Problem statement and notation

There are N jobs, K customers, M manufacturers, and V vehicles. There is a set J = {J₁,J₂,⋯,J_N}of N independent jobs to be processed by M manufacturers. Overall, there are n types of jobs, and the set of the lth type of jobs is indicated by J^l, where J = J¹∪J²∪⋯∪J^l∪⋯∪Jⁿ, l = 1,2,⋯n. The processing time and size of jobs, denoted by p_i and s_i (i = 1,2,⋯N), respectively, may vary due to differences in types. The same types of jobs are to be partitioned into the same batch, and a batch is processed and transported together.

The following assumptions are considered for the problem formulation:

Production

1. All machines and vehicles are available at time zero.
2. A setup time is required before a job is processed on a machine.
3. The setup times of machines are independent of the job sequences.
4. All jobs in a batch should share the same setup time, and the same job types are processed in a batch.

Distribution

1. Two distribution centers are very close to the manufacturers. As such, the difference in storage costs between them is not considered.
2. Vehicles have weight and capacity limits.
3. The speed of each vehicle is the same.
4. Each vehicle must finally return to its logistics center.

The corresponding parameters are defined as follows:

1. N = Total number of jobs
2. n = Total number of job types
3. M = Total number of manufacturers
4. K = Total number of customers
5. H_total = Total number of human resources
6. R = Total number of human resources level
7. U = Total number of distribution centers
8. V = Total number of vehicles
9. π_l = Total number of jobs belonging to the lth job type
10. i,f = Index of jobs, i,f = 1,2,⋯,N
11. l = Index of job types, l = 1,2,⋯,n
12. j = Index of manufacturers, j = 1,2,⋯,M
13. v = Index of vehicles,v = 1,2,⋯,V
14. s_i = Size of job i
15. ϖ = Capacity of batching machines and vehicles
16. χ = Index of customers, χ = 1,2,⋯,K
17. u = Index of distribution centers, u = 1,2,⋯,U
18. r = Index of human resource levels, r = 1,2,⋯,R
19. h = Index of human resources, h = 1,2,⋯,H_total
20. e, g = Index of integrated customers and distribution centers’ series, e, g = 1,2,⋯,K+U
21. ζ_i = Fuzzy due date of job i
22. ξ_i = Product material cost of job i
23. κⁱ = Variable cost of the ith job processing
24. δⁱ = Fixed cost of the ith job processing
25. = Unit wage cost of worker belonging to the rth level
26. = Setup time of the lth type job on the machine of manufacturer j
27. q_fj = Setup time of the fth job produced on the machine of manufacturer j
28. θ = A large enough positive constant
29. = Due time window of the ith job in the first stage
30. ,,, = Key time nodes of due time window
31. = Total weight of production demanded by customer χ
32. = Total volume of production demanded by customer χ
33. ρ_v = Maximum load weight of vehicle v,
34. = Maximum volume of production demanded by customer χ
35. η_v = Maximum volume of vehicle v,
36. fc_u,v = Fixed cost of vehicle v belonging to the distribution center u
37. ω_u,v = Velocity of vehicle v from distribution center u
38. [ET_χ, LT_χ] = Service time window for customer χ
39. ET_χ = Earliest available service time for customer χ
40. LT_χ = Latest available service time for customer χ
41. UT_χ = Unloading time for customer χ
42. SE_χ = Required service start time of customer χ
43. P_E = Early penalty coefficient when product arrival time is earlier than ET_χ
44. P_L = Delay penalty coefficient when product arrival time is later than LT_χ
45. ε_v = Overweight penalty coefficient of vehicle v

Decision variables

1. T_ij = Processing time of the ith job produced on the machine of manufacturer j
2. T_ijh(F_ijh) = Operation time (completion time) of the ith job produced by the hth worker on the machine of manufacturer j
3. τ_ij(F_ij) = Starting time (completion time) of the ith job processed on the machine of manufacturer j
4. μ_i(F_i) = Fuzzy membership of the ith job
5. σ_eg = Distribution cost between e and g
6. F = Completion time of all jobs in the first stage
7. F_i = Completion time of job i
8. Cost = Total production cost
9. Q = Customer satisfaction
10. Cost_dis = Total distribution cost
11. tr_mc = Transportation time between the manufacturers and customers

Auxiliary binary variables

1. z_ifj = 1, if the ith job is processed before the fth job on the machine of manufacturer j; otherwise 0
2. , if the ith job is processed by the hth worker before the fth job on the machine of manufacturer j; otherwise 0
3. w_ij = 1, if the ith job is processed by the machine of manufacturer j; otherwise 0
4. β_u,v = 1, if the vth vehicle of distribution center u is used to distribute products; otherwise 0
5. , if the distribution task between e and g is accomplished by the vth vehicle of distribution center u; otherwise 0

Indicator variables

1. W_ijr = 1, if the ith job is processed by the worker belonging to the rth level on the machine of manufacturer j; otherwise 0
2. , if the ith job belongs to the lth job type; otherwise 0

Production scheduling sub-problem

In this paper, mathematical models with multi-objectives (including optimizing the production cost, processing time, and customer satisfaction) are developed to comply with the operational constraints commonly encountered in industry. These constraining factors include setup times, optional processing machines, and worker flexibility.

