An Advanced Optimization Technique for Smart Production Using α-Cut Based Quadrilateral Fuzzy Number

Abstract

In the design phase of a new smart product, production costs are unpredictable due to location, transport, and engineering design. In these situations, consequently, cost optimization becomes ambiguous. This paper presents a methodology to obtain optimization through a fuzzy linear programming problem (FLPP) in which fuzzy numbers signify the right-side parameters. The comparative investigation of modeling and optimizing creation cost through a new α-cut based quadrilateral fuzzy number is proposed to solve the fuzzy linear programming and the necessary operations on the proposed number. Due to the probabilistic increase and decrease in the accessibility of the various constraints, the actual expected total cost fluctuates. In this respect, a unique situation of instability is incorporated, and reasonable models to reduce the cost of eradication in the creation process are presented. The main endeavor is made to look at the credibility of optimized cost utilizing the α-cut based quadrilateral FLPP models, and the outcome is contrasted with its augmentation. The data of the production cost of RCF Kapurthala is taken, and the creation expenses of various mentors from the year 2010–2011 are considered as input parameters. The aggregate cost is focused on the objective function. The least low, lower, upper, and most upper bounds are computed for each situation, and then systems of optimized fuzzy LPP are constructed. The credibility of quadrilateral fuzzy LPP concerning all situations is obtained and using this membership grade, the minimum, and highest minimum costs are illustrated.

Appendices

Appendix 1

Let $\tilde{B}_{i}^{p}$ and $\tilde{B}_{i}^{q}$ be quadrilateral fuzzy numbers with different confidence levels such that $\beta^{p} \le \beta^{q}$. Take $\beta^{s} \in [\beta^{p} ,\beta^{q} ]$ , i.e., $\beta^{s} = \beta^{p}$ then $\alpha^{s}$-cut of $\tilde{B}_{i}^{p}$ and $\tilde{B}_{i}^{q}$ are

When $\alpha^{p} \le \beta^{p} ,\alpha^{p} , \beta^{p} \ne 0$

$\tilde{B}_{i}^{p} = \left[ {\beta_{i}^{p} - \varepsilon_{i}^{p} , \beta_{i}^{p} - \varepsilon_{i}^{p} + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }}} \right) \cup \left[ {\beta_{i}^{p} , \beta_{i}^{p} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right)} \right) \cup \left[ {\beta_{i}^{p*} , \alpha^{p} \varepsilon_{i}^{p*} + \left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right)} \right]$ $\tilde{B}_{i}^{q} = \left[ {\beta_{i}^{q} - \varepsilon_{i}^{q} , \beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q} }}{{\beta^{q} }}} \right) \cup \left[ {\beta_{i}^{q} , \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right) \cup \left[ {\beta_{i}^{q*} , \alpha^{q} \varepsilon_{i}^{q*} + \left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right]$

$$\tilde{B}_{i}^{q + } = \left[ {\beta_{i}^{q} - \varepsilon_{i}^{q} , \beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right) \cup \left[ {\beta_{i}^{q} , \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right) \cup \left[ {\beta_{i}^{q*} , \alpha^{q} \varepsilon_{i}^{q*} + \left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right]\,\beta_{i}^{q} - \varepsilon_{i}^{q + } = (\beta_{i}^{q} - \varepsilon_{i}^{q} ) + \frac{{\beta^{s} }}{{\beta^{q} }}\varepsilon_{i}^{q} \Rightarrow \varepsilon_{i}^{q + } = \beta_{i}^{q} - (\beta_{i}^{q} - \varepsilon_{i}^{q} ) - \frac{{\beta^{s} }}{{\beta^{q} }}\varepsilon_{i}^{q}$$

