e_{k}}^{\mathrm {-1}}\left( \mathrm {1-}\delta _{k} \right) } \le 0$.

• (iv) From Theorem A.1 and Theorem A.4, the crisp transformation of constraint

  $$\begin{aligned}&\mathcal {M}\left\{ \left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {\left( v_{i}^{p}+\xi _{c_{ijk}^{p}} \right) x}_{ijk}^{p}\right. \right. \\&\quad \left. \left. +\,\xi _{f_{ijk}^{p}} y_{ijk}^{p} \right\} -\xi _{B_{j}^{p}}\le 0 \right\} \ge \rho _{j} \qquad \hbox { is equivalently becomes}, \\&\quad \left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K \left( v_{i}^{p} +{\Phi }_{c_{ijk}^{p}}^{\mathrm {-1}}\left( \rho _{j} \right) \right) x_{ijk}^{p} +{\Phi }_{f_{ijk}^{p}}^{\mathrm {-1}}\left( \rho _{j} \right) y_{ijk}^{p} \right\} \\&\quad -\,\Phi _{B_{j}}^{-1}\left( {1-\rho }_{j} \right) {\le 0}. \end{aligned}$$

Therefore, considering (i), (ii), (iii) and (iv) shown above, the crisp equivalent of model (10) follows directly, the model (B1).

Corollary B.1

If $\xi _{c_{ijk}^{p}}$, $\xi _{f_{ijk}^{p}}$, $\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}^{p}}$ and $\xi _{B_{j}^{p}}$ are the independent zigzag uncertain variables of the form $\mathcal {Z}\left( g,h,l \right)$ with $g<h<l.$ Then, according to Theorem B.1 and the inverse uncertainty distribution of zigzag uncertain variables, we can conclude the following.

(i) For all chance levels $<0.5$, model (B1) becomes

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \bar{Z}_{1}=~ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-\left( \left( 1-2{\alpha }_{\mathrm {1}} \right) g_{c_{ijk}^{p}}\right. \right. \right. \\ \quad \quad \quad \qquad \left. \left. \left. +2{\alpha }_{\mathrm {1}}h_{c_{ijk}^{p}} \right) \right) x_{ijk}^{p}-~ \left( \left( 1-2{\alpha }_{\mathrm {1}} \right) g_{f_{ijk}^{p}}+2{\alpha }_{\mathrm {1}}h_{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \\ \mathrm{Min}~ \bar{Z}_{2}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( \left( 1-2{\alpha }_{\mathrm {2}} \right) g_{t_{ijk}^{p}}+2{\alpha }_{\mathrm {2}}h_{t_{ijk}^{p}} \right) y_{ijk}^{p} \right] ~ \\ \mathrm{subject~ to} \\ \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( 2\beta _{i}^{p}h_{a_{i}^{p}}+\left( 1-2\beta _{i}^{p} \right) l_{a_{i}^{p}} \right) } \le 0,~\\ \quad \quad \quad \qquad i=1,2,\ldots ,m,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( {\left( 1-2\gamma _{j}^{p} \right) }g_{b_{j}^{p}}+2\gamma _{j}^{p}h_{b_{j}^{p}} \right) \ge 0} ,\\ \quad \quad \quad \qquad j=1,2,\ldots ,n,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m {\sum \nolimits _{j=1}^n x_{ijk}^{p} -\left( 2\delta _{k}h_{e_{k}}+\left( 1-2\delta _{k} \right) l_{e_{k}} \right) } \le 0,\\ \quad \quad \quad \qquad k=1,2,\ldots ,K~ \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {\left[ v_{i}^{p}+\left( \left( 1-2\rho _{j} \right) g_{c_{ijk}^{p}}+2\rho _{j}^{p}h_{c_{ijk}^{p}} \right) \right] }x_{ijk}^{p}\\ \quad \quad \quad \qquad +\left[ \left( \left( 1-2\rho _{j} \right) g_{f_{ijk}^{p}}+2\rho _{j}h_{f_{ijk}^{p}} \right) \right] y_{ijk}^{p}\\ \quad \quad \quad \qquad -\left( {2\rho _{j}h}_{B_{j}}+\left( 1-2\rho _{j} \right) l_{B_{j}} \right) {\le 0,~ j=1,2,\ldots ,n} \\ x_{ijk}^{p}\ge 0,~ y_{ijk}^{p}=\left\{ {\begin{array}{l} 1;~~x_{ijk}^{p}>0~ \\ 0;~~\mathrm{otherwise}~ \\ \end{array}} \right. ~ \forall ~ p,i,j,k.~ \\ \end{array}} \right. \end{aligned}$$

