A hybrid differential evolution algorithm with estimation of distribution algorithm for reentrant hybrid flow shop scheduling problem

Abstract

This paper proposes a reentrant hybrid flow shop scheduling problem where inspection and repair operations are carried out as soon as a layer has completed fabrication. Firstly, a scheduling problem domain of reentrant hybrid flow shop is described, and then, a mathematical programming model is constructed with an objective of minimizing total weighted completion time. Then, a hybrid differential evolution (DE) algorithm with estimation of distribution algorithm using an ensemble model (eEDA), named DE–eEDA, is proposed to solve the problem. DE–eEDA incorporates the global statistical information collected from an ensemble probability model into DE. Finally, simulation experiments of different problem scales are carried out to analyze the proposed algorithm. Results indicate that the proposed algorithm can obtain satisfactory solutions within a short time.

Appendix

This section proposes how to construct the lower bound of our problem. The lower bound is got by the Lagrangian relaxation algorithm. The details are as follows.

1.1 Relaxing machine capacity constraints

The machine capacity constraint (5) can be relaxed by using nonnegative Lagrangian multipliers $\lambda_{k,u}$($k = 1, \ldots ,K,$ $u = 1, \ldots ,\left| M \right|$). Then, the relaxed problem RP is as follows:

$$\hbox{min} \left\{ {\sum\limits_{i = 1}^{N} {w_{i} C_{{i,L_{i} ,J_{i} }} + \sum\limits_{u = 1}^{M} {\sum\limits_{k = 1}^{K} {\lambda_{k,u} \times \left[ {\sum\limits_{{(i,j,l) \in O_{u} }} {(\phi (k - C_{i,l,j} + p_{i,l,j} - 1) - \phi (k - C_{i,l,j} - 1)) - 1} } \right]} } } } \right\}$$

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$$\lambda_{k,u} \ge 0,\quad k = 1, \ldots ,K\quad u = 1, \ldots ,\left| M \right|$$

(23)

s.t. (2), (3), (4) and (23).

The objective function of problem RP can be written as:

$$\sum\limits_{i = 1}^{N} {\hbox{min} \left\{ {w_{i} C_{{i,L_{i} ,J_{i} }} + \sum\limits_{l = 1}^{{L_{i} }} {\sum\limits_{j = 1}^{{J_{i} }} {\sum\limits_{{k = C_{i,l,j} - p_{i,l,j} + 1}}^{{C_{i,l,j} }} {\lambda_{{k,m_{i,l,j} }} } } } } \right\} - \sum\limits_{u = 1}^{M} {\sum\limits_{k = 1}^{K} {\lambda_{k,u} } } }$$

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Then, the problem RP can be decomposed into independent job-level subproblems. The job-level subproblem denoted as SP _i can be presented as follows:

$$\hbox{min} \left\{ {w_{i} C_{{i,L_{i} ,J_{i} }} + \sum\limits_{l = 1}^{{L_{i} }} {\sum\limits_{j = 1}^{{J_{i} }} {\sum\limits_{{k = C_{i,l,j} - p_{i,l,j} + 1}}^{{C_{i,l,j} }} {\lambda_{k,u} } } } } \right\}$$

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s.t. (2), (3), (4) and (23).

1.2 Solving job-level subproblems

The subproblems are solved by dynamic programming with precedence constraints embedded in the dynamic programming recursion.

$h_{i,l,j} (u,t)$ represents the cost for completing the operation of job i at layer l at station j in time t on machine u. It is defined as follows:

$$h_{i,l,j} (u,t) = \left\{ \begin{array}{ll} \sum\nolimits_{{x = t - p_{i,l,j} + 1}}^{t} \lambda_{x,u} + w_{i} t&\quad\;{\text{if}}\;l = L_{i} ,j = J_{i} \hfill \\ \sum\nolimits_{{x = t - p_{i,l,j} + 1}}^{t} \lambda_{x,u}&\quad{\text{otherwise}} \hfill \\ \end{array} \right.$$

$$i = 1, \ldots ,N,\quad l = 1, \ldots ,L_{i} ,\quad j = 1, \ldots ,J_{i} ,\quad t = T_{i,l,j} , \ldots ,K,\quad u \in M_{j}$$