Let $\tilde{B}_{i}^{s} = \tilde{B}_{i}^{p} + \tilde{B}_{i}^{q + } = \{ y{\mid }y \in \tilde{B}_{\alpha }^{s} \} \,\forall \alpha^{s} \in \left[ {0,1} \right]$. Here $\tilde{B}_{\alpha }^{s} = \left[ {\tilde{B}_{\alpha }^{sL} \left( {\alpha^{s} } \right), \tilde{B}_{\alpha }^{sU} \left( {\alpha^{s} } \right)} \right]$ be its $\alpha^{s}$-cuts such that $\tilde{B}_{\alpha }^{sL} \left( {\alpha^{s} } \right) = \tilde{B}_{i}^{pL} (\alpha^{p} ) + \tilde{B}_{i}^{qL} (\alpha^{q} )\,{\text{and}}\,\tilde{B}_{\alpha }^{sU} \left( {\alpha^{s} } \right) = \tilde{B}_{i}^{pU} (\alpha^{p} ) + \tilde{B}_{i}^{qU} (\alpha^{q} )\,\,{\text{i}}.{\text{e}}.$

$$\tilde{B}_{\alpha }^{s} = \left[ {\tilde{B}_{i}^{pL} \left( {\alpha^{p} } \right) + \tilde{B}_{i}^{qL} \left( {\alpha^{q} } \right) , \tilde{B}_{i}^{pU} \,\left( {\alpha^{p} } \right) + \tilde{B}_{i}^{qU} \left( {\alpha^{q} } \right)} \right]$$

$$\tilde{B}_{\alpha }^{s} = I_{1}^{s} \cup I_{2}^{s} \cup I_{3}^{s} , where$$

$$I_{1}^{s} = \left[ {\beta_{i}^{p} - \varepsilon_{i}^{p} + \beta_{i}^{q} - \varepsilon_{i}^{q} ,\beta_{i}^{p} - \varepsilon_{i}^{p} + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }} + \beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right)$$

$$I_{2}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} ,\beta_{i}^{p} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right) + \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right]$$

$$I_{3}^{s} = \left[ {\beta_{i}^{p*} + \beta_{i}^{q*} ,\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) - \alpha^{p} \varepsilon_{i}^{p*} + \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - \alpha^{q} \varepsilon_{i}^{q*} } \right]$$

Now

$$I_{1}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} - (\varepsilon_{i}^{p} + \varepsilon_{i}^{q} ) ,\beta_{i}^{p} + \beta_{i}^{q} - (\varepsilon_{i}^{p} + \varepsilon_{i}^{q} ) + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q + } }}} \right)$$

$$\alpha^{p} = \alpha^{q} = \alpha^{s}$$

$$I_{1}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} - (\varepsilon_{i}^{p} + \varepsilon_{i}^{q} ) ,\beta_{i}^{p} + \beta_{i}^{q} - (\varepsilon_{i}^{p} + \varepsilon_{i}^{q} ) + \alpha^{s} \left( {\frac{{\varepsilon_{i}^{p} }}{{\beta^{p} }} + \frac{{\varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right)} \right)$$

$$\beta_{i}^{p} + \beta_{i}^{q} - (\varepsilon_{i}^{p} + \varepsilon_{i}^{q} ) + \alpha^{s} \left( {\frac{{\varepsilon_{i}^{p} }}{{\beta^{p} }} + \frac{{\varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right) - y = 0$$

$$f_{{B^{s} }}^{U} \left( y \right) = \frac{{y - \left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right) + (\beta_{i}^{q} - \varepsilon_{i}^{q} )}}{{\left( { \frac{{\varepsilon_{i}^{p} }}{{\beta^{p} }} + \frac{{\varepsilon_{i}^{q + } }}{{\beta^{q + } }} } \right) }}$$

$$when \, \beta^{p} = \beta^{q + } = \beta^{s}$$

$$g_{{B^{s} }}^{U} \left( y \right) = \beta^{s} \left( {\frac{{y - \left[ {\left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right) + (\beta_{i}^{q} - \varepsilon_{i}^{q} } \right)]}}{{\varepsilon_{i}^{p} + \varepsilon_{i}^{q + } }}} \right)$$

