Multi-Objective Multi-Stage Optimize Scheduling Algorithm for Nonlinear Virtual Work-Flow Based on Pareto

Abstract

Work-flow scheduling is for finding the allocation method to achieve optimal resource utilization. In the scheduling process, constraints, such as time, cost and quality, need to be considered. How to balance these parameters is a NP-hard problem, and the nonlinear manufacturing process increases the difficulty of scheduling, so it is necessary to provide an effective heuristic algorithm. Aiming at these problems, a multi-objective nonlinear virtual work-flow model was set up, and a multi-objective staged scheduling optimization algorithm with the objectives of minimizing cost and time and maximizing quality was proposed. The algorithm includes three phases: the virtualization phase abstracts tasks and services into virtual nodes to generate a virtual work-flow model; the virtual scheduling phase divides optimized segments and obtains the solution set through reverse iteration; the generation phase obtains the scheduling path according to the Pareto dominance. The proposed algorithm performed 10.5% better in production quality than the minimum critical path algorithm, reduced the time to meet the time constraint by 9.1% and saves 13.7% more of the cost than the production accuracy maximization algorithm.

1. Introduction

Work-flow technology is a kind of automatic program flow that is used often for commercial and research purposes because of its efficient and clear organization of the work process. Exploiting work-flow technology, the manufacturing process can be interconnected, sharing data between production machines and analytics systems [1,2]. This increased flow of data enables a clear view in the manufacturing process, making it possible to automatically and better schedule processes.

With the development of technology and the change of consumer demand, enterprises are inclined toward personalized and non-standardized design and manufacturing [3]. In this kind of complex manufacturing system, the process route presents the characteristics of being nonlinear and highly selective. Current research focuses on solving scheduling problems from two aspects: problem models and solution methods [4,5,6]. Scheduling processes directly affect the efficiency and cost of manufacturing. Nonlinear manufacturing processes are constrained by the requirements of the production time, product quality and overhead costs. In addition, each customer may have different quality of service (QoS) requirements based on their workload. How to dynamically balance these parameters is an NP-hard problem [7].

Multi-objective optimization has become a hot research topic over the last decades, as it has been proven to be an easy and effective approach for solving real-world optimization problems [8]. The conflict within multi-objective optimization limits the scheduling direction of feasible solutions, and the existing algorithms have difficulty balancing the parameters or cannot find the optimal solution in solving the multi-objective nonlinear scheduling problem [9,10]. This paper proposes a heuristic approach to address this restriction in scheduling nonlinear manufacturing. We intend to determine a schedule for the tasks to select a proposal service. To obtain this schedule, we formulate a multi-objective optimization problem: the enterprises may seek a tradeoff among processing time, cost and quality. In this work, we establish the virtual work-flow model of the multi-objective nonlinear manufacturing process and propose a three-stage approach to solve the problem. The first stage is the virtualization stage; the core of this stage is simplification, and the nonlinear work-flow instance is abstracted to a non-linear virtual work-flow model. The second stage is the virtual scheduling stage; the work-flow is segmented, and each segment is scheduled through a backward reduction iteration to obtain the feasible solutions. The third stage is the generation stage; the solution set is obtained by forward scheduling, and the scheduling sequence is generated by the Pareto domination relation.

2. Related Work

According to the optimization objective, work-flow scheduling problems can be divided into single-objective optimization and multi-objective optimization. For single-objective optimization problems, scholars often use precise algorithms, heuristic algorithms and intelligent algorithms to solve problem. At present, most multi-objective optimization problems consider 2–3 optimization objectives. To solve these different problems, scholars usually use intelligent algorithms, heuristic algorithms, hybrid algorithms, etc. The problem of resource provisioning and scheduling has been extensively addressed in the literature by considering different perspectives. To achieve the optimal resource scheduling scheme, various scheduling algorithms have been proposed.

