An ant colony-based algorithm for integrated scheduling on batch machines with non-identical capacities

Abstract

In this paper, a production–distribution scheduling problem with non-identical batch machines and multiple vehicles is considered. In the production stage, n jobs are grouped into batches, which are processed on m parallel non-identical batch machines. In the distribution stage, there are multiple vehicles with identical capacities to deliver jobs to customers after the jobs are processed. The objective is to minimize the total weighted tardiness of the jobs. Considering the NP-hardness of the studied problem, an algorithm based on ant colony optimization is presented. A new local optimization strategy called LOC is proposed to improve the local exploitation ability of the algorithm and further search the neighborhood solution to improve the quality of the solution. Moreover, two interval candidate lists are proposed to reduce the search for the feasible solution space and improve the search speed. Furthermore, three objective-oriented heuristics are developed to accelerate the convergence of the algorithm. To verify the performance of the proposed algorithm, extensive experiments are carried out. The experimental results demonstrate that the proposed algorithm can provide better solutions than the state-of-the-art algorithms within a reasonable time.

Appendices

Appendix A: Abbreviations

Abbreviation	Method
ABC	Artificial bee colony
ACO	Ant colony optimization
AL	Access list
ALNS	Adaptive large neighborhoodsearch
BFLPT	Best fit longest processing time
BPM	Batch processing machine
BSP	Batch scheduling problem
CRO	Chemical reaction optimization
DOE	Design of experiment
EI	Emergent intelligence
E/T	Earliness-tardiness
FF	First fit
FFD	First-fit-decreasing
FFLPT	First-fit longest processing time
GA	Genetic algorithm
GRASP	Greedy randomized adaptive search procedure
GS	Global best solution
ICL	Interval candidate list
JT	Job transportation
LOC	Local optimization strategy
LACO	Ant colony optimization with localoptimization
MARB	Minimum attribute ratio of thebatch
MIP	Mixed-integer programming
MMAS	Max-min ant system
MTO	Make-to-order
PSO	Particle swarm optimization
RKGA	Random-keys genetic algorithm
RV	Response variable
SA	Simulated annealing
SC	Solution construction
TCT/TWCT	Total (weighted) completion time
TWT	Total weighted tardiness
VNS	Variable neighborhood search
WACO	LACO without local optimal strategy

Appendix B: Mixed-integer programming model

To present the MIP model of the studied problem, the parameters and decision variables are listed as follows:

Parameters:

j	index of each job, j = 1, 2,…,n
i	index of each machine, i = 1, 2,…,m
k	index of each batch, k = 1, 2,…,b
l	index of each departure time, l = 1, 2,…,d
pj	processing time of job Jj
sj	size of job Jj
dj	due date of job Jj
Si	capacity of machine Mi
wj	weight of job Jj
CV	capacity of vehicle
DTl	the l-th departure time
V Al	number of vehicles depart at the l-th departure time
Dj	departure time of job Jj
Pki	processing time of the k-th batch on machine Mi
CTki	completion time of the k-th batch on machine Mi
STki	start time of processing for the k-th batch on machine Mi
cj	completion time of job Jj
Ci	completion time of machine Mi

Decision variables:

$$ X_{ki}=\left\{ \begin{aligned} 1, & \text{~~~~} \text{If batch}~ B_{k} ~\text{is assigned to machine}~M_{i}\\ 0, & \text{~~~~otherwise. } \end{aligned} \right. $$

(11)

$$ Y_{jki}=\left\{ \begin{aligned} 1, & \text{~~~~} \text{If job}~ J_{j} ~\text{is assigned to the \textit{k}th batch on machine}~M_{i}\\ 0, & \text{~~~~otherwise. } \end{aligned} \right. $$

(12)

$$ Z_{jl}=\left\{ \begin{aligned} 1, & \text{~~~~} \text{If job}~ J_{j} ~\text{is transported at the \textit{l}th departure time $DT_{l}$}\\ 0, & \text{~~~~otherwise. } \end{aligned} \right. $$

(13)

The MIP model of this problem is formulated as follows:

$$ \text{Minimize}{~~~~}TWT = \sum\limits_{j=1}^{n}w_{j}\cdot T_{j} $$

(14)

Subject to.

$$ \sum\limits_{i=1}^{m} {X_{ki}} = 1~~~k=1,2,\ldots,b $$

(15)

$$ \sum\limits_{i=1}^{m}\sum\limits_{k=1}^{b} {Y_{jki}} = 1~~~j=1,2,\ldots,n $$

(16)

$$ \sum\limits_{j=1}^{n} {s_{j}}\cdot{Y_{jki}}\leq S_{i}~~~k=1,2,\ldots,b;i=1,2,\ldots,m $$

(17)

$$ P_{ki} =p_{j}\cdot Y_{jki}~~~j=1,2,\ldots,n;k=1,2,\ldots,b;i=1,2,\ldots,m $$

(18)

$$ X_{ki}\geq Y_{jki}~~~j=1,2,\ldots,n;k=1,2,\ldots,b;i=1,2,\ldots,m $$

(19)

$$ ST_{ki} = CT_{k-1,i}~~~k=1,2,\ldots,b;i=1,2,\ldots,m $$

(20)

$$ CT_{ki} = ST_{ki} + P_{ki}~~~k=1,2,\ldots,b;i=1,2,\ldots,m $$

(21)

$$ ST_{1i} = 0~~~i=1,2,\ldots,m $$

(22)

$$ CT_{ki}\leq DT_{d}~~~k=1,2,\ldots,b;i=1,2,\ldots,m $$

(23)

$$ \sum\limits_{l=1}^{d} Z_{jl} = 1~~~j=1,2,\ldots,n $$

(24)

$$ \sum\limits_{j=1}^{n} {s_{j}}\cdot Z_{jl}\leq CV\cdot VA_{l}~~~l=1,2,\ldots,d $$

(25)

$$ \begin{array}{@{}rcl@{}} DT_{l} &=& D_{j}\cdot X_{ki}\cdot Y_{jki}\cdot Z_{jl}~~~j=1,2,\ldots,n;\\ k&=&1,2,\ldots,b;i=1,2,\ldots,m;l=1,2,\ldots,d \end{array} $$

(26)

$$ \sum\limits_{k=1}^{b} X_{ki} \geq 1~~~i=1,2,\ldots,m $$

(27)

$$ T_{j} = max \{0,D_{j}-d_{j}\}~~~j=1,2,\ldots,n $$

(28)

$$ X_{ki},Y_{jki},Z_{jl}\in \{0,1\}~~~ $$

(29)

The objective function (14) is used to minimize the TWT of jobs. Constraints (15) ensure that each batch can only be assigned to one machine. Constraints (16) ensure that each job can only be assigned into one batch on one machine. Constraints (17) indicate that the total size of all jobs in one batch cannot exceed the capacity of the machine that processes this batch. Constraints (18) define the processing time of the batch. Constraints (19) guarantee that each job can only be assigned into a batch after it has been created. Constraints (20) define the start processing time of a batch. Constraints (21) formulate that the completion time of each batch equals the sum of the start time and processing time of the batch. Constraints (22) define that the start time of the first batch on each machine is zero. Constraints (23) ensure that the completion time of each batch does not exceed the last delivery time. Constraints (24) indicate that each job can only be transported once. Constraints (25) guarantee that the sum of the size of jobs transported at each departure does not exceed the total capacity of the corresponding vehicles. Constraints (26) determine the departure time of each job. Constraints (27) ensure that the number of the batches processed on each machine is not less than one. Constraints (28) define the tardiness time of each job. Constraint (29) define the binary restriction on the decision variables.