$$g_{{B^{s} }}^{U} \left( y \right) = \beta^{s} \left( {\frac{{y - \left[ {\left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right) + (\beta_{i}^{q} - \varepsilon_{i}^{q} } \right)]}}{{\varepsilon_{i}^{p} + \beta_{i}^{q} - (\beta_{i}^{q} - \varepsilon_{i}^{q} ) - \frac{{\beta^{r} }}{{\beta^{p} }}\varepsilon_{i}^{q} }}} \right)$$

$$g_{{B^{s} }}^{U} \left( y \right) = \beta^{s} \left( {\frac{{y - \left[ {\left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right) + (\beta_{i}^{q} - \varepsilon_{i}^{q} } \right)]}}{{\beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q} - \left[ {\left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right) + (\beta_{i}^{q} - \varepsilon_{i}^{q} } \right)]}}} \right)$$

$$g_{{B^{s} }}^{U} \left( y \right) = \beta^{s} \left( {\frac{{y - \left( {\beta_{i}^{s} - \varepsilon_{i}^{s} } \right)}}{{\beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q} - \left( {\beta_{i}^{s} - \varepsilon_{i}^{s} } \right)}}} \right)$$

$${\text{where}}\, \beta_{i}^{s} = \beta_{i}^{p} + \beta_{i}^{q} ,\varepsilon_{i}^{s} = \varepsilon_{i}^{p} + \varepsilon_{i}^{q}$$

$$\left( {\beta_{i}^{s} - \varepsilon_{i}^{s} } \right) \le y \le \beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q}$$

Now

$$I_{2}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} ,\beta_{i}^{p} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right) + \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right]$$

$$I_{2}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} ,\beta_{i}^{p} + \beta_{i}^{q} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right) + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right]$$

$${\text{here }}\frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }} = \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }} = \frac{{\overline{{\alpha^{s} }} - \beta^{s} }}{{\overline{{\beta^{s} }} }}$$

$$I_{2}^{s} = \left[ {\beta_{i}^{p} + \beta_{i}^{q} ,\beta_{i}^{p} + \beta_{i}^{q} + \left( {\overline{{\alpha^{s} }} - \beta^{s} } \right)\left( {\frac{{\beta_{i}^{p*} - \beta_{i}^{p} }}{{\overline{{\beta^{p} }} }} + \frac{{\beta_{i}^{q*} - \beta_{i}^{q} }}{{\overline{{\beta^{q} }} }}} \right)} \right]$$

$$\beta_{i}^{p} + \beta_{i}^{q} + \left( {\overline{{\alpha^{s} }} - \beta^{s} } \right)\left( {\frac{{\beta_{i}^{p} - \beta_{i}^{p} }}{{\overline{{\beta^{p} }} }} + \frac{{\beta_{i}^{q} - \beta_{i}^{q} }}{{\overline{{\beta^{q} }} }}} \right) - y = 0$$

$${\text{when}}\, \overline{{\beta^{p} }} = \overline{{\beta^{q} }} = \overline{{\beta^{s} }}$$

$$g_{{B^{s} }}^{U} \left( y \right) = \overline{{\alpha^{s} }} - \beta^{s} = \left( {\frac{{y - \left( {\beta_{i}^{p} + \beta_{i}^{q} } \right)}}{{\beta_{i}^{p*} - \beta_{i}^{p} + \beta_{i}^{q*} - \beta_{i}^{q} }}} \right) \times \overline{{\beta^{s} }}$$

$$\overline{{\alpha^{s} }} = \beta^{s} + \left( {\frac{{y - \left( {\beta_{i}^{p} + \beta_{i}^{q} } \right)}}{{\beta_{i}^{p*} - \beta_{i}^{p} + \beta_{i}^{q*} - \beta_{i}^{q} }}} \right) \times \overline{{\beta^{s} }} \Rightarrow \overline{{\alpha^{s} }} = \beta^{s} + \left( {\frac{{y - \beta_{i}^{s} }}{{\beta_{i}^{s*} - \beta_{i}^{s} }}} \right) \times \overline{{\beta^{s} }}$$

$${\text{where }}\,\beta_{i}^{s*} = \beta_{i}^{p*} + \beta_{i}^{q*} ,\beta_{i}^{s} = \beta_{i}^{p} + \beta_{i}^{q}$$

