Integrated rescheduling and preventive maintenance for arrival of new jobs through evolutionary multi-objective optimization

Abstract

In this paper, we study a rescheduling problem in response to arrival of new jobs in single machine layout, where preventive maintenance should be determined. Preventive maintenance together with controllable processing time could alleviate the inherent deteriorating effect in manufacturing system. Processing sequence of original and new jobs, compression of each job, and position of maintenance should be optimized simultaneously with regards to total operational cost (job’s total completion times, maintenance cost and compression cost) and total completion time deviation. An improved elitist non-dominated sorting genetic algorithm (NSGA-II) has been proposed to solve the rescheduling problem. To address the key problem of balancing between exploration and exploitation, we hybridize differential evolution mutation operation with NSGA-II to enhance diversity, constitute high-quality initial solution based on assignment model for exploitation, and incorporate analytic property of non-dominated solutions for exploration. Finally computational study is designed by randomly generating various instances with regards to the problem size from given distributions. By use of existing performance indicators for convergence and diversity of Pareto fronts, we illustrate the effectiveness of the hybrid algorithm and the incorporation of domain knowledge into evolutionary optimization in rescheduling.

Appendices

Appendix A. Proof of Lemma 1

Proof

The total operational cost $\mathop {\sum }C_j +\mathop {\sum }c_j u_j \,+\,qM$ could be formulated as follows:

$$\begin{aligned}&{\mathop {\sum }C_j +\mathop {\sum }c_j u_j +qM} =\mathop {\sum }\limits _{j=1}^m [( {n_O -j+1})\\&\quad +\,\gamma (n_O-m+q)]p_{[j]}+\mathop {\sum }\limits _{j=m+1}^{n_{O}} (n_O -j+1)p_{[j]}\\&\quad +\mathop {\sum }\limits _{j=1}^{n_O } c_{[j]} u_{[j]} ={\mathop {\sum }\limits _{j=1}^{n_O } \tilde{w}_{j} p_{[j]}+\mathop {\sum }\limits _{j=1}^{n_O } c_{[j]} u_{[j]} } \\ \end{aligned}$$

where the weight for the $j$th position is

$$\begin{aligned} \tilde{w}_j =\left\{ \begin{array}{l@{\quad }l} ( {n_O -j+1})+\gamma ( {n_O -m+q}), &{} 1\le j\le m \\ n_O -j+1, &{} m+1\le j\le n_O \\ \end{array}\right. \end{aligned}$$

(1) For the optimal resource allocation

$$\begin{aligned}&\mathop {\sum }\limits _{j=1}^m \tilde{w}_j p_{[j]} +\mathop {\sum }\limits _{j=m+1}^{n_O } \tilde{w}_j p_{[j]} +\mathop {\sum }\limits _{j=1}^{n_O } c_{[j]} u_{[j]} =\mathop {\sum }\limits _{j=1}^{n_O } W_j ( {\overline{p}_{[j]} -b_{[j]} u_{[j]} })\\&+\mathop {\sum }\limits _{j=1}^{n_O } c_{[j]} u_{[j]} =\mathop {\sum }\limits _{j=1}^{n_O } W_j \overline{p}_{[j]} +\mathop {\sum }\limits _{j=1}^{n_O } ( {c_{[j]} -W_j b_{[j]} })u_{[j]} \end{aligned}$$

where $W_j =\left\{ {{\begin{array}{l} {\mathop {\sum }\limits _{i=j}^m \mathop {\sum }\limits _{k=0}^{i-j} R_{i-j+1,k+1} \alpha ^k\tilde{w}_i ,\quad 1\!\le \! j\le m} \\ \mathop {\sum }\limits _{i=j}^{n_O } \mathop {\sum }\limits _{k=0}^{i-j} R_{i-j+1,k+1} \alpha ^k\tilde{w}_i ,\quad m\!+\!1\!\le \!j\le n_O\\ \end{array} }} \right.$, and $R$ is a matrix of size $n_O \times n_O$. We have $R_{1,1} =1$ and $R_{1,j} = 0, R_{i,1} =0, R_{i,j} =R_{i-1,j-1} +R_{i-1,j}$ for $i,j=2,3,\ldots ,n_O$.

