An optimization model for a manufacturing-inventory system with rework process based on failure severity under multiple constraints

Abstract

The present work investigates a manufacturing-inventory system with a single machine and multiple products, featuring returns on sales and backorders. In the proposed model, some imperfect items, including scrapped and defective items, are produced by the manufacturer. Such items can be classified, based on the severity of the failure, into several categories; as a result, the rework process is carried out at different rates. Moreover, the implementation of the quality control policy requires monitoring and checking the items during the production and reworking processes via an inspection process. The present study is aimed to calculate and obtain the optimal values of the cycle length and backorders quantity for every product in order to achieve the minimum total cost of system considering machine capacity, service level, warehouse space, and budget constraints. To solve the presented model, given as a Nonlinear Programming (NLP) problem, the GAMS software as well as four commonly used algorithms, which are categorized among the meta-heuristic algorithms, is used. These algorithms include the GA (Genetic Algorithm), IWO (Invasive Weed Optimization), GWO (Grey Wolf Optimizer) and HHO (Harris Hawks Optimization) algorithms. Along with these algorithms, the Response Surface Methodology (RSM) is applied to calibrate the parameters of the proposed algorithms. Finally, several numeric problems are solved, the results of which are then compared with each other. Moreover, an analytical hierarchy process (AHP) technique for order performance by similarity to ideal solution (TOPSIS), which is a hybrid method of decision making with multiple attributes, is used for ranking the algorithms.

Appendices

Appendices

Appendix A Determining the length of the cycle

In accordance with Eq. (8), we will have:

$$T= {t}_{i}^{0}+ {t}_{i}^{1}+ {t}_{i}^{2}+\dots +{t}_{i}^{m+1}+ {t}_{i}^{m+2}+ {t}_{i}^{m+3}$$

(A.1)

$$T= \frac{{B}_{i}}{\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]{ P}_{i}-{D}_{i}}+ \frac{{Q}_{i}}{{P}_{i}}- \frac{{B}_{i}}{\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]{ P}_{i}-{D}_{i}}+ \left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}+\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]\frac{{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}+ \dots + \left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]\frac{{Q}_{i}}{{V}_{i}^{m}{ P}_{i}} + \frac{{H}_{i}^{m}}{{D}_{i}} + \frac{{B}_{i}}{{D}_{i}}$$

(A.2)

$$T= \frac{{Q}_{i}}{{P}_{i}}+ \left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}+ \left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]\frac{{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}+\dots + \left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]\frac{{Q}_{i}}{{V}_{i}^{m}{ P}_{i}} + \frac{\left[\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]{ P}_{i}-{D}_{i}\right]\left(\frac{{Q}_{i}}{{P}_{i}}\right)- {B}_{i}}{{D}_{i}}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]\left({\gamma }_{i}^{1}{V}_{i}^{1}{P}_{i}-{D}_{i}\right)\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}}{{D}_{i}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]\left({\gamma }_{i}^{2}{V}_{i}^{2}{P}_{i}-{D}_{i}\right)\frac{{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}}{{D}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]\left({\gamma }_{i}^{m}{V}_{i}^{m}{P}_{i}-{D}_{i}\right)\frac{{Q}_{i}}{{V}_{i}^{m}{ P}_{i}}}{{D}_{i}} + \frac{{B}_{i}}{{D}_{i}}$$

(A.3)

$$T= \frac{{Q}_{i}}{{P}_{i}}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{Q}_{i}}{{V}_{i}^{m}{ P}_{i}}+ \frac{\left[\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]{ P}_{i}-{D}_{i}\right]{Q}_{i}}{{P}_{i}{D}_{i}}- \frac{{B}_{i}}{{D}_{i}}+\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]\left({\gamma }_{i}^{1}{V}_{i}^{1}{P}_{i}-{D}_{i}\right){Q}_{i}}{{{V}_{i}^{1}{ P}_{i} D}_{i}}+\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]\left({\gamma }_{i}^{2}{V}_{i}^{2}{P}_{i}-{D}_{i}\right){Q}_{i}}{{{V}_{i}^{2}{ P}_{i} D}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]\left({\gamma }_{i}^{m}{V}_{i}^{m}{P}_{i}-{D}_{i}\right){Q}_{i}}{{{V}_{i}^{m}{ P}_{i} D}_{i}} + \frac{{B}_{i}}{{D}_{i}}$$