$$I_{3}^{s} = \left[ {\beta_{i}^{p*} + \beta_{i}^{q*} ,\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) - \alpha^{p} \varepsilon_{i}^{p*} + \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - \alpha^{q} \varepsilon_{i}^{q*} } \right]$$

$$\alpha^{p} = \alpha^{q} = \alpha^{s}$$

$$I_{3}^{s} = \left[ {\beta_{i}^{p*} + \beta_{i}^{q*} ,\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) + \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - \alpha^{s} \left( {\varepsilon_{i}^{p*} + \varepsilon_{i}^{q*} } \right)} \right]$$

$$y - \left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) - \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) + \alpha^{s} \left( {\varepsilon_{i}^{p*} + \varepsilon_{i}^{q*} } \right) = 0$$

$$g_{{B^{s} }}^{U} \left( y \right) = \alpha^{s} = \left( {\frac{{\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) + \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - y}}{{\varepsilon_{i}^{p*} + \varepsilon_{i}^{q*} }}} \right) \Rightarrow g_{{B^{s} }}^{U} \left( y \right) = \left( {\frac{{\left( {\beta_{i}^{s*} + \varepsilon_{i}^{s*} } \right) - y}}{{\varepsilon_{i}^{s*} }}} \right)$$

$${\text{where}} ,\, \beta_{i}^{s*} = \beta_{i}^{p*} + \beta_{i}^{q*} \,,\,\,\varepsilon_{i}^{s*} = \varepsilon_{i}^{p*} + \varepsilon_{i}^{q*}$$

$$B_{i}^{s} \left( y \right) = \left\{ {\begin{array}{*{20}c} {\beta^{s} + \left( {\frac{{y - \beta_{i}^{s} }}{{\beta_{i}^{s*} - \beta_{i}^{s} }}} \right) \times \overline{{\beta^{s} }} } & {{\text{When}}\, \beta_{i}^{s} \le y \le \beta_{i}^{s*} } \\ {\beta^{s} \left( {\frac{{y - \left( {\beta_{i}^{s} - \varepsilon_{i}^{s} } \right)}}{{\beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q} - \left( {\beta_{i}^{s} - \varepsilon_{i}^{s} } \right)}}} \right)} & {{\text{When }}\,\beta_{i}^{s} - \varepsilon_{i}^{s} \le y \le \beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q} } \\ {\frac{{\left( {\beta_{i}^{s*} + \varepsilon_{i}^{s*} } \right) - y}}{{\varepsilon_{i}^{s*} }}} & {{\text{When}}\, \beta_{i}^{s*} \le y \le \beta_{i}^{s*} + \varepsilon_{i}^{s*} } \\ 0 & {\text{otherwise}} \\ \end{array} } \right.$$

$$\beta_{i}^{s} - \varepsilon_{i}^{s} = (\beta_{i}^{p} - \varepsilon_{i}^{p} ) + (\beta_{i}^{q} - \varepsilon_{i}^{q} ).$$

$$\beta_{i}^{s} = \beta_{i}^{p} + \beta_{i}^{q} - \frac{{\beta^{s} }}{{\beta^{p} }}\varepsilon_{i}^{q} .$$

$$\beta_{i}^{s*} = \beta_{i}^{p*} + \beta_{i}^{q*} .$$

$$\beta_{i}^{s*} + \varepsilon_{i}^{s*} = \left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) + \left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right).$$

Appendix 2

Proof As the quadrilateral membership functions of $\tilde{B}_{i}^{p}$ and $\tilde{B}_{i}^{q}$ are given in Eqs. (17) and (18), respectively. Thus, in order to find the membership of $\tilde{B}_{i}^{s}$ = $\tilde{B}_{i}^{p} \times \tilde{B}_{i}^{q}$ = ($\beta_{i}^{s} - \varepsilon_{i}^{s}$, $\beta_{i}^{s} , \beta_{i}^{s*}$, $\beta_{i}^{s*} + \varepsilon_{i}^{s*}$)