1. Production cost minimization
   The term production cost includes material and processing costs. The processing costs are further divided into machine and labor cost. Meanwhile, machine costs are composed of setup and operational costs. (1)
2. Processing time minimization (2)
3. Customer satisfaction maximization (3)
   Here, μ_i(F_i) follows the trapezoidal membership function distribution. (4)
   Subject to (5) (6) (7)

The objective function (1) minimises the weighted sum of the following: the total weighted artificial cost, material cost, preparation cost and operational costs. Objective (2) represents the total processing time. Objective (3) represents the total degree of customer satisfaction. Constraint (5) ensures that another job can be produced (after completion of the previous processing job) by the same manufacturer. Constraint (6) ensures that two different jobs cannot be simultaneously processed by the same worker. Constraint (7) ensures that no worker can process more than one job at the same time.

Vehicle routing sub-problem

The second stage deals with a multi-objective optimization problem. The ultimate goal is to reduce operational costs and improve customer service levels from an overall perspective. The operation scheme takes into consideration multi-constraint conditions, such as the delivery time window, customer service time, vehicle overload penalty, vehicle service time, vehicle load limits and vehicle capacity constraints. In order to solve the problems imposed by these constraints, an improved GAA is designed.

In order to conveniently compute the distribution cost, this paper forms an integrated customers and distribution center series, where 1 to K represent the customers, and K+1 to K+U represent the distribution centers.

(8)

(9)

(10)

(11)

Objective function (8) minimises the weighted sum of the total distribution cost, early penalty, delay penalty and overweight penalty, here, if SE_χ < ET_χ, namely, the required service start time is earlier than the earliest available service time ET_χ, if SE_χ > LT_χ, namely, the required service start time is later than the latest available service time LT_χ. Regardless of whether the required service start time is advanced or delayed, the penalty value is as follows: . In addition, if the total weight of production demanded by customer χ is greater than the maximum load weight of vehicle, vehicle overload penalty should be taken into account. Constraint (9) implies that a vehicle starts out from the distribution center (to distribute products) and returns to the same distribution center. Constraint (10) implies that the volume and weight of each customer’s demanded products are less than the maximum weight and volume of any vehicle. If the above conditions are not met, a punishment is introduced. Constraint (11) ensures that the distribution task meets the requirement of the time window. If not, a punishment is again introduced.

Approach design

The SC scheduling problem (IPDS) is known to be NP-hard. To address this scheduling issue, an efficient algorithm to solve the SC scheduling problem is required [29–33]. The NSGA-II and genetic annealing algorithm are two famous heuristic algorithms. The NSGA-II is suitable for use in solving multi-objective optimization problems. The GAA combines the advantages of both genetic algorithms and simulated annealing algorithms. These algorithms are especially effective for solving single-objective complex problems. Based on the above facts, the production scheduling and distribution scheduling problems are optimized by using an improved NSGA-II algorithm and an improved GAA algorithm, respectively [34–38].

The problem is divided into two parts. Part 1 is the optimization of production scheduling. Part 2 is the optimization of distribution scheduling. The solution of Part 1 determines the manufacturer’s production order, as well as when the finished products will be transported to the nearest temporary distribution centers. Part 2 optimizes the transportation scheme according to customer demand.

Non-dominated sorting genetic algorithm-II

Many decision making problems in real life involve the simultaneous optimization of two or more multiple conflicting objectives. This means that improvements in terms of one objective value result in the degradation of others. The NSGA-II algorithm has been chosen for the optimization solution (Deb et al. [39]). To this end, a method is required to find the trade-off among the three conflicting objectives of 1) cost, 2) total processing time and 3) customer satisfaction. Therefore, we developed a specific and improved NSGA-II algorithm to address the production scheduling problem. The improved NSGA-II algorithm is presented in Fig 2. Several improvement strategies were introduced. A better crowding density sorting method was used to improve the compositor level of individuals within the same individual ranking. In addition, a modified elitism strategy was adopted to ensure population diversity and enhance search performance.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Improved NSGA-II.