(A.4)

$$T= \frac{{Q}_{i}}{{P}_{i}}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{Q}_{i}}{{V}_{i}^{m}{ P}_{i}}+ \frac{\left[\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]-\left(\frac{{D}_{i}}{{P}_{i}}\right)\right]{P}_{i}{Q}_{i}}{{D}_{i}{P}_{i}}- \frac{{B}_{i}}{{D}_{i}}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{\gamma }_{i}^{1}{V}_{i}^{1}{P}_{i}{Q}_{i}}{{{V}_{i}^{1}{ P}_{i} D}_{i}} - \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{D}_{i}{Q}_{i}}{{{V}_{i}^{1}{ P}_{i} D}_{i}} + \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{\gamma }_{i}^{2}{V}_{i}^{2}{P}_{i}{Q}_{i}}{{{V}_{i}^{2}{ P}_{i} D}_{i}} - \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{D}_{i}{Q}_{i}}{{{V}_{i}^{2}{ P}_{i} D}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{\gamma }_{i}^{m}{V}_{i}^{m}{P}_{i}{Q}_{i}}{{{V}_{i}^{m}{ P}_{i} D}_{i}} - \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{D}_{i}{Q}_{i}}{{{V}_{i}^{m}{ P}_{i} D}_{i}} + \frac{{B}_{i}}{{D}_{i}}$$

(A.5)

$$T= \frac{{Q}_{i}}{{P}_{i}}+ \frac{\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]{Q}_{i}}{{D}_{i}}-\frac{{Q}_{i}}{{P}_{i}} + \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{\gamma }_{i}^{1}{Q}_{i}}{{ D}_{i}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{\gamma }_{i}^{2}{Q}_{i}}{{ D}_{i}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{\gamma }_{i}^{m}{Q}_{i}}{{ D}_{i}}$$

(A.6)

$$T= \frac{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right]+ \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right){Q}_{i}}{{ D}_{i}}$$

(A.7)

Appendix B Calculating the cost of holding

Considering Eq. (23), we have:

$$CH={CH}_{a}+{CA}_{b}+{CH}_{c}+{CH}_{d}$$

(B.1)

According to Eq. (24), ${CH}_{a}$ is:

$${CH}_{a}=\frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[{ I}_{i}\left(\frac{{Q}_{i}}{{P}_{i}}- \frac{{B}_{i}}{{a}_{i}}\right)+ \left({2I}_{i}+\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}\right)\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}\right)+ \left({2H}_{i}^{1}+ \left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}\right)\left(\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}\right)+ \dots +\left({2H}_{i}^{m-1}+ \left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}\frac{{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}\right)\left(\frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{Q}_{i}}{{V}_{i}^{m}{ P}_{i}}\right)+ \frac{{{(H}_{i}^{m})}^{2}}{{D}_{I}}\right]$$

(B.2)

Based on Eqs. (9–12) we have:

$${I}_{i}= {a}_{i}\left(\frac{{Q}_{i}}{{P}_{i}}\right)- {B}_{i}$$

(B.1)

$${{(I}_{i})}^{2}=\frac{{({a}_{i})}^{2}{({Q}_{i})}^{2}}{{({P}_{i})}^{2}}+\boldsymbol{ }{({B}_{i})}^{2}-\boldsymbol{ }2\left(\frac{{a}_{i}}{{P}_{i}}\right){Q}_{i}{B}_{i}$$

(B.2)

$${{(I}_{i})}^{2}={\left(\frac{{a}_{i}}{{P}_{i}}\right)}^{2}{{Q}_{i}}^{2}+\boldsymbol{ }{{(B}_{i})}^{2}-\boldsymbol{ }2\left(\frac{{a}_{i}}{{P}_{i}}\right){Q}_{i}{B}_{i}$$