In paper [11], the authors proposed two scheduling heuristics for complex application jobs, which reduce scheduling overhead by reducing queue rearrangement and processing task priority constraints. In paper [12], the authors investigated the single-machine scheduling problem that considers the due date assignment and past-sequence-dependent setup times simultaneously, and they designed an algorithm that can jointly find the optimal sequence and optimal due dates. In paper [13], the authors proposed a Pareto-based, multi-objective, discrete Ant Lion Optimization Algorithm that involves a new encoding scheme for ants and antlions to address the discrete nature of work-flow scheduling. Subsequently, they used the Pareto dominance and crowding distance approach to tackle the optimization of multiple objectives and to achieve the optimal solutions. In paper [14], the authors used the multi-objective Black Widow Optimization Algorithm to solve the multi-objective optimization problem of a power system; the weighting factor method was adopted to handle the multi-objective optimal scheduling problem by simultaneously maximizing profit and minimizing emissions, while satisfying the related constraints. In paper [15], to deal with the combinatorial problem, the authors presented the three main phases of hybrid, discrete Particle Swarm Optimization, which combined with the Hill Climbing technique at the third phase to enhance the overall performance. In paper [16], the authors formulated the cyber–physical cloud systems work-flow scheduling problem under reliability, security and time constrains, and proposed a dependable hybrid scheduling scheme with statically scheduled tasks and dynamically assigned recoveries. In paper [17], the authors proposed a dynamic, deadline- and budget-aware work-flow scheduling algorithm that is designed for multiple instances or dynamic behavior work-flow in WaaS environments. In paper [18], a variable neighborhood search-based method and a hybrid heuristic method are developed to solve problems with practical order picking and a distribution system. In paper [19], the authors proposed an efficient priority and relative distance algorithm to minimize the task scheduling length for precedence-constrained work-flow applications without violating the end-to-end deadline constraint. In paper [20], based on a generalized backward strategy, the authors built timed Petri net models for two transient processes and derived two linear programs to search a feasible schedule with a minimal makespan. In paper [21], combining the merits of an artificial bee colony (ABC) and cuckoo search (CS) in a multi-cloud environment, the authors proposed a new resource provisioning model using the hybrid CS algorithm; the approach can provide seamless service to the users in an efficient manner. In paper [22], the authors propose an enhanced, multi-objective, harmony search algorithm and a Gaussian mutation to solve flexible flow shop scheduling problems with a sequence-based setup time, transportation time and probable rework. In paper [23], the authors proposed a Dynamic Multi-Stage Schedule Optimization Algorithm to seek a tradeoff among processing time, energy consumption and other cost functions in flexible manufacturing. In paper [24], after analyzing the combined effects of cloud uncertainty and probabilistic constraints, the authors proposed a customized Genetic Algorithm to minimize the expected work-flow execution time and monetary cost under probabilistic constraints on the deadline and budget. In paper [25], the authors proposed a multi-objective optimal dispatch model of an interconnected gas–electricity energy system, and an improved cuckoo algorithm is used to optimize the multi-objective scheduling model. In paper [26], the authors proposed a semi-multi-objective problem by considering intermediate storage with a tolerable maximum completion time, and a two-step evolutionary algorithm solution was proposed to solve the problem. In paper [27], the authors proposed an improved multi-objective Ant Lion Algorithm that aims to minimize the maximum completion time and energy consumption, and simultaneously optimize the maximum completion time, energy consumption cost and carbon emissions. In paper [28], based on the reliability assessment results, the authors designed a migration strategy, with reliability as the optimization goal while considering costs, and proposed a service function chain reliability evaluation method and reliability optimization algorithm. In paper [29], based on an improved version of the Rainfall Algorithm and the Salp Swarm Algorithm, the authors presented a novel scheduling scheme for real-time home energy management systems.