(B2)

(ii) For all chance levels $\ge ~ 0.5$, model (B1) can be described as given in (B3).

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \bar{Z}_{1}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-\left( \left( 2-2{\alpha }_{\mathrm {1}} \right) h_{c_{ijk}^{p}}\right. \right. \right. \\ \quad \quad \quad \qquad \left. \left. \left. +\left( 2{\alpha }_{\mathrm {1}}-1 \right) l_{c_{ijk}^{p}} \right) \right) x_{ijk}^{p}-\left( \left( 2-2{\alpha }_{\mathrm {1}} \right) h_{f_{ijk}^{p}}\right. \right. \\ \quad \quad \quad \qquad \left. \left. +{\left( 2{\alpha }_{\mathrm {1}}-1 \right) l}_{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \\ \mathrm{Min}~ \bar{Z}_{2}=~ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( \left( 2-2{\alpha }_{\mathrm {2}} \right) h_{t_{ijk}^{p}}\right. \right. \\ \quad \quad \quad \qquad \left. \left. +\left( 2{\alpha }_{\mathrm {2}}-1 \right) l_{t_{ijk}^{p}} \right) y_{ijk}^{p} \right] ~ \\ \mathrm{subject~ to} \\ \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( \left( 2\beta _{i}^{p}-1 \right) g_{a_{i}^{p}}+\left( 2-2\beta _{i}^{p} \right) h_{a_{i}^{p}} \right) } \le 0,\\ \quad \quad \quad \qquad i=1,2,\ldots ,m,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( \left( 2-2\gamma _{j}^{p} \right) h_{b_{j}^{p}}+\left( 2\gamma _{j}^{p}-1 \right) l_{b_{j}^{p}} \right) \ge 0} ,\\ \quad \quad \quad \qquad j=1,2,\ldots ,n,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m {\sum \nolimits _{j=1}^n x_{ijk}^{p} -\left( \left( 2\delta _{k}-1 \right) g_{e_{k}}+\left( 2-2\delta _{k} \right) h_{e_{k}} \right) } \le 0,\\ \quad \quad \quad \qquad k=1,2,\ldots ,K \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {\left[ v_{i}^{p}+\left( \left( 2-2\rho _{j} \right) h_{c_{ijk}^{p}}+\left( 2\rho _{j}-1 \right) l_{c_{ijk}^{p}} \right) \right] }x_{ijk}^{p}\\ \quad \quad \quad \qquad +\left[ \left( 2-2\rho _{j} \right) h_{f_{ijk}^{p}}+\left( 2\rho _{j}-1 \right) l_{f_{ijk}^{p}} \right] y_{ijk}^{p} ~ \\ \quad \quad \quad \qquad -\left( \left( 2\rho _{j}-1 \right) g_{B_{j}}+\left( 2-2\rho _{j} \right) h_{B_{j}} \right) {\le 0,~ j=1,2,\ldots ,n} \\ x_{ijk}^{p}\ge 0,~ y_{ijk}^{p}=\left\{ {\begin{array}{l} 1{~ ;}x_{ijk}^{p}>0~ \\ 0~ ;\mathrm{otherwise}~ \\ \end{array}} \right. ~ \forall ~ p,i,j,k.~ \\ \end{array}} \right. \end{aligned}$$

(B3)

Corollary B.2

If $\xi _{c_{ijk}^{p}}$, $\xi _{f_{ijk}^{p}}$, $\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}}$ and $\xi _{B_{j}}$ are independent normal uncertain variables of the form $\mathcal {N}(\mu ,\sigma )$, such that $\mu ,~ \sigma \in \mathcal {R}$ and $\sigma >0.$ Then, according to Theorem B.1 and the inverse uncertainty distribution of normal uncertain variables, model (B1) can be written as follows.