(B.3)

and

$${H}_{i}^{1}={I}_{i}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\frac{{Q}_{i}}{{ P}_{i}}$$

(B.4)

$${({H}_{i}^{1})}^{2}={({I}_{i})}^{2}+ {(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}})}^{2}{(\frac{{Q}_{i}}{{ P}_{i}})}^{2}+ 2(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\frac{{Q}_{i}}{{ P}_{i}})({I}_{i})$$

(B.5)

$${({H}_{i}^{1})}^{2}={\left(\frac{{a}_{i}}{{P}_{i}}\right)}^{2}{{Q}_{i}}^{2}+ {{(B}_{i})}^{2}- 2\left(\frac{{a}_{i}}{{P}_{i}}\right){Q}_{i}{B}_{i}+ {(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}})}^{2}{(\frac{{Q}_{i}}{{ P}_{i}})}^{2}+ 2{a}_{i}\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\right){\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}- 2(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}})(\frac{{Q}_{i}}{{ P}_{i}}){B}_{i}$$

(B.6)

Also

$${H}_{i}^{2}={H}_{i}^{1}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{y}_{i}^{2}}{{V}_{i}^{2}}\frac{{Q}_{i}}{{ P}_{i}}$$

(B.7)

$${{(H}_{i}^{2})}^{2}={{(H}_{i}^{1})}^{2}+{(\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{y}_{i}^{2}}{{V}_{i}^{2}})}^{2} ({\frac{{Q}_{i}}{{ P}_{i}})}^{2}+ 2 \left(\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{y}_{i}^{2}}{{V}_{i}^{2}}\right)\left(\frac{{Q}_{i}}{{ P}_{i}}\right){(H}_{i}^{1})$$

(B.8)

$${{(H}_{i}^{2})}^{2}= {\left(\frac{{a}_{i}}{{P}_{i}}\right)}^{2}{{Q}_{i}}^{2}+ {{(B}_{i})}^{2}- 2\left(\frac{{a}_{i}}{{P}_{i}}\right){Q}_{i}{B}_{i}+ {(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}})}^{2}{(\frac{{Q}_{i}}{{ P}_{i}})}^{2}+ 2{a}_{i}\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\right){\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}- 2\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\right)\left(\frac{{Q}_{i}}{{ P}_{i}}\right){B}_{i}+ {(\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{y}_{i}^{2}}{{V}_{i}^{2}})}^{2} ({\frac{{Q}_{i}}{{ P}_{i}})}^{2}+ 2{a}_{i}\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\right){\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}-2\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}}{{V}_{i}^{1}}\right)\left(\frac{{Q}_{i}}{{ P}_{i}}\right){B}_{i}+ 2\left(\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{y}_{i}^{1}\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{y}_{i}^{2}}{{V}_{i}^{1}{V}_{i}^{2}}\right){\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}$$

(B.9)

Therefore

$$({{H}_{i}^{m})}^{2}= \left({\left(\frac{{a}_{i}}{{P}_{i}}\right)}^{2}{{Q}_{i}}^{2}+ {{(B}_{i})}^{2}- 2\left(\frac{{a}_{i}}{{P}_{i}}\right){Q}_{i}{B}_{i}+{\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}+ \sum_{j=1}^{m}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)}^{2}+2{a}_{i}{\left(\frac{{Q}_{i}}{{ P}_{i}}\right)}^{2}\sum_{j=1}^{m}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)}^{2}\right)$$

(B.10)

Then, based on Eqs. (2–6), we have:${CH}_{a}=\sum_{i=1}^{n}\left(\left(\frac{{{h}_{i}a}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+\left({{h}_{i}a}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)\right)\left(T\right)+\left(\frac{{h}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{i=1}^{m}\left({\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}{y}_{i}^{j}\right)\right)\left(T\right)+ \left({h}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+ \left(\frac{{a}_{i}{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left(T\right)+ \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{1}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}{\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)}^{2}\right)\left(T\right)+ \frac{{h}_{i}}{2{D}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right) + \frac{{h}_{i}}{2{a}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right)- \left(\frac{{h}_{i}{a}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right) - \left(\frac{{h}_{i}}{{ P}_{i} (1-{\theta }_{i})}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left({B}_{i}\right)- \left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right)- \left(\left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\sum_{j=1}^{{\varvec{m}}}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)}^{2}\right)\left({B}_{i}\right)\right)$