Let $\tilde{B}_{i}^{s} = \tilde{B}_{i}^{p} \times \tilde{B}_{i}^{q + } = \left\{ {y{\mid }y \in \tilde{B}_{\alpha }^{s} } \right\}\,\forall \alpha^{s} \in \left[ {0,1} \right]$. Here $\tilde{B}_{\alpha }^{s} = \left[ {\tilde{B}_{\alpha }^{sL} \left( {\alpha^{s} } \right), \,\tilde{B}_{\alpha }^{sU} \left( {\alpha^{s} } \right)} \right]$ be its $\alpha^{s}$-cuts such that $\tilde{B}_{\alpha }^{sL} \left( {\alpha^{s} } \right) = \tilde{B}_{i}^{pL} (\alpha^{p} ) \times \tilde{B}_{i}^{qL} (\alpha^{q} )\,{\text{and}}\,\tilde{B}_{\alpha }^{sU} \left( {\alpha^{s} } \right) = \tilde{B}_{i}^{pU} (\alpha^{p} ) \times \tilde{B}_{i}^{qU} (\alpha^{q} )\,\,{\text{i}} . {\text{e}}$

$$\tilde{B}_{\alpha }^{s} = \left[ {\tilde{B}_{i}^{pL} \left( {\alpha^{p} } \right) \times \tilde{B}_{i}^{qL} \left( {\alpha^{q} } \right) , \tilde{B}_{i}^{pU} (\alpha^{p} ) \times \tilde{B}_{i}^{qU} (\alpha^{q} )} \right]$$

$$\tilde{B}_{\alpha }^{s} = I_{1}^{s} \cup I_{2}^{s} \cup I_{3}^{s} ,\,{\text{where}}$$

$$I_{1}^{s} = \left[ {(\beta_{i}^{p} - \varepsilon_{i}^{p} )(\beta_{i}^{q} - \varepsilon_{i}^{q} ),\,\left\{ {\beta_{i}^{p} - \varepsilon_{i}^{p} + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }}} \right\}\left\{ {\beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right\}} \right)$$

$$I_{2}^{s} = \left[ {\beta_{i}^{p} \beta_{i}^{q} ,\left\{ {\beta_{i}^{p} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right)} \right\} \left\{ { \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right\}} \right]$$

$$I_{3}^{s} = \left[ {\beta_{i}^{p*} \beta_{i}^{q*} ,\left\{ {\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) - \alpha^{p} \varepsilon_{i}^{p*} } \right\}\left\{ {\left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - \alpha^{q} \varepsilon_{i}^{q*} } \right\}} \right]$$

Now,

$$I_{1}^{s} = \left[ {\left( {\beta_{i}^{p} - \varepsilon_{i}^{p} } \right)\left( { \beta_{i}^{q} - \varepsilon_{i}^{q} } \right) ,\left\{ {\beta_{i}^{p} - \varepsilon_{i}^{p} + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }}} \right\}\left\{ {\beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right\}} \right)$$

$$\alpha^{p} = \alpha^{q} = \alpha^{s} \, {\text{and}} \, \beta^{p} = \beta^{q + } = \beta^{s}$$

$$\left\{ {\beta_{i}^{p} - \varepsilon_{i}^{p} + \frac{{\alpha^{p} \varepsilon_{i}^{p} }}{{\beta^{p} }}} \right\}\left\{ {\beta_{i}^{q} - \varepsilon_{i}^{q} + \frac{{\alpha^{q} \varepsilon_{i}^{q + } }}{{\beta^{q} }}} \right\} - y = 0$$

$$\left( {\alpha^{s} } \right)^{2} \frac{{\varepsilon_{i}^{p} \left( {\varepsilon_{i}^{q + } - \varepsilon_{i}^{q} } \right)}}{{\overline{{\beta^{s} }} \overline{{\beta^{q} }} }} + \alpha^{s} \left\{ {\frac{{(\beta_{i}^{p} - \varepsilon_{i}^{p} )\left( {\varepsilon_{i}^{q + } - \varepsilon_{i}^{q} } \right)}}{{\overline{{\beta^{q} }} }} + \frac{{(\beta_{i}^{q} - \varepsilon_{i}^{q} )\varepsilon_{i}^{p} }}{{\overline{{\beta^{s} }} }}} \right\} + (\beta_{i}^{p} - \varepsilon_{i}^{p} )(\beta_{i}^{q} - \varepsilon_{i}^{q} ) - y = 0$$