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \bar{Z}_{1}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-\left( \mu _{c_{ijk}^{p}}\right. \right. \right. ~\\ \quad \quad \quad \qquad \left. \left. \left. +\frac{\sqrt{3} \sigma _{c_{ijk}^{p}}}{\pi }ln\frac{{\alpha }_{\mathrm {1}}}{1-{\alpha }_{\mathrm {1}}} \right) \right) x_{ijk}^{p}\right. ~\\ \quad \quad \qquad \qquad \left. -\left( \mu _{f_{ijk}^{p}}+\frac{\sqrt{3} \sigma _{f_{ijk}^{p}}}{\pi }ln\frac{{\alpha }_{\mathrm {1}}}{1-{\alpha }_{\mathrm {1}}} \right) y_{ijk}^{p} \right] ~ \\ ~ \mathrm{Min}~ \bar{Z}_{2}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( \mu _{t_{ijk}^{p}}\right. \right. ~\\ \quad \quad \qquad \qquad \left. \left. +\frac{\sqrt{3} \sigma _{t_{ijk}^{p}}}{\pi }ln\frac{{\alpha }_{\mathrm {2}}}{1-{\alpha }_{\mathrm {2}}} \right) y_{ijk}^{p} \right] ~ \\ \mathrm{subject~ to} \\ ~\sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( \mu _{a_{i}^{p}}-\frac{\sqrt{3} \sigma _{a_{i}^{p}}}{\pi }ln\frac{\beta _{i}^{p}}{1-\beta _{i}^{p}} \right) } \le 0,~\\ \quad \quad \quad \quad i=1,2,\ldots ,m,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {x_{ijk}^{p}-\left( \mu _{b_{j}^{p}}+\frac{\sqrt{3} \sigma _{b_{j}^{p}}}{\pi }ln\frac{\gamma _{j}^{p}}{{1-\gamma }_{j}^{p}} \right) {\ge 0}} ,~\\ \quad \quad \quad \quad j=1,2,\ldots ,n,~ p=1,2,\ldots ,r~ \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m {\sum \nolimits _{j=1}^n x_{ijk}^{p} -\left( \mu _{e_{k}}-\frac{\sqrt{3} \sigma _{e_{k}}}{\pi }ln\frac{\delta _{k}}{1-\delta _{k}} \right) } \le 0,~\\ \quad \quad \quad \quad k=1,2,\ldots ,K~ \\ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{k=1}^K {\left[ v_{i}^{p}+\mu _{c_{ijk}^{p}}+\frac{\sqrt{3} \sigma _{c_{ijk}^{p}}}{\pi }ln\frac{\rho _{j}}{1-\rho _{j}} \right] }x_{ijk}^{p} \\ \quad \quad \quad \quad +\left[ \mu _{f_{ijk}^{p}}+\frac{\sqrt{3} \sigma _{f_{ijk}^{p}}}{\pi }ln\frac{\rho _{j}}{1-\rho _{j}} \right] y_{ijk}^{p}\\ \quad \quad \quad \quad -\left( \mu _{B_{j}}-\frac{\sqrt{3} \sigma _{B_{j}}}{\pi }ln\frac{\rho _{j}}{1-\rho _{j}} \right) \le 0,~ ~ j=1,2,\ldots ,n \\ x_{ijk}^{p}\ge 0,~ y_{ijk}^{p}=\left\{ {\begin{array}{l} 1;\quad x_{ijk}^{p}>0~ \\ 0;\quad \mathrm{otherwise}~ \\ \end{array}} \right. ~ \forall ~ p,i,j,k.~ \\ \end{array}} \right. \end{aligned}$$

(B4)

Crisp equivalents of dependent chance-constrained model (DCCM)

For DCCM, the following model in (B5) is considered as a general case of crisp equivalent for DCCM corresponding to models (B6) and (B7), respectively, for zigzag and normal uncertain variables.