From the above formulation, we could easily conclude that for the schedule where maintenance takes place after the completion of the $m$th job, if $c_{\left[ j \right] } \ge W_j b_{\left[ j \right] }$, the optimal resource allocation is $u_{\left[ j \right] } =0$; and if $c_{\left[ j \right] } <W_j b_{\left[ j \right] }$, the optimal resource allocation is $u_{\left[ j \right] } =\overline{u}_{\left[ j \right] }$.

(2) For the optimal maintenance position and jobs’ processing sequence

Let

$$\begin{aligned} C_{jr} =\left\{ {{\begin{array}{ll} W_r \overline{p}_j ,&{}\quad r=1,2,\ldots ,n_O ;c_j \ge W_r b_j \\ W_r \overline{p}_j +( {c_j -W_r b_j })\overline{u}_j, &{}\quad r=1,2,\ldots ,n_O ;c_j <W_r b_j \\ \end{array} }} \right. \end{aligned}$$

where $W_r \!=\!\left\{ {{\begin{array}{ll} \mathop {\sum }\limits _{i=r}^m \mathop {\sum }\limits _{k=0}^{i-r} R_{i\!-\!r\!+\!1,k\!+\!1} \alpha ^k\tilde{w}_i, &{}\quad 1\le r\le m\\ \mathop {\sum }\limits _{i\!=\!r}^{n_O } \mathop {\sum }\limits _{k=0}^{i-r} R_{i\!-\!r\!+\!1,k\!+\!1} \alpha ^k\tilde{w}_i, &{}\quad m+1\le r\le n_O\\ \end{array}}}\right.$, and the weight for position $i$ is

$\tilde{w}_i =\left\{ {{\begin{array}{ll} ( {n_O -i+1})\!+\!\gamma ( {n_O \!-\!m\!+\!q}),&{}\quad 1\le i\le m\\ n_O -i+1, &{}\quad m\!+\!1\le i\!\le \! n_O\\ \end{array}}} \right.$.

Here, we introduce binary variable $x_{jr}$ to denote whether job $J_j$ is arranged at the $r$th position: $x_{jr} =1$ means ‘yes’ and $x_{jr} =0$ means ‘no’, $j,r=1,2,\ldots ,n_O$. Then the problem could be transferred into the following assignment problem $A(m)$:

$$\begin{aligned} \min&\mathop {\sum }\limits _{r=1}^{n_O } \mathop {\sum }\limits _{j=1}^{n_O } C_{jr} x_{jr}\\ s.t.&\mathop {\sum }\limits _{r=1}^{n_O } x_{jr} =1,\qquad j=1,2,\ldots ,n_O\\&\mathop {\sum }\limits _{j=1}^{n_O } x_{jr} =1, \qquad r=1,2,\ldots ,n_O\\&x_{jr} \in \left\{ {0,1} \right\} ,\qquad \qquad r,j=1,2,\ldots ,n_O \end{aligned}$$

The time complexity of solving the above model is $O(n_O^3 )$. Note that the obtained optimal objective value is based on fixing $m$. Let $m$ take the value of 1,2,...$n_O$ and solve $n_O$ assignment, respectively, then pick out the optimal objective value. The corresponding maintenance position, jobs’ processing sequence, and job’s compression could be integrated to form the optimal baseline schedule. The time complexity of the entire procedure would be $O(n_O^4 )$, therefore, Lemma 1 holds. $\square$