https://doi.org/10.1371/journal.pone.0206806.g002

Improved crowd density sorting method

Although the traditional NSGA-II algorithm has improved in terms of performance (compared to the original NSGA), the NSGA-II algorithm still requires significantly more time. This paper proposes a crowd density sorting method based on improved niche dimensions. Conducting density measurement can be conducted via two available methods, namely the decision space measurement and object space measurement. Deb et al. [39] demonstrated that the performance of the object space measurement is superior to the decision space measurement. This paper further improves the performance of the object space measurement. With this method, an individual location is uniquely determined by fitness values in three-dimensional space. We assume that there is a problem with λ (λ = 1, 2,⋯,ξ) objective functions. Here,ξ = 3, t is the number of genetic iterations, γ (γ = 1,2,⋯,popsize) is the index of a member of the population, and f_tγλ represents the λth objective value of the γth individual at the tth iteration. The formula of niche dimensions is as follows: (12)

Improved elitist strategy

This paper presents a modified elitist strategy, to ensure population diversity. This strategy will further improve the ability of the algorithm to find the optimal solution.

1. Step1. Execute an evolutionary (crossover and mutation) operation on population P_t with N individuals, and then obtain population . Combine P_t and to generate a new population , whose size is 2N.
2. Step2. Execute the non-dominated sorting operation on Q_t. Then, obtain the non-dominated solution set {F₁,F₂,⋯}. Set P_t+1 = Ф, i = 0. If|P_t+1|+|F_i|≤N, P_t+1 = P_t+1∪F_i[1:(|F_i|-1)], and |F_i|-1 individuals are copied to P_t+1, and then let i = i+1. If|P_t+1|+|F_i|>N, calculate the crowding distance of individuals in population F_i, select N-|P_t+1| individuals copied to P_t+1 in the order of from sparse to dense.
3. Step3.t = t+1. If t≥T, (T is the maximum number of iterations), stop computations. If t<T, go to Step 2.

Initialization

This paper adopts a three layer encoding system, which consists of 1) jobs, 2) the assignment of manufacturers, and 3) staff allocation. An example of chromosome structure is shown in Fig 3. The first layer specifies the same symbol for all jobs of the same type. The second layer is the number of manufacturers who can process the job in the first layer. The third layer is the number of workers who can operate the machine in the second layer. This three layer encoding method optionally meets the constraints of process flexibility, workers, and manufacturers. In addition, the three layer chromosomes can generate feasible scheduling.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. An example of chromosome structure.

https://doi.org/10.1371/journal.pone.0206806.g003

Crossover and mutation

According to the structure of chromosome, a corresponding composite crossover method is designed based on the combination of jobs, manufacturer assignment and staff allocation. First, the crossover position of jobs is randomly selected. For the crossover of manufacturer assignment, the manufacturer must be guaranteed for process availability. A single-point crossover method is used. For two parent individuals, the crossover s positions are randomly selected, then this method need exchange the manufacturers assigned by the jobs and owned by the two individuals. For the crossover based on the staff allocation, the staff's availability to the manufacturer must be guaranteed, then, this method need exchange the staffs assigned by the manufacturers and shared by the two parent individuals.

The corresponding composite mutation method is also designed based on the combination of jobs, manufacturer assignment and staff allocation. First, an individual is selected as the parent, and then a job is randomly selected and inserted into another position in the chromosome. For the manufacturer assignment, since each job can be completed by multiple manufacturers, the manufacturer can be selected differently from the previous manufacturer. For staff allocation, the same approach as manufacturer assignment is used.

Decoding

1. Step1. Extract the job chromosome P (length matrix:), the manufacturer chromosome Ma(length matrix:), and the worker chromosome Pe (length matrix: ). Get the job number i, the corresponding processing manufacturer number j and the worker number h.
2. Step2. Read ξ_i (product material cost of job i),(setup time of job i processed by manufacturer j), δⁱ (Fixed cost of the ith job), T_ij (processing time of the ith job produced on the machine of manufacturer j), κⁱ (Variable cost of the ith job), Hc_r × W_ijr (unit wage cost of the rth level worker belonging to manufacturer j who can process the ith job).
3. Step3. Calculate the processing cost according to Formulation (1).
4. Step4. Calculate the processing time
   1. Determine TMval (starting processing time of the manufacturer), TPval (completion time of the last job), and TPeval (completion time of the worker).
   2. Select the maximum time to assign to the variable val, val = max([TMval, TPval, TPeval]).
   3. Calculate the starting time A and completion time B of the ith job, A = val, B = val+T_ij.
   4. Record the correlation time, process time and personnel time, TMval = B, TPval = B, TPeval = B.
   5. Calculate the total processing time.
5. Step5. Calculate customer satisfaction f_sum.