3. Problem Description

3.1. Work-Flow Definitions

Definition 1

(Original work-flow model W). Defined as W = <P,T,L,S>, where P is the process nodes set, denoted as P = (p₀, p₁,…,p_n); T is the transfer node set, denoted as T = (t₀, t₁, …, t_m); L is the directed edges set, denoted as L = (l₀, l₁, …, l_y); S is the resources set, denoted as S = (s₀, s₁, …, s_z), which corresponds to the optional resources of each transfer node, and each optional resource s_k corresponds to the production attribute a_i = (q_i, c_i, h_i), including three parts, namely, production quality q_i, production cost c_i and production time h_i.

Definition 2

(Virtual node p′). Tasks in work-flow can be reconstructed to a virtual node, p′_[i–j] denotes the virtual node is reconstructed from node p_i to p_j, and p′_[i,j] denotes the virtual node is reconstructed of the adjacent node p_i and p_j. In addition, the virtual nodes p_b and p_e are set in the virtual work-flow model to represent the beginning and end of the business process.

Definition 3

(Virtual work-flow VW). Defined as VW(W, DP, P′, T′, Id, Od), where VW is obtained from the original work-flow model W through virtualization; DP is the abstract set of detection nodes set in the model according to the process requirements, denoted as DP = (dp₁, dp₂ … dp_i … dp_k); dp_i = (σ_iq, σ_ic) indicates that when the work-flow is executed to detection department i, the quality pass rate σ_iq and cost pass rate σ_ic are set in this node according to the process requirements, and the cumulative production quality f_w(p_j, h_j) and cumulative production cost f_c(p_j, h_j) of the precursor process node p_j are detected respectively. If the σ_iq is met, the work-flow will continue to execute; otherwise, it will enter the feedback path for adjustment until the σ_iq is met. Id is the abstract set of in-degree of each node, denoted as Id = (id₁, id₂ … id_i … id_z); Od is the abstract set of the out-degree of each node, denoted as Od = (od₁, od₂ … od_i … od_z). P′ is the abstract process nodes set through reconstruction, denoted as P′ = (p₁′, p₂′ … p_i′ … p_n′); T′ is the virtual transfer node set after reconstruction, denoted as T′ = (t₁′, t₂′ … t_i′ … t_m′).

Definition 4

(Virtual work-flow diagram VG). Defined as VG(VW, E), where E is the abstract set of directed edges reflecting the dependencies of the nodes in VW, denoted as E = (e₁, e₂ … e_i … e_n).

Definition 5

(Manufacturing expectation ME). Refers to the product parameters that the manufacturing process represented by the nonlinear work-flow expects to obtain when the product is delivered. Set ME = (H, C, Q), where H is the time constraint, the latest completion time of the work-flow; C is the cost constraint, the cost limit of the selected processing service; Q is the production quality constraint, the minimum quality to be achieved when the product is delivered.

Definition 6

(Task interval TF_i). TF_i reflects the time range in which process node p_i can choose the optimal manufacturing service, and is denoted as TF_i[FB_i,FE_i], which can be calculated by Equation (1), where FB_i is the earliest start time of process node p_i, FE_i is the latest start time of process node p_i, FB_q is the direct predecessor of process node p_i, and FE_p is the direct successor.

$$\left\{\begin{array}{l}F{B}_i=\mathrm{Max}\left\{F{B}_{i-1}+\mathrm{Min}\left(h_{ij}^{}\right)\right\},F{B}_{i-1}\in \underset{k}{\underbrace{\left\{\dots ,F{B}_q,\dots \right\}}}\\ F{E}_i=\mathrm{Min}\left\{F{E}_{i-1}-\mathrm{Min}\left(h_{ij}^{}\right)\right\},F{E}_{i-1}\in \underset{l}{\underbrace{\left\{\dots ,F{E}_p,\dots \right\}}}\\ F{B}_1=0,F{E}_n=ME.H\\ s.t.k=1,2,\dots ,i{d}_i;l=1,2,\dots ,o{d}_i\end{array}\right.$$

(1)