Appendix B. Proof of Lemma 2

Proof

Given resources allocation, in the optimal schedule $\sigma$ of problem (P), let $J_{\left[ j \right] }$ denote the job with minimum position that does not satisfy the above SPT rule, and let $J_{[i]}$ denote the job which immediately precedes $J_{\left[ j \right] }$ in $\sigma$. We have $j>i$ but $\overline{p}_{\left[ j \right] } -b_{\left[ j \right] } u_{\left[ j \right] } <\overline{p} _{\left[ i \right] } -b_{\left[ i \right] } u_{\left[ i \right] }$. Let the starting time of $J_{[i]}$ be $t_0$. By exchanging the processing order of these two jobs, we could obtain a new schedule $\tilde{\sigma }$, and we have $C_{[j]} ( {\tilde{\sigma }})=t_0 +\overline{p}_{[j]} +\alpha t_0 -b_{[j]} u_{[j]} <C_{[i]} ( \sigma )$. We also have $t_{[i]} ( {\tilde{\sigma }})<t_{[j]} ( \sigma )$, where $t_{[i]} ( {\tilde{\sigma }})$ is the starting time of job $J_{[i]}$ in $\tilde{\sigma }$, and $t_{[j]} ( \sigma )$ is the starting time of job $J_{[i]}$ in $\sigma$. Then it holds that $t_0 +\overline{p}_{[j]} +\alpha t_0 -b_{[j]} u_{[j]} +\overline{p} _{[i]} +\alpha t_{[i]} ( {\tilde{\sigma }})-b_{[i]} u_{[i]} <t_0 +\overline{p}_{[i]} +\alpha t_0 -b_{[i]} u_{[i]} +\overline{p}_{[j]} +\alpha t_{[j]} ( \sigma )-b_{[j]} u_{[j]}$, meaning $C_{[i]} ( {\tilde{\sigma }})<C_{[j]} ( \sigma )$. Therefore, we could conclude that after the exchanging, the two jobs’ completion times have both been reduced, and this is contradictory to the optimality of schedule $\sigma$. Lemma 2 holds. $\square$

Appendix C. Proof of Theorem 1

Proof

In the Pareto optimal solution denoted by $\sigma$ with given resource allocation and maintenance position $m$, let $J_{\left[ j \right] } (j<m)$ be the original job with minimum position that violates SPT rule, and let $J_{[i]}$ be the last original job that is arranged before $J_{\left[ j \right] }$. We have $j>i$ and $\overline{p}_{\left[ j \right] } -b_{\left[ j \right] } u_{\left[ j \right] } <\overline{p}_{\left[ i \right] } -b_{\left[ i \right] } u_{\left[ i \right] }$. Denote the starting time of $J_{[i]}$ in $\sigma$ as $t_0$, and $\tilde{J}_1,\tilde{J}_2,\ldots , \tilde{J}_h$ stand for the new jobs arranged between $J_{[i]}$ and $J_{[j]}$ in $\sigma$. By exchanging the positions of $J_{[i]}$ and $J_{[j]}$, we could obtain a new schedule $\tilde{\sigma }$, and analyze the change of total operational cost and deviation after exchanging as follows.

(1) For the change of total operational cost.

First we have $C_{[j]} ( {\tilde{\sigma }})=t_0 +\overline{p} _{[j]} +\alpha t_0 -b_{[j]} u_{[j]} <C_{[i]} ( \sigma )$, and jobs $\widetilde{J_1 },\widetilde{J_2 },\ldots ,\widetilde{J_h }$ could start earlier in $\tilde{\sigma }$, therefore, would be completed earlier compared with in $\sigma$. Let $t_{[i]} ( {\tilde{\sigma }})$ and $t_{[j]} ( \sigma )$ be the starting times of job $J_{[i]}$ in $\tilde{\sigma }$ and job $J_{[j]}$ in $\sigma$, respectively, and we have $t_{[i]} ( {\tilde{\sigma }})<t_{[j]} ( \sigma )$. Then it holds that $t_0 \,+\,\overline{p}_{[j]} +\alpha t_0\, -\,b_{[j]} u_{[j]} +\overline{p}_{[i]} \,+\,\alpha t_{[i]} ( {\tilde{\sigma }})-b_{[i]} u_{[i]} <t_0 +\overline{p}_{[i]} +\alpha t_0 -b_{[i]} u_{[i]} +\overline{p}_{[j]} +\alpha t_{[j]} ( \sigma )-b_{[j]} u_{[j]}$. Therefore, $C_{[i]} ( {\tilde{\sigma }})<C_{[j]} ( \sigma )$. As to the optimal maintenance position, $C_{[m]} ( {\tilde{\sigma }})<C_{[m]} ( \sigma )$ and the duration of maintenance could be reduced, leading to lower maintenance cost. To summarize, from $\sigma$ to $\tilde{\sigma }$ with compression staying constant, the total operational cost could be reduced.

(2) For the change of deviation.