Set TVal1 = 0;

For i = 1:N

 time = F_i

  Ftime = ζ_i

if , Z = 0

  elseif

  elseif

   Z = 1

  elseif else

   Z = 0

end

 TVal1 = TVal1+Z

end

Improved genetic annealing algorithm

An improved genetic annealing algorithm is constructed to solve the distribution scheduling problem. The algorithm steps are as follows:

1. Step1. Assume popsize represents the population size, and k is the index of iteration. Set k = 0. Calculate the function values, and then select the maximum value f_0max and the minimum value f_0min. Then calculate the temperature t₀ = (f_0max-f_0min)/popsize. Generate the initial population pop(popsize), which meets the constraints of weight and volume. Then calculate the function f(i), set i* = i and f* = f, where i is one of the chromosomes in this iteration, and i* is the optimal chromosome that ensures the model obtains the minimum objective function value f*.
2. Step2. If all constraints are met, then the output is i* and f*. If all constraints are not met, then randomly select the neighbourhood state j of chromosome i. Calculate the acceptance probability of the simulated annealing algorithm. (13)
   Make a decision to accept or reject chromosome j, according to Eq (13), where f_i(t_k) and f_j(t_k)represent the objective function values of chromosomes i and j, respectively, at temperature t_k. Execute popsize times the iteration to generate a new population pop1(popsize).
3. Step3. Calculate the fitness value fitness(t_k) = 1/f(t_k) of the population pop1(popsize). Sort the chromosomes in population pop1(popsize), according to the fitness values. Then select the chromosomes that ensure maximum fitness to copy to the next generation. The remaining popsize-1 chromosomes are produced at random, so the new population pop2(popsize) is formed.
4. Step4. Perform crossover (mutation) functions on pop2(popsize), according to the adaptive crossover rate p_c (mutation rate p_m), respectively.
5. Step5. Compute the objective function values of every individual in pop2(popsize). Select the chromosome χ that ensures the minimum objective function values. Then find the corresponding fitness value λ. If λ<f*, then set i* = χ and f* = λ.μ clarifies the annealing probability, t_k+1 = μt_k, k = k+1. Then go to Set 2.

Initialization

A natural number coding method is constructed to express the DC, vehicle number (V) and distribution sort value (val-number). Take the chromosome [1,3,0.1||2,1,0.5||1,3,0.2] as an example. This implies that the demand of the first customer is transported by the third vehicle of the first distribution center. The distribution order value is 0.1. In the second and third section, the demand of the second (or third) customer is delivered by the first (or third) vehicle of the second (or first) distribution center. The order values are 0.5 and 0.2, respectively. From this example, we can conclude that the third vehicle in the first distribution center is used to distribute products for Customers 1 and 3, and the vehicle’s distribution path is Center 1- Customer 3- Customer 1- Center 1. The distribution routing of the first vehicle from Distribution Center 2 is Center 2- Customer 2- Center 2. The initialization method, which meets the weight and volume constraints, is constructed as follows:

Input

 λ: the vehicle matrix of every distribution center

 β: the volume constraint matrix of each vehicle

 τ:the weight constraint matrix of each vehicle

while i< = popsize

 for χ = 1:K

 initial-code(i,3*χ-2) = floor(U*rand)+1; % select the distribution center

  initial-code(i,3*χ-1) = floor(λ(initial-code(i,3*χ-2))*rand)+1;% select vehicle

   initial-code(i,3*χ) = rand;%distribution sort

end

 for j = 1:U

for g = 1:(λ(j))

 U^'(j,K) = 0; P(j,K) = 0;

for x = 1:K

if initial-code(i,3*x-2) = = j&initial-code(i,3*x-1) = = g

U^'(j,g) = U^'(j,g)+τ(x);

P(j,g) = P(j,g)+β(x);

    end

   end

  end

end

  W = U^'-τ; V = P-β;

 if max(max(W))>0|max(max(V))>0

i = i;

else

i = i+1;

end

end

Adaptive crossover and mutation operation

In this paper, we propose an adaptive crossover and mutation operation. According to the objective function values of all candidate solutions, the crossover (mutation) rate is dynamically adjusted with the increase of iterations. The adaptive crossover (mutation) rate is adopted, in order to increase the diversity of candidate solutions and enhance the exploration capacity of the solution space.

1. Adaptive crossover operation: (14)
2. Adaptive mutation operation: (15)
   Here, P_C, P_C1 and P_C2 represent the crossover rate, and P_C1 and P_C2 represent the maximum and minimum crossover rate values, respectively. In addition, P_M, P_M1 and P_M2 represent mutation rates, while P_M1 and P_M2 represent the maximum and minimum mutationrate values respectively. In addition, f_max is the maximum fitness value of the population, f_avg is the average fitness value of every iteration of population adaptation, f is the larger fitness value of the two selected crossover individuals, and f*describes the fitness value of the selected mutation individual.