Definition 7

(The execution domain θ). θ = [θ_min,θ_max] is the manufacturing interval of the virtual node transformed by the partial virtual node P’(P’ ∈ P) in the reconstructed virtual work-flow graph VG, which can be obtained from Equation (2).

$$\left\{\begin{array}{l}{\theta }_{\min }=\max \{F{B}_j-F{B}_i\}\\ {\theta }_{\max }=\min \{F{E}_j-F{B}_i\}\end{array}\right.$$

(2)

Definition 8

(Special different path SDP). If there exists a node p_i in the virtual work-flow graph VG with Od > 1, which can be reconstructed with node p_j as a virtual node p’_[i,j], but another out-degree of node p_i can be reconstructed with node p_k as another virtual node p’_[i,k], then the path formed by p_i and p_k is SDP.

Definition 9

(Manufacturing attribute A). Defined as A = (A_Qp, A_Cp, A_Hp), it represents the cumulative production attributes at node p, including the cumulative production quality A_Qp, cumulative production cost A_Cp and cumulative production time A_Hp. It can be calculated by Equation (3), where q is the predecessor node of node p, when l_ij = 1 denotes the selection of the corresponding resource in the manufacturing resource set, and when p = E denotes that the scheduling is complete; A_E denotes the final manufacturing attribute.

$$\left\{\begin{array}{l}{A}_{Hp}=\mathrm{Max}\underset{k}{\underbrace{\left\{\dots ,A_{Hp},\dots \right\}}}+l_{pj}{h}_{qj}\le ME.H\\ A_{Cp}=\mathrm{Max}\underset{k}{\underbrace{\left\{\dots ,A_{Cp},\dots \right\}}}+l_{pj}{c}_{pj}\le ME.C\\ A_{Qp}={\displaystyle \prod _{n_i\in N^{\prime }}{l}_{ij}{w}_{ij}}\ge ME.Q\\ s.t.{\displaystyle \sum _{j=1}^k{l}_{ij}=1,}{l}_{ij}\in \left\{0,1\right\},q<p,k=1,2,\dots ,i{d}_p\end{array}\right.$$

(3)

3.2. Nonlinear Virtual Work-Flow Model VWF Generation Algorithm

The essence of the work-flow scheduling problem is to establish the mapping relationship between multiple tasks and optional services. In order to improve the resource waste and time loss caused by irrational process selection in the nonlinear manufacturing process, this paper proposes a virtual stage scheduling model of nonlinear work-flow, which is composed of three parts: virtual work-flow model, virtual work-flow diagram and manufacturing resource set, which can be defined as VWF(Re, Map, VW, VG), where Re is the manufacturing resources set consisting of the set P and the set T; Map is the set of mapping relations from the set of manufacturing resources to the model VW, which can be expressed as Map = (map₁, map₂ … map_i …, map_{n + m}).

When users submit work-flow tasks and requirements, the available resources in each task and resource collection are abstracted to form a virtual manufacturing resource collection, which is mapped to the virtual work-flow after virtualization. Tasks with associated dependencies are divided according to the manufacturing process characteristics and actual requirements, and detection nodes are preset to schedule tasks in each phase one by one to generate a virtual work-flow graph. Combined with previous research [7], the design modeling algorithm of the nonlinear virtual work-flow model VWF is as follows.

Step 1

Abstract the production as the process node set P and T, the optional resources corresponding to the conversion form as resource set S, and the partial order relationship of each process node as directed edge set L to construct the original work-flow model W.

Step 2

Add the set of detection nodes DP and the virtual nodes P_B and P_E to the work-flow model W and update the set of directed edges L.

Step 3

Input process nodes with od or id greater than 1 into the Queue, except for the feedback predecessor structure.

Step 4

Output the process node p_j with od > 1 in Queue, find the nearest feedback in-degree process node p_i, and delete all process nodes between p_i and p_j from the queue.

Step 5

Reconstruct all process nodes between nodes p_i and p_j, save node p_i′ temporarily to the virtual process node set P′, reconstruct the transfer nodes involved in this process as virtual transfer nodes, and save them to the virtual transfer node set T′.