According to Eq. (25), ${CH}_{b}$ is:

$${CH}_{b }=\frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+ {\theta }_{i}\right){Q}_{i }\left({t}_{i}^{0}+ {t}_{i}^{1}\right)+ \left(\left(1-{\sigma }_{i}\right){e1}_{i}^{j}+ {\alpha }_{i}^{1}\right)\left(1- {\gamma }_{i}^{1}\right){Q}_{i}{t}_{i}^{2}+ \left(\left(1-{\sigma }_{i}\right){e1}_{i}^{j}+ {\alpha }_{i}^{2}\right)\left(1- {\gamma }_{i}^{2}\right){Q}_{i}{t}_{i}^{3}+ \dots + \left(\left(1-{\sigma }_{i}\right){e1}_{i}^{j}+ {\alpha }_{i}^{m}\right)\left(1- {\gamma }_{i}^{m}\right){Q}_{i}{t}_{i}^{m+1}\right]$$

(B.12)

Base on Eqs. (1)–(5) we have:

$${CH}_{b }= \frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+ {\theta }_{i}\right)\frac{{{Q}_{i }}^{2}}{{ P}_{i}}+\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{1}+ {\alpha }_{i}^{1}\right)\left(1- {\gamma }_{i}^{1}\right)\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{{Q}_{i}}^{2}}{{V}_{i}^{1}{ P}_{i}} + \left(\left(1-{\sigma }_{i}\right){e1}_{i}^{2}+ {\alpha }_{i}^{2}\right)\left(1- {\gamma }_{i}^{2}\right)\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{{Q}_{i}}^{2}}{{V}_{i}^{2}{ P}_{i}}+ \dots + \left(\left(1-{\sigma }_{i}\right){e1}_{i}^{m}+ {\alpha }_{i}^{m}\right)\left(1- {\gamma }_{i}^{m}\right)\frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{{Q}_{i}}^{2}}{{V}_{i}^{m}{ P}_{i}}\right]$$

(B.13)

$${CH}_{b }= \frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+ {\theta }_{i}\right)\frac{{{Q}_{i }}^{2}}{{ P}_{i}}+{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{1}+ {\alpha }_{i}^{1}\right)}^{2}\left(1- {\gamma }_{i}^{1}\right)\frac{{{Q}_{i}}^{2}}{{V}_{i}^{1}{ P}_{i}} + {\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{2}+ {\alpha }_{i}^{2}\right)}^{2}\left(1- {\gamma }_{i}^{2}\right)\frac{{{Q}_{i}}^{2}}{{V}_{i}^{2}{ P}_{i}}+ \dots + {\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{m}+ {\alpha }_{i}^{m}\right)}^{2}\left(1- {\gamma }_{i}^{m}\right)\frac{{{Q}_{i}}^{2}}{{V}_{i}^{m}{ P}_{i}}\right]$$

(B.14)

$${CH}_{b }= \frac{1}{2T}\sum_{i=1}^{n}\frac{{h}_{i }{{Q}_{i}}^{2}}{{ P}_{i}}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+ {\theta }_{i}\right)+\frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{1}+ {\alpha }_{i}^{1}\right)}^{2}\left(1- {\gamma }_{i}^{1}\right)}{{V}_{i}^{1}} + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{2}+ {\alpha }_{i}^{2}\right)}^{2}\left(1- {\gamma }_{i}^{2}\right)}{{V}_{i}^{2}}+ \dots + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{m}+ {\alpha }_{i}^{m}\right)}^{2}\left(1- {\gamma }_{i}^{m}\right)}{{V}_{i}^{m}}\right]$$

(B.15)