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \upsilon _{Z_{1}^{'}}~ \\ \mathrm{Max}~ \upsilon _{Z_{2}^{'}}~ \\ \mathrm{subject~ to~ the~ constraints~of}~ \left( \mathrm {B}1 \right) . \\ \end{array}} \right. \end{aligned}$$

(B5)

Theorem B.2

Let $\xi _{c_{ijk}^{p}}{,}~ \xi _{f_{ijk}^{p}}{,}~ \xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}}$ and $\xi _{B_{j}}$ are the independent zigzag uncertain variables denoted as $\mathcal {Z}\left( g_{c},h_{c},l_{c} \right)$ with $c\in \left\{ \xi _{c_{ijk}^{p}}\mathrm {,~ }\xi _{f_{ijk}^{p}}\mathrm {,~ }\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}}\mathrm {,~ }\xi _{B_{j}} \right\}$ and $0.5\le \eta \le 1,$ where

$\eta \in \left\{ {~ \beta }_{i}^{p},{~ \gamma }_{j}^{p},\delta _{k},\rho _{j} \right\} .$ Then, the crisp equivalent of DCCM, presented in model (11), is equivalent to model (B6).

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \nu _{Z_{1}^{'}}~ \\ \mathrm{Max}~ \nu _{Z_{2}^{'}}~ \\ \mathrm{subject~ to~ the~ constraints~of}~ \left( \mathrm {B3} \right) , \\ \end{array}} \right. \end{aligned}$$

(B6)

where

$$\begin{aligned} \nu _{Z_{1}^{'}}= & {} \left\{ {\begin{array}{l@{\quad }l} 1,&{} \mathrm{if}~ Z_{1}^{'}\le \bar{g}~ \\ \frac{2\bar{h}-\bar{g}-Z_{1}^{'}}{2\left( \bar{h}-\bar{g} \right) },&{} \mathrm{if}~ \bar{g}<Z_{1}^{'}\le \bar{h} \\ \frac{\bar{l}-Z_{1}^{'}}{2\left( \bar{l}-\bar{h} \right) },&{} \mathrm{if}~ \bar{h}<Z_{1}^{'}\le \bar{l}~ \\ 0,&{} \mathrm{if}~ Z_{1}^{'}>\bar{l}~ \\ \end{array}} \right. {\mathrm{and} } \\ \nu _{Z_{2}^{'}}= & {} \left\{ {\begin{array}{l@{\quad }l} 0,&{} \mathrm{if}~ Z_{2}^{'}\le \overline{\overline{g}} \\ \frac{Z_{2}^{'}-\bar{\bar{g}}}{2\left( \bar{\bar{h}}-\bar{\bar{g}} \right) },&{} \mathrm{if}~ \overline{\overline{g}}<Z_{2}^{'}\le \overline{\overline{h}} \\ \frac{Z_{2}^{'}+\overline{\overline{l}}-2\overline{\overline{h}}}{2\left( \overline{\overline{l}}-\overline{\overline{h}} \right) },&{} \mathrm{if}~ \overline{\overline{h}}<Z_{2}^{'}\le \overline{\overline{l}} \\ 1,&{} \mathrm{if}~ Z_{2}^{'}>\overline{\overline{l}}, \\ \end{array}} \right. \end{aligned}$$