Step 6

Iterate virtual process node p_i′ in the set P′ and determine whether there is an SDP generated due to virtual reconstruction. If so, divide the virtual process node p_i′ and the virtual transfer node t_i′ and restore the process nodes and transfer nodes that constitute the virtual node. Otherwise, delete the p_i′ and t_i′, repeat until the set P′ or T′ is empty, and then output the VW.

Step 7

Traverse model VW; count all paths from adjacent process node p_i to p_j(p_i ≠ p_j); establish directed edge e_jk, where k = 1, 2,…, id_pj; calculate the parameters h_jk, c_jk and q_jk of directed edge e_jk; and mark these parameters to form a virtual work-flow diagram VG.

3.3. Virtual Work-Flow Example

The production process of a sheet metal workshop contains process classification, part programming, machine setup, cutting, mass production, quality inspection and other links. It can be extracted as a set of 12 process nodes P and the corresponding set of transfer nodes T, as shown in

Table 1. Sheet metal manufacturing process node sets P and T.

Node	Meaning	Node	Meaning	Node	Meaning
p1	Process classification	p2	Programming	p3	Machine adjustment
p4	Profile processing	p5	Material calculation	p6	Material collection
p7	Cutting	p8	Bulk processing	p9	Adjustment
p10	Process allocation	p11	Batch production	p12	Warehouse
t1	Category ready	t2	Program ready	t3	Machine ready
t4	Processing ready	t5	Material preparation	t6	Distribution
t7	Cutting ready	t8	Processing ready	t9	Adjustment ready
t10	Allocation ready	t11	Production ready	t12	Storage ready

.

The production process presents nonlinear characteristics. A nonlinear virtual work-flow can be used to optimally schedule its production time, production quality and production cost, with multiple objectives, and input the information of each node into the model generation algorithm VWF to obtain the work-flow model shown in Figure 1.

According to the requirements of users, the principal tasks and the corresponding services are extracted from the production process as a process node set P and a transfer node set T, and their execution order is mapped to the original work-flow graph. On this basis, according to the virtualization rules, the nodes are transformed into virtual nodes, and the directed edge relationship is reconstructed to generate a virtual work-flow graph to decide the execution order of each task.

4. Multi-Objective Nonlinear Virtual Work-Flow Staged Scheduling Algorithm Based on Pareto Optimization

Definition 10

(Pareto optimal). Refers to an ideal state of manufacturing resource allocation, which is the equilibrium point of the optimal strategy. In the process of moving from one allocation state to another, making one attribute improve without making another attribute decrease is called Pareto improvement, and the state where there is no Pareto improvement is called Pareto optimal.

Definition 11

(Pareto domination). Suppose there are functions ƒ_q with the objective of maximizing the cumulative production quality A_Qp and functions ƒc and ƒh with the objectives of minimizing the cumulative production time A_Hp and cumulative production cost A_Cp. x_i = (p₁/t_1Sk, p₂/t_2Sk, …, p_n/t_nSk) and x_j = (p₁/t_1Sk, p₂/t_2Sk, …, p_n/t_nSk) are two sets of solution vectors consisting of the process node p_i and the transfer node t_iSk on this path. When f_q (x_i) ≥ f_q (x_j), f_h (x_i) ≤ f_h (x_j), f_c (x_i) ≤ f_c (x_j) and at least one of the strict inequalities holds, the solution vector x_i dominates the solution vector x_j, denoted as x_i > x_j.

Definition 12

(Pareto-optimal solution). Let X be the set of feasible solutions of a multi-objective optimization problem, and the set of solutions of all Pareto feasible solutions constitutes the Pareto optimal solution set of this multi-objective optimization problem. x_i is a Pareto optimal solution if there is no x_j in X, such that all the objective attributes corresponding to x_i are elevated.