Using Eq. (14), we have:

$${CH}_{b }= \frac{T}{2}\sum_{i=1}^{n}\frac{{h}_{i }{{D}_{i}}^{2}}{{\left\{\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right\}}^{2}}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+{\theta }_{i}\right)+\frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{1}+ {\alpha }_{i}^{1}\right)}^{2}\left(1- {\gamma }_{i}^{1}\right)}{{V}_{i}^{1}} + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{2}+ {\alpha }_{i}^{2}\right)}^{2}\left(1- {\gamma }_{i}^{2}\right)}{{V}_{i}^{2}}+ \dots + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{m}+ {\alpha }_{i}^{m}\right)}^{2}\left(1- {\gamma }_{i}^{m}\right)}{{V}_{i}^{m}}\right]$$

(B.16)

According to Eq. (26), ${CH}_{c}$ is:

$${CH}_{c}= \frac{1}{2}\sum_{i=1}^{n}{h}_{i}\left[{\alpha }_{i}{e2}_{i}^{j}{Q}_{i}\right]$$

(B.17)

Using Eq. (14), we have:

$${CH}_{c}= \frac{T}{2}\sum_{i=1}^{n}{h}_{i}\left[\frac{{\alpha }_{i}{e2}_{i}^{j}{D}_{i}}{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right]$$

(B.18)

According to Eq. (27), ${CH}_{d}$ is:

$${CH}_{d}= \frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[\left[\left(\left(1- {\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\right)+ \left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)\right]{Q}_{i}{t}_{i}^{2}+ \left[\left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)+ \left(\sum_{j=3}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=3}^{m}{e1}_{i}^{j}\right)\right]{Q}_{i}{t}_{i}^{3}+\dots + \left[\sum_{j=m}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=m}^{m}{e1}_{i}^{j}\right]{Q}_{i}{t}_{i}^{m+1}\right]$$

(B.19)

According to Eqs. (3–5), we have:

$${CH}_{d}= \frac{1}{2T}\sum_{i=1}^{n}{h}_{i}\left[\left[\left(\left(1- {\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\right)+ \left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)\right]\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]\frac{{{Q}_{i}}^{2}}{{V}_{i}^{1}{ P}_{i}}+ \left[\left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)+ \left(\sum_{j=3}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=3}^{m}{e1}_{i}^{j}\right)\right]\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]\frac{{{Q}_{i}}^{2}}{{V}_{i}^{2}{ P}_{i}}+\dots + \left[\sum_{j=m}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=m}^{m}{e1}_{i}^{j}\right]\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]\frac{{{Q}_{i}}^{2}}{{V}_{i}^{m}{ P}_{i}}\right]$$

(B.20)

Then, using Eq. (14), we have:

$${CH}_{d}= \frac{T}{2}\sum_{i=1}^{n}\frac{{h}_{i}{{D}_{i}}^{2}}{{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}^{2}}\left[\left[\left(\left(1- {\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\right)+ \left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)\right]\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]}{{V}_{i}^{1}} + \left[\left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)+ \left(\sum_{j=3}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=3}^{m}{e1}_{i}^{j}\right)\right]\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]}{{V}_{i}^{2}} +\dots + \left[\sum_{j=m}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=m}^{m}{e1}_{i}^{j}\right]\frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]}{{V}_{i}^{m}}\right]$$

(B.21)