${\text{Take}}\,M_{2} = \frac{{\varepsilon_{i}^{p} \left( {\varepsilon_{i}^{q + } - \varepsilon_{i}^{q} } \right)}}{{\overline{{\beta^{s} }} \overline{{\beta^{q} }} }},\,\,K_{2} = \frac{{(\beta_{i}^{p} - \varepsilon_{i}^{p} )\left( {\varepsilon_{i}^{q + } - \varepsilon_{i}^{q} } \right)}}{{\overline{{\beta^{q} }} }} + \frac{{(\beta_{i}^{q} - \varepsilon_{i}^{q} )\varepsilon_{i}^{p} }}{{\overline{{\beta^{s} }} }},\,\,{\text{and}}\,N_{2} = (\beta_{i}^{p} - \varepsilon_{i}^{p} )(\beta_{i}^{q} - \varepsilon_{i}^{q} )\,\left( {\alpha^{s} } \right)^{2} {{ M}}_{2} + \,\alpha^{s} {{ K}}_{2} + \,{{ N}}_{2} - y = 0.$

$$\alpha^{s} = \frac{{ - K_{2} \pm \sqrt {K_{2}^{2} + 4M_{2} \left( {y - N_{2} } \right)} }}{{2M_{2} }}$$

Similarly, $In \, I_{2}^{s} = \left[ {\beta_{i}^{p} \beta_{i}^{q} ,\left\{ {\beta_{i}^{p} + \frac{{\overline{{\alpha^{p} }} - \beta^{p} }}{{\overline{{\beta^{p} }} }}\left( {\beta_{i}^{p*} - \beta_{i}^{p} } \right)} \right\} \left\{ { \beta_{i}^{q} + \frac{{\overline{{\alpha^{q} }} - \beta^{q} }}{{\overline{{\beta^{q} }} }}\left( {\beta_{i}^{q*} - \beta_{i}^{q} } \right)} \right\}} \right]$

$$\alpha^{s} = \beta^{s} + \frac{{ - K_{1} + \sqrt {K_{1}^{2} + 4M_{1} \left( {y - N_{1} } \right)} }}{{2M_{1} }}$$

And $I_{3}^{s} = \left[ {\beta_{i}^{p*} \beta_{i}^{q*} ,\left\{ {\left( {\beta_{i}^{p*} + \varepsilon_{i}^{p*} } \right) - \alpha^{p} \varepsilon_{i}^{p*} } \right\}\left\{ {\left( {\beta_{i}^{q*} + \varepsilon_{i}^{q*} } \right) - \alpha^{q} \varepsilon_{i}^{q*} } \right\}} \right]$

$$\alpha^{s} = \frac{{K_{3} + \sqrt {K_{3}^{2} + 4M_{3} \left( {y - N_{3} } \right)} }}{{2M_{3} }}$$

Appendix 3

3.1 Objective Function

Let $y_{1} , y_{2} , \ldots ,y_{20}$ be variables for different constraints.

Minimize Z = 66.4y₁ + 61.58y₂ + 64.47y₃ + 264.12y₄ + 69.17y₅ + 130.48y₆ + 46.03y₇ + 164.11y₈ + 262.29y₉ + 129.41y₁₀ + 52.16y₁₁ + 202.79y₁₂ + 206.74y₁₃ + 142.17y₁₄ + 236.53y₁₅ + 302.08y₁₆ + 234.01y₁₇ + 236.54y₁₈ + 98.67y₁₉ +119.13y₂₀.