such that

$$\begin{aligned} \bar{g}= & {} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-l_{\xi _{c_{ijk}^{p}}} \right) x_{ijk}^{p}\right. \\&\quad \left. -\left( l_{\xi _{f_{ijk}^{p}}} \right) y_{ijk}^{p} \right] ,\\ \bar{h}= & {} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-h_{\xi _{c_{ijk}^{p}}} \right) x_{ijk}^{p}\right. \\&\quad \left. -\left( h_{\xi _{f_{ijk}^{p}}} \right) y_{ijk}^{p} \right] ,\\ \bar{l}= & {} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-g_{\xi _{c_{ijk}^{p}}} \right) x_{ijk}^{p}\right. \\&\quad \left. -\left( g_{\xi _{f_{ijk}^{p}}} \right) y_{ijk}^{p} \right] \\ \overline{\overline{g}}= & {} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( g_{\xi _{t_{ijk}^{p}}} \right) y_{ijk}^{p} \right] ,\\&\quad \overline{\overline{h}}=\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( h_{\xi _{t_{ijk}^{p}}} \right) y_{ijk}^{p} \right] \hbox { and} \\ \overline{\overline{l}}= & {} \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( l_{\xi _{t_{ijk}^{p}}} \right) y_{ijk}^{p} \right] . \end{aligned}$$

Proof

Considering the objective $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n\right. \left. \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \right) x_{ijk}^{p}-\left( \xi _{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \ge Z_{1}^{'} \right\}$, $\xi _{c_{ijk}^{p}}$ and $\xi _{f_{ijk}^{p}}$ are independent zigzag uncertain variables. $x_{ijk}^{p},~ s_{j}^{p}$ and $v_{i}^{p}$ are greater or equal to zero, and $y_{ijk}^{p}$ are binary variables. Consequently, $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K\right. \left. \left[ \left( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \right) x_{ijk}^{p}-\left( \xi _{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \ge Z_{1}^{'} \right\}$ follows zigzag uncertainty distribution and therefore is a zigzag uncertain variable say $\mathcal {Z}\left( \bar{g},\bar{h},\bar{l} \right)$, where $\bar{g},\bar{h}$ and $\bar{l}$ are defined above in (B6). Then, from Definition 2.2, and theorems A.2 and A.4, we write

$$\begin{aligned}&\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \right) x_{ijk}^{p}\right. \right. \\&\left. \left. -\left( \xi _{f_{ijk}^{p}} \right) y_{ijk}^{p} \right] \ge Z_{1}^{'} \right\} \\&\quad \Leftrightarrow 1-\mathcal {M}\left\{ \mathcal {Z}\left( \bar{g},\bar{h},\bar{l} \right) \le Z_{1}^{'} \right\} =\nu _{Z_{1}^{'}}\\&\quad = \left\{ \begin{array}{l@{\quad }l} 1, &{} \mathrm{if}~ Z_{1}^{'}\le \bar{g}~ \\ \frac{2\bar{h}-\bar{g}-Z_{1}^{'}}{2\left( \bar{h}-\bar{g} \right) },&{} \mathrm{if}~ \bar{g}<Z_{1}^{'}\le \bar{h} \\ \frac{\bar{l}-Z_{1}^{'}}{2\left( \bar{l}-\bar{h} \right) },&{} \mathrm{if}~ \bar{h}<Z_{1}^{'}\le \bar{l} \\ 0,&{} \mathrm{if}~ Z_{1}^{'}>\bar{l}. \\ \end{array} \right. \end{aligned}$$