Therefore,

$$CH=\sum_{i=1}^{n}\left(\left(\frac{{{h}_{i}a}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+\left({{h}_{i}a}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)\right)\left(T\right)+\left(\frac{{h}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{i=1}^{m}\left({\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}{y}_{i}^{j}\right)\right)\left(T\right)+ \left({h}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+ \left(\frac{{a}_{i}{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left(T\right)+ \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{1}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}{\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)}^{2}\right)\left(T\right)+ \frac{{h}_{i}}{2{D}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right) + \frac{{h}_{i}}{2{a}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right)- \left(\frac{{h}_{i}{a}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right) - \left(\frac{{h}_{i}}{{ P}_{i} (1-{\theta }_{i})}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left({B}_{i}\right)- \left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right)- \left(\left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\sum_{j=1}^{{\varvec{m}}}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)}^{2}\right)\left({B}_{i}\right)\right)+ \frac{T}{2}\sum_{i=1}^{n}\frac{{h}_{i }{{D}_{i}}^{2}}{{\left\{\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right\}}^{2}}\left[\left(\left(1-{\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\left(1- {e2}_{i}^{j}\right)+{\theta }_{i}\right)+\frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{1}+ {\alpha }_{i}^{1}\right)}^{2}\left(1- {\gamma }_{i}^{1}\right)}{{V}_{i}^{1}} + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{2}+ {\alpha }_{i}^{2}\right)}^{2}\left(1- {\gamma }_{i}^{2}\right)}{{V}_{i}^{2}}+ \dots + \frac{{\left(\left(1-{\sigma }_{i}\right){e1}_{i}^{m}+ {\alpha }_{i}^{m}\right)}^{2}\left(1- {\gamma }_{i}^{m}\right)}{{V}_{i}^{m}}\right]+ \frac{T}{2}\sum_{i=1}^{n}{h}_{i}\left[\frac{{\alpha }_{i}{e2}_{i}^{j}{D}_{i}}{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right]+ \frac{T}{2}\sum_{i=1}^{n}\frac{{h}_{i}{{D}_{i}}^{2}}{{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}^{2}}\left[\left[\left(\left(1- {\sigma }_{i}\right){e1}_{i}+ {\alpha }_{i}\right)+ \left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)\right]\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]}{{V}_{i}^{1}} + \left[\left(\sum_{j=2}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=2}^{m}{e1}_{i}^{j}\right)+ \left(\sum_{j=3}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=3}^{m}{e1}_{i}^{j}\right)\right]\frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]}{{V}_{i}^{2}} +\dots + \left[\sum_{j=m}^{m}{\alpha }_{i}^{j}+ \left(1- {\sigma }_{i}\right)\sum_{j=m}^{m}{e1}_{i}^{j}\right]\frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]}{{V}_{i}^{m}}\right]$$

(B.22)

Appendix C: Determining the machine capacity constraint

$$\sum_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{ t}}_{\mathrm{i}}^{0}+ {\mathrm{t}}_{\mathrm{i}}^{1}+ {\mathrm{t}}_{\mathrm{i}}^{2}+ {\mathrm{t}}_{\mathrm{i}}^{3}+ \dots + {\mathrm{t}}_{\mathrm{i}}^{\mathrm{m}+1}\right)+ \sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{i}} \le \mathrm{T}$$

(C.1)

Based on Eqs. (1)–(5), we have:

$$\sum_{i=1}^{n}\left(\frac{{Q}_{i}}{{P}_{i}}+ \frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]{Q}_{i}}{{V}_{i}^{1}{ P}_{i}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]{Q}_{i}}{{V}_{i}^{2}{ P}_{i}}\quad +\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]{Q}_{i}}{{V}_{i}^{m}{ P}_{i}}\right)+ \sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{i}} \le \mathrm{T}$$

(C.2)

$$\sum_{i=1}^{n}\frac{{Q}_{i}}{{P}_{i}}\left(1+\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]}{{V}_{i}^{1}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]}{{V}_{i}^{2}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]}{{V}_{i}^{m}}\right)+ \sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{i}} \le \mathrm{T}$$

(C.3)

By Inserting Eq. (14), the Machine capacity constraint is determined as follow.

$$\sum_{i=1}^{n}\frac{{D}_{i} T }{\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right){ P}_{i}}\left(1+\frac{\left[{\alpha }_{i}^{1}+\left(1-{\sigma }_{i}\right){e1}_{i}^{1}\right]}{{V}_{i}^{1}}+ \frac{\left[{\alpha }_{i}^{2}+\left(1-{\sigma }_{i}\right){e1}_{i}^{2}\right]}{{V}_{i}^{2}}+\dots + \frac{\left[{\alpha }_{i}^{m}+\left(1-{\sigma }_{i}\right){e1}_{i}^{m}\right]}{{V}_{i}^{m}}\right)+ \sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{i}} \le \mathrm{T}$$

(C.4)