Subjected to constraints:

4.38y₁ + 4.07y₂ + 4.04y₃ + 9.88y₄ + 4.10y₅ + 7.38y₆ + 2.5y₇ + 8.11y₈ + 14.81y₉ + 7.22y₁₀ + 3.05y₁₁ + 9.18y₁₂ + 10.70y₁₃ + 6.38y₁₄ + 10.38y₁₅ + 10.51y₁₆ + 11.24y₁₇ + 11.57y₁₈ + 5.91y₁₉ + 7.79y₂₀ ≥ B_lab.

45.7y₁ + 41.99y₂ +44.49y₃ +211.93y₄ + 49.13y₅ +94.06y₆ +33.62y₇ +124.32y₈ +190.08y₉ +93.76y₁₀ +37.29y₁₁ +156.57y₁₂ +153.98y₁₃ +110.86y₁₄ +184.34y₁₅ +246.6y₁₆ +178.25y₁₇ +179.28y₁₈ + 70.13y₁₉ + 81.84y₂₀ ≥ B_mat.

7.33y₁ + 6.81y₂ + 6.77y₃ + 16.55y₄ + 6.87y₅ + 12.36y₆ + 4.58y₇ + 14.35y₈ + 24.4y₉ + 13.17y₁₀ + 5.55y₁₁ + 15.27y₁₂ + 17.84y₁₃ + 10.59y₁₄ + 16.98y₁₅ + 17.19y₁₆ + .18.42y₁₇ + 18.97y₁₈ + 9.70y₁₉ + 12.86y₂₀ ≥ B_foh.

5.8y₁ + 5.39y₂ +5.35y₃ +13.09y₄ +5.43y₅ + 9.78y₆ +2.91y₇ +9.14y₈ + 19.3y₉ + 8.39y₁₀ +3.54y₁₁  + 12.08y₁₂ + 14.11y₁₃ + 8.38y₁₄ + 13.43y₁₅ + 13.59y₁₆ + 14.57y₁₇ + 15.01y₁₈ + 7.67y₁₉ + 10.17y₂₀ ≥ $B_{\text{aoh}}$.

1.17y₁ + 1.09y₂ + 1.08y₃ + 2.65y₄ + 1.10y₅ + 1.98y₆ + 0.75y₇ + 2.35y₈ + 3.91y₉ + 2.16y₁₀ + 0.91y₁₁ + 2.45y₁₂ + 2.86y₁₃ + 1.70y₁₄ + 2.72y₁₅ + 2.75y₁₆ + 2.95y₁₇ + 3.04y₁₈ + 1.55y₁₉ + 2.06y₂₀ ≥ B_tooh.

0.37y₁ +0.34y₂ +0.36y₃ +1.74y₄ +0.40y₅ +0.77y₆ +0.23y₇ +0.85y₈ +1.56y₉ +0.64y₁₀ +0.25y₁₁ + 1.28y₁₂ + 1.26y₁₃ + 0.91y₁₄ + 1.51y₁₅ + 2.02y₁₆ + 1.46y₁₇ + 1.47y₁₈ + 0.58y₁₉ + 0.67y₂₀ ≥ B_soh.

14.69y₁ + 13.63y₂ + 13.57y₃ + 34.04y₄ + 13.80y₅ + 24.89y₆ + 8.47y₇ + 26.68y₈ + 49.16y₉ + 24.35y₁₀ + 10.26y₁₁ + 31.09y₁₂ + 36.07y₁₃ + 21.58y₁₄ + 34.64y₁₅ + 35.56y₁₆ + 37.39y₁₇ + 38.49y₁₈ + 19.49y₁₉ + 25.76y₂₀ ≥ B_toh.

1.63y₁ + 1.89y₂ + 2.37y₃ + 8.27y₄ + 2.13y₅ + 4.14y₆ + 1.44y₇ + 5y₈ + 8.23y₉ + 4.07y₁₀ + 1.57y₁₁ + 5.96y₁₂ + 5.99y₁₃ + 3.35y₁₄ + 7.17y₁₅ +9.42y₁₆ + 7.12y₁₇ + 7.20y₁₈ + 3.14y₁₉ + 3.74 ≤ B_proof.

For all $y_{i}$ ≥ 0.