Similarly, for the second objective of model (11), $\xi _{t_{ijk}^{p}}$are independent zigzag uncertain variables. Hence, $\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( \xi _{t_{ijk}^{p}} \right) y_{ijk}^{p} \right] \le Z_{2}^{'} \right\}$ follows zigzag uncertainty distribution for a zigzag uncertain variable say $\mathcal {Z}\left( \bar{\bar{g}},\bar{\bar{h}},\bar{\bar{l}} \right)$, where $\bar{\bar{g}},\bar{\bar{h}}$ and $\bar{\bar{l}}$ are defined above in model (B6). So, from Definition 2.2 and Theorem A.2 we have

$$\begin{aligned}&\mathcal {M}\left\{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \left[ \left( \xi _{t_{ijk}^{p}} \right) y_{ijk}^{p} \right] \le Z_{2}^{'} \right\} =\nu _{Z_{2}^{'}}\\&\quad =\left\{ {\begin{array}{l@{\quad }l} 0,&{} \mathrm{if}~ Z_{2}^{'}\le \overline{\overline{g}} \\ \frac{Z_{2}^{'}-\bar{\bar{g}}}{2\left( \bar{\bar{h}}-\bar{\bar{g}} \right) },&{} if\, \overline{\overline{g}}<Z_{2}^{'}\le \overline{\overline{h}} \\ \frac{Z_{2}^{'}+\overline{\overline{l}}-2\overline{\overline{h}}}{2\left( \overline{\overline{l}}-\overline{\overline{h}} \right) },&{} \mathrm{if}~ \overline{\overline{h}}<Z_{2}^{'}\le \overline{\overline{l}} \\ 1,&{} \mathrm{if}~ Z_{2}^{'}>\overline{\overline{l}}. \\ \end{array}} \right. \end{aligned}$$

Moreover, from Corollary B.1 (ii) the crisp transformations of the constraint set of model (11) become same to that of the constraint set of model (B3). Hence, it directly follows model (B6).

Theorem B.3

Le $\xi _{c_{ijk}^{p}}$, $\xi _{f_{ijk}^{p}}$, $\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}^{p}}$ and $\xi _{B_{j}^{p}}$ are the independent normal uncertain variables of the form $\mathcal {N}\left( \mu _{q},\sigma _{q} \right)$ with $q\in \left\{ \xi _{c_{ijk}^{p}}\mathrm {,~ }\xi _{f_{ijk}^{p}}\mathrm {,~ }\xi _{t_{ijk}^{p}}{,}~ \xi _{a_{i}^{p}}{,}~ \xi _{b_{j}^{p}}{,}~ \xi _{e_{k}^{p}}\mathrm {,~ }\xi _{B_{j}^{p}} \right\}$. Then the crisp equivalent of model (11) is given in model (B7).

$$\begin{aligned} \left\{ {\begin{array}{l} \mathrm{Max}~ \nu _{Z_{1}^{'}}={1-\left( 1+\mathrm {exp}\left( \frac{\pi \left( \mu _{1}-Z_{1}^{'} \right) }{\sqrt{3} ~ \sigma _{1}} \right) \right) }^{-1}~ \\ Max{~ \nu }_{Z_{2}^{'}}=~ \left( 1+\mathrm {exp}\left( \frac{\pi \left( \mu _{2}-Z_{2}^{'} \right) }{\sqrt{3} ~ \sigma _{2}} \right) \right) ^{-1}~ \\ \mathrm{subject~ to~ the~ constraints}~\mathrm{of}~\mathrm{(B4)}.\\ \end{array}} \right. \end{aligned}$$

(B7)