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$T_{i,l,j}$ represents the earliest completion time of the operation of job i at layer l at station j. Let $f_{i,l,j} (u,t)$ be the optimal criterion value of state $(u,t)$ for the operation of job i at layer l at station j. Then, the dynamic programming recursion for solving each job-level subproblem is expressed as:

$$\begin{aligned} & f_{i,l,j} (u,t) = \left\{ \begin{array}{ll} h_{i,l,j} (u,t)\;if\;l = 1,j = 1 \hfill \\ h_{i,l,j} (u,t) + \mathop {\hbox{min} }\limits_{{v \in M_{{i,l - 1,J_{i} }} ,T_{{i,l - 1,J_{i} }} \le x \le t - p_{{i,l - 1,J_{i} }} }} f_{{i,l - 1,J_{i} }} (v,x)\quad {\text{if}}\quad l \ne 1,j = 1 \hfill \\ h_{i,l,j} (u,t) + \mathop {\hbox{min} }\limits_{{v \in M_{i,l,j - 1} ,T_{i,l,j - 1} \le x \le t - p_{i,l,j - 1} }} f_{i,l,j - 1} (v,x)&\quad{\text{otherwise}} \hfill \\ \end{array} \right. \\ & i = 1, \ldots ,N,l = 1, \ldots ,L_{i} ,\quad j = 1, \ldots ,J_{i} ,\quad t = T_{i,l,j} , \ldots ,K,\quad u \in M_{j} \\ \end{aligned}$$

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The optimal machine selection and completion time for the operation of job i at layer L _i at station J _i can be obtained recursively by $(u_{{_{{i,L_{i} ,J_{i} }} }}^{ * } ,t_{{_{{i,L_{i} ,J_{i} }} }}^{ * } ) = \arg \mathop {\hbox{min} }\limits_{{u \in M_{{i,L_{i} ,J_{i} }} ,T_{{i,L_{i} ,J_{i} }} \le t \le K}} f_{{i,L_{i} ,J_{i} }} (u,t)$

The optimal machine selection and completion time for the operation of job i at layer $L_{i} - 1$, $L_{i} - 2$,…, 1 at station $J_{i} - 1$, $J_{i} - 2, \ldots ,1$ can be derived recursively by:

$$(u_{{_{i,l,j} }}^{ * } ,t_{{_{i,l,j} }}^{ * } ) = \left\{ \begin{array}{ll} \arg \mathop {\hbox{min} }\limits_{{u \in M_{i,l,j} ,T_{i,l,j} \le t \le t_{i,l + 1,0}^{ * } - p_{i,l + 1,0} }} f_{i,l,j} (u,t)&\quad {\text{if}}\quad l \ne L_{i} ,j = J_{i} \hfill \\ \arg \mathop {\hbox{min} }\limits_{{u \in M_{i,l,j} ,T_{i,l,j} \le t \le t_{i,l,j + 1}^{ * } - p_{i,l,j + 1} }} f_{i,l,j} (u,t)&\quad\;{\text{otherwise}} \hfill \\ \end{array} \right.$$

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1.3 Construction of a feasible solution

The priority list of jobs are created by sorting the job number according to the ascending order of the completion time of the operation for the job at the first station of the first layer from the solution of the relaxed problem. Assign the jobs to the first machine that becomes available successively. Then, for the following stations, update the ready times in each station to be the completion times of the previous station. Arrange the jobs in increasing order of ready times and assign the jobs to the first available machine successively.

1.4 Subgradient algorithm

The Lagrange multipliers are updated by:

$$\lambda_{k,u} = \hbox{max} \left\{ {0,\lambda_{k,u} + \frac{UB - LB}{{\sum\limits_{u = 1}^{M} {\sum\limits_{k = 1}^{K} {h_{k,u}^{2} } } }} \times h_{k,u} } \right\}$$

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$$\begin{aligned} h_{k,u} & = \sum\limits_{{(i,l,j) \in O_{u} }} {\left\{ {\phi (k - C_{i,l,j} + p_{i,l,j} - 1) - \phi (k - C_{i,l,j} - 1)} \right\}} - 1 \\ & \quad k = 1, \ldots ,K,\quad u = 1, \ldots ,\left| M \right| \\ \end{aligned}$$

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where UB and LB represents the upper bound and the lower bound, respectively.

1.5 Lagrangian relaxation algorithm

• Step 1: Set the number of iterations n = 0, set the Lagrange multipliers $\lambda_{k,u} = 0$.

• Step 2: Each job-level subproblem (SP _i ) is solved by the dynamic programming recursion. The lower bound LB is calculated.

• Step 3: Construct a feasible solution. The upper bound UB is calculated.

• Step 4: If n < 300, go to Step 5. Otherwise, stop.

• Step 5: Update the Lagrange multipliers and n = n + 1 and then return to Step 2.

1.6 The value of lower bound

See Table 4.

Table 4 The value of lower bound