We have $T_{[i]} ( {\tilde{\sigma }})=\mathrm{Max} \left\{ \! {C_{[i]} ( {\tilde{\sigma }})-\overline{C}_{[i]} ,0} \right\} ,T_{[j]} ( {\tilde{\sigma }})\!=\!\mathrm{Max} \left\{ \! {C_{[j]} ( {\tilde{\sigma }})-\overline{C} _{[j]} ,0} \right\} , T_{[i]} ( \sigma )\!=\!\mathrm{Max} \left\{ \! {C_{[i]} ( \sigma )-\overline{C}_{[i]} ,0} \right\}$, $T_{[j]} ( \sigma )\!=\!\mathrm{Max}\left\{ {C_{[j]} ( \sigma )-\overline{C}_{[j]} ,0} \right\}$, then $T_{[i]} ( {\tilde{\sigma }})+T_{[j]} ( {\tilde{\sigma }})-(T_{\left[ i \right] } ( \sigma )+T_{[j]} ( \sigma ))=\mathrm{Max} \left\{ {C_{[i]} ( {\tilde{\sigma }})-\overline{C}_{[i]} ,0} \right\} -\mathrm{Max} \left\{ {C_{\left[ j \right] } ( \sigma )-\overline{C} _{\left[ j \right] } ,0} \right\} +\mathrm{Max} \left\{ {C_{\left[ j \right] } ( {\tilde{\sigma }})-\overline{C}_{\left[ j \right] } ,0} \right\} -\mathrm{Max} \left\{ {C_{[i]} ( \sigma )-\overline{C}_{[i]} ,0} \right\}$. Combining $C_{[j]} ( {\tilde{\sigma }})\!<\!C_{[i]} ( \sigma )\!<\!C_{[i]} ( {\tilde{\sigma }})<C_{[j]} ( \sigma )$ with $\overline{C}_{[j]} <\overline{C}_{[i]}$ derived from $\overline{p}_{\left[ j \right] } -b_{\left[ j \right] } u_{\left[ j \right] } <\overline{p} _{\left[ i \right] } -b_{\left[ i \right] } u_{\left[ i \right] }$ (Lemma 2), we could easily conclude that $T_{[i]} ( {\tilde{\sigma }})+T_{[j]} ( {\tilde{\sigma }})-(T_{\left[ i \right] } ( \sigma )+T_{[j]} ( \sigma ))\le 0$. As to jobs between $J_{[i]}$ and $J_{[j]}$ and after position $m$, since they are all completed earlier after exchanging, their virtual tardiness would not be increased.

To summarize, by repeating the above procedures, we could finally obtain a partial SPT schedule for original jobs before maintenance. The two objectives would not be increased during this procedure. By use of similar analysis, we could obtain similar conclusions for new jobs before maintenance, original and new jobs after maintenance, respectively. Therefore, Theorem 1 holds.

Appendix D. Proof of Theorem 2

Proof

Assume that in the optimal solution, the resource allocation for the job in the $k$th position is $u_{\left[ k \right] } \in \left[ {0,\overline{u}_{\left[ k \right] } } \right]$. Given a small resource allocation deviation $\delta \in \left[ {-u_{\left[ k \right] } ,\overline{u}_{\left[ k \right] } -u_{\left[ k \right] } } \right]$, we would analyze the change of two objectives from $u_{\left[ k \right] }$ to $u_{\left[ k \right] } +\delta$. Denote the optimal maintenance position as $m$, and for the purpose of convenience, denote $\overline{C}_j =+\infty$ for $J_j \in \left\{ {J_{n_O +1} ,J_{n_O +2} ,\ldots ,J_{n_O +n_N } } \right\}$.

(1) The change of $\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } C_j +\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } c_j u_j +qM$ from $u_{\left[ k \right] }$ to $u_{\left[ k \right] } +\delta$ contains two cases:

Case 1: $m<k$,

$$\begin{aligned} \begin{array}{l} -b_{\left[ k \right] } \delta +\left[ {-( {\alpha +1})b_{\left[ k \right] } \delta } \right] +\left[ {-( {\alpha +1})^2b_{\left[ k \right] } \delta } \right] \\ \quad +\cdots +\left[ {-( {\alpha +1})^{n_O +n_N -k}b_{\left[ k \right] } \delta } \right] +c_{\left[ k \right] } \delta \\ \quad =\left( {c_{\left[ k \right] } -\mathop {\sum }\limits _{j=k}^{n_O +n_N } ( {\alpha +1})^{j-k}b_{\left[ k \right] } }\right) \delta \\ \end{array} \end{aligned}$$