Proof

Considering the first objective of model (11), i.e., $\mathcal {M}\Big \{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \Big ( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \Big )x_{ijk}^{p}-\Big ( \xi _{f_{ijk}^{p}} \Big )y_{ijk}^{p} \Big ] \ge Z_{1}^{'} \Big \}{,}~ \xi _{c_{ijk}^{p}}$ and $\xi _{f_{ijk}^{p}}$ are independent normal uncertain variables. $x_{ijk}^{p},~ s_{j}^{p}$ and $v_{i}^{p}$ are greater or equal to zero, and $y_{ijk}^{p}$ are binary variables. Then $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \Big ( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \Big )x_{ijk}^{p}-\Big ( \xi _{f_{ijk}^{p}} \Big )y_{ijk}^{p} \Big ]$ can be considered as a normal uncertain variable $\mathcal {N}\Big ( \mu _{1},\sigma _{1} \Big )$, such that $\mu _{1}$ and $\sigma _{1}$ are, respectively, $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \Big ( s_{j}^{p}-v_{i}^{p}{-\mu }_{\xi _{c_{ijk}^{p}}} \Big )x_{ijk}^{p}+\Big ( \mu _{\xi _{f_{ijk}^{p}}} \Big )y_{ijk}^{p} \Big ]$ and $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ {\sigma _{\xi _{c_{ijk}^{p}}}}x_{ijk}^{p}+\sigma _{\xi _{f_{ijk}^{p}}}y_{ijk}^{p} \Big ] .$ Therefore, from Definition 2.3, and theorems A.3 and  A.4, $\mathcal {M}\Big \{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \Big ( s_{j}^{p}-v_{i}^{p}-\xi _{c_{ijk}^{p}} \Big )x_{ijk}^{p}-\Big ( \xi _{f_{ijk}^{p}} \Big )y_{ijk}^{p} \Big ] \ge Z_{1}^{'} \Big \}={1-\Big (1}{+\exp \Big ( \frac{\pi \Big ( \mu _{1}-Z_{1}^{'} \Big )}{\sqrt{3} ~ \sigma _{1}} \Big ) \Big )}^{-1}$.

Table 10 Unit purchase costs of items 1 and 2 at two different sources

Table 11 Unit selling prices of items 1 and 2 at three different destinations

Table 12 Transportation costs for item 1 and item 2 represented as zigzag and normal uncertain variables

Table 13 Fixed charge costs for item 1 and item 2 represented as zigzag and normal uncertain variables

Table 14 Transportation time for item 1 and item 2 represented as zigzag and normal uncertain variables

Similarly, for the second objective of model (11), $\xi _{t_{ijk}^{p}}$ are the independent normal uncertain variables. Therefore, $\mathcal {M}\Big \{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \xi _{t_{ijk}^{p}}y_{ijk}^{p} \Big ] \le Z_{2}^{'} \Big \}$ is a normal uncertain variable, $\mathcal {N}\Big ( \mu _{2},\sigma _{2} \Big )$ such that $\mu _{2}$ and $\sigma _{2}$ are, respectively, $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \mu _{\xi _{t_{ijk}^{p}}}y_{ijk}^{p} \Big ]$ and $\sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \sigma _{\xi _{t_{ijk}^{p}}}y_{ijk}^{p} \Big ]$.

Accordingly, from Definition 2.3, and theorems A.3 and A.4, $\mathcal {M}\Big \{ \sum \nolimits _{p=1}^r \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^n \sum \nolimits _{k=1}^K \Big [ \xi _{t_{ijk}^{p}}y_{ijk}^{p} \Big ] \le Z_{2}^{'} \Big \}=\Big ( 1+\mathrm {exp}\Big ( \frac{\pi \Big ( \mu _{2}-Z_{2}^{'} \Big )}{\sqrt{3} ~ \sigma _{2}} \Big ) \Big )^{-1}.$ Further, from Corollary B.2 the crisp transformations of the constraints of model (11) becomes same to that of the constraint set of model (B4). Hence, the model (B7) follows directly.

Appendix C

Data tables for input parameters

The input parameters related to UMMFSTPwB are reported in tables 10, 11, 12, 13, 14, 15, 16, 17 and 18. The parameters shown in tables 10 and 11 are crisp. The parameters, presented in tables 12, 13, 14, 15, 16, 17 and 18 are uncertain. These uncertain parameters are represented as: (i) zigzag uncertain variables and (ii) normal uncertain variables.

Table 15 Available amounts of item 1 and item 2 represented as zigzag and normal uncertain variables

Table 16 Demands for item 1 and item 2 represented as zigzag and normal uncertain variables

Table 17 Transportation capacities of two conveyances expressed as zigzag and normal uncertain variables

Table 18 Budget availability at destinations represented as zigzag and normal uncertain variables