Case 2: $m\ge k$,

$$\begin{aligned} \begin{array}{l} -b_{\left[ k \right] } \delta +\left[ {-( {\alpha +1})b_{\left[ k \right] } \delta } \right] +\left[ {-( {\alpha +1})^2b_{\left[ k \right] } \delta } \right] \\ \quad +\cdots +\left[ {-( {\alpha +1})^{m-k}b_{\left[ k \right] } \delta } \right] +\left[ {-( {1+\gamma })( {\alpha +1})^{m+1-k}b_{\left[ k \right] } \delta } \right] \\ \quad +\cdots +\left[ {-( {1+\gamma })( {\alpha +1})^{n_O +n_N -k}b_{\left[ k \right] } \delta } \right] +c_{\left[ k \right] } \delta \\ \quad -q\gamma ( {\alpha +1})^{m-k}b_{\left[ k \right] } \delta \\ =\left( c_{\left[ k \right] } -\mathop {\sum }\limits _{j=k}^m ( {\alpha +1})^{j-k}b_{\left[ k \right] }\right. \\ \quad \left. -\mathop {\sum }\limits _{j=m+1}^{n_O +n_N } ( {1+\gamma })( {\alpha +1})^{j-k}b_{\left[ k \right] } -q\gamma ( {\alpha +1})^{m-k}b_{\left[ k \right] } \right) \delta \\ \end{array} \end{aligned}$$

(2) The change of $\mathop {\sum }\nolimits _{j=1}^{n_O } T_j$ contains 4 cases:

Case 1: $m<k$ and $\delta \ge 0$,

$$\begin{aligned} \begin{array}{l} \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k]} -C_{\left[ k \right] } ,0} \right\} ,-b_{\left[ k \right] } \delta } \right\} \\ \qquad +\,\mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k+1]} \!-\!C_{\left[ {k+1} \right] } ,0} \right\} ,\!-\!( {\alpha \!+\!1})b_{\left[ k \right] } \delta } \right\} \\ \qquad +\cdots +\mathrm{Max}\left\{ \mathrm{min}\left\{ {\overline{C}_{[n_O +n_N ]} -C_{[n_O +n_N ]} ,0} \right\} ,\right. \\ \qquad \left. -( {\alpha +1})^{n_O +n_N -k}b_{\left[ k \right] } \delta \right\} \\ \quad =\mathop {\sum }\limits _{j=k}^{n_O +n_N } \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[j]} -C_{[j]} ,0} \right\} ,-( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta } \right\} \\ \end{array} \end{aligned}$$

Case 2: $m<k$ and $\delta <0$,

$$\begin{aligned} \begin{array}{l} \mathrm{Max}\left\{ {\mathrm{min}\left\{ {C_{\left[ k \right] } -\overline{C}_{\left[ k \right] } ,0} \right\} -b_{\left[ k \right] } \delta ,0} \right\} \\ \qquad +\mathrm{Max}\left\{ {\mathrm{min}\left\{ {C_{\left[ {k+1} \right] } -\overline{C} _{\left[ {k+1} \right] } ,0} \right\} -( {\alpha +1})b_{\left[ k \right] } \delta ,0} \right\} \\ \qquad +\cdots +\mathrm{Max}\left\{ \mathrm{min}\left\{ {C_{[n_O +n_N ]} -\overline{C}_{\left[ {n_O +n_N } \right] } ,0} \right\} \right. \\ \qquad \left. -( {\alpha \!+\!1})^{n_O +n_N -k}b_{\left[ k \right] } \delta ,0 \right\} \\ \quad \!=\!\mathop {\sum }\limits _{j=k}^{n_O +n_N } \mathrm{Max}\left\{ {\mathrm{min}\left\{ {C_{[j]} -\overline{C}_{\left[ j \right] } ,0} \right\} -( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta ,0} \right\} \\ \end{array} \end{aligned}$$

Case 3: $m\ge k$ and $\delta \ge 0$,

$$\begin{aligned} \begin{array}{l} \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k]} -C_{\left[ k \right] } ,0} \right\} ,-b_{\left[ k \right] } \delta } \right\} + \\ \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k+1]} -C_{\left[ {k+1} \right] } ,0} \right\} ,-( {\alpha +1})b_{\left[ k \right] } \delta } \right\} \\ \quad +\cdots +\mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ m \right] } -C_{\left[ m \right] } ,0} \right\} ,-( {\alpha +1})^{m-k}b_{\left[ k \right] } \delta }\right\} \\ \quad +\mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ {m+1} \right] } -C_{\left[ {m+1} \right] } ,0} \right\} ,-( {1+\gamma })( {\alpha +1})^{m+1-k}b_{\left[ k \right] } \delta } \right\} \\ \quad +\cdots +\mathrm{Max}\left\{ \mathrm{min}\left\{ {\overline{C}_{\left[ {n_O +n_N } \right] } -C_{[n_O +n_N ]} ,0} \right\} ,\right. \\ \quad \left. -( {1+\gamma })( {\alpha +1})^{n_O +n_N -k}b_{\left[ k \right] } \delta \right\} \\ =\mathop {\sum }\limits _{j=k}^m \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ j \right] } -C_{\left[ j \right] } ,0} \right\} ,-( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta } \right\} \\ \quad +\mathop {\sum }\limits _{j=m+1}^{n_O +n_N } \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ j \right] } -C_{\left[ j \right] } ,0} \right\} ,-( {1+\gamma })( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta } \right\} \\ \end{array} \end{aligned}$$

Case 4: $m\ge k$ and $\delta <0$,

$$\begin{aligned} \begin{array}{l} \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k]} -C_{\left[ k \right] } ,0} \right\} -b_{\left[ k \right] } \delta ,0} \right\} \\ \quad +\mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{[k+1]} -C_{\left[ {k+1} \right] } ,0} \right\} -( {\alpha +1})b_{\left[ k \right] } \delta ,0} \right\} +\cdots \\ \quad +\mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ m \right] } -C_{\left[ m \right] } ,0} \right\} -( {\alpha +1})^{m-k}b_{\left[ k \right] } \delta ,0} \right\} \\ \quad +\mathrm{Max}\left\{ \mathrm{min}\left\{ {\overline{C}_{\left[ {m+1} \right] } -C_{\left[ {m+1} \right] } ,0} \right\} \right. \\ \quad \left. \quad -( {1+\gamma })( {\alpha +1})^{m+1-k}b_{\left[ k \right] } \delta ,0 \right\} \\ \quad +\cdots +\mathrm{Max}\left\{ \mathrm{min}\left\{ {\overline{C}_{\left[ {n_O +n_N } \right] } -C_{[n_O +n_N ]} ,0} \right\} \right. \\ \quad \left. -( {1+\gamma })( {\alpha +1})^{n_O +n_N -k}b_{\left[ k \right] } \delta ,0 \right\} \\ =\mathop {\sum }\limits _{j=k}^m \mathrm{Max}\left\{ {\mathrm{min}\left\{ {\overline{C}_{\left[ j \right] } -C_{\left[ j \right] } ,0} \right\} -( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta ,0} \right\} \\ \quad +\mathop {\sum }\limits _{j=m+1}^{n_O +n_N } \mathrm{Max}\left\{ \mathrm{min}\left\{ {\overline{C}_{\left[ j \right] } -C_{\left[ j \right] } ,0} \right\} \right. \\ \quad \left. -( {1+\gamma })( {\alpha +1})^{j-k}b_{\left[ k \right] } \delta ,0 \right\} \\ \end{array} \end{aligned}$$

To summarize the two cases of $\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } C_j +\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } c_j u_j +qM$ and four cases of $\mathop {\sum }\nolimits _{j=1}^{n_O } T_j$, from $u_{\left[ k \right] }$ to $u_{\left[ k \right] } +\delta$, the change of $( {1-a})( {\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } C_j +\mathop {\sum }\nolimits _{j=1}^{n_O +n_N } c_j u_j +qM})+a( {\mathop {\sum }\nolimits _{j=1}^{n_O } T_j })$ would be a linear function of $\delta$. The optimal objective value would be taken at two extreme points $\delta =-u_{\left[ k \right] }$ or $\delta =\overline{u}_{\left[ k \right] } -u_{\left[ k \right] }$. $J_{\left[ k \right] }$ is either compressed to its upper bound or not compressed at all. Therefore, Theorem 2 holds. $\square$