Assuming that the functions ƒ_q(p_i,h_pi) and ƒ_c(p_i,h_pi) denote the highest production quality and lowest cost, respectively, of process node p_i(p_i ∈ P′) selected in its task interval TF_i[FB_i,FE_i] at moment h_pi for the virtual work-flow, ƒ_q(p_i,h_pi) and ƒ_c(p_i,h_pi) can be calculated using Equation (4).

$$\left\{\begin{array}{l}{ƒ}_q(p_i,h_{pi})=\max \left\{q_{{}_{pi}}^k\right\}\\ {ƒ}_c(p_i,h_{pi})=\min \left\{c_{{}_{pi}}^k\right\}\\ h_{pi}\in T{F}_{pi}\left[F{B}_{pi},F{E}_{pi}\right],0<k\le i{d}_{pi}\\ s.t.h_{pi}+h_{pik}\le ME.H,c_{pi}+c_{pik}\le ME.C,w_{pi}\times w_{pik}\ge ME.Q\end{array}\right.$$

(4)

Let p_j be the precursor node of process node p_i(p_i ∈ P′,p_j ∈ P′) and use Equation (5) to calculate the cumulative production quality and cost of node p_j.

$$\left\{\begin{array}{l}{ƒ}_q(p_j,h_{pj})=\max \{f_q(p_i,h_{pj}+h_{pi})\times q_{{}_{pj}}^k\}\\ {ƒ}_c(p_i,h_{pj})=\min \{f_c(p_i,h_{pj}+h_{pi})+c_{{}_{pj}}^k\}\\ h_{pj}\in T{F}_{pj}\left[F{B}_{pj},F{E}_{pj}\right],0<k\le i{d}_{pj}\\ s.t.h_{pj}+h_{pjk}\le ME.H,c_{pj}+c_{pjk}\le ME.C,w_{pj}\times w_{pjk}\ge <ME.Q\end{array}\right.$$

(5)

The calculation of each attribute of the last task of each layer can be completed using Equation (4), and then combined with Equation (5), can be iterated backwards to calculate the cumulative attributes obtained by scheduling resources for each task within the maximum range allowed by the task interval TF_i.

The Multi-objective Nonlinear Virtual Work-flow Staged Scheduling Algorithm (VWFA) uses virtual technology to transform tasks into virtual nodes, which are used as the basis for the subsequent simplification and merging of serial and feedback paths. The relative execution order of tasks in different stages in the optimization segment is limited by the task interval to ensure that the constraints are met and the available resources are scheduled in the maximum range. The results of the backward iteration are integrated to form a feasible solution set, and the feasible solutions are ranked according to the Pareto dominance relation, which can avoid the local optimal solution or the unreasonable local resource allocation caused by the limitation of local parameters, and select the optimal scheduling scheme that meets the needs of users. Based on the above strategy, the steps of algorithm VWFA are as follows.

Step 1

Invoke the modeling algorithm VWF to process each parameter of the nonlinear manufacturing process and form a nonlinear virtual work-flow model.

Step 2

Combine the manufacturing expectations ME and manufacturing characteristics to reverse the work-flow hierarchy, mark the last node of each layer, traverse backwards to the virtual node P_B, and analyze and calculate the task interval TF_i of each process node p_i formed by each layer using Equation (1).

Step 3

Compute and mark the ƒ_w(E₀,h_E0) and ƒ_c(E₀,h_E0) of the process node E₀ of the ending layer at each moment within its active interval TF_i using Equation (4).

Step 4

Traverse the process nodes set P and determine whether there is an SDP. If not, then process the nodes at each stage, combine the process nodes on the serial path as virtual nodes pi, use Equation (4) to calculate the ƒ_q(p_i,h_pi) and ƒ_c(p_i,h_pi) available to the virtual node at each moment in its execution domain, and record the local feasible solution path. If it exists, however, turn to Step 5.

Step 5

If SDP exists, first calculate the execution domain θ of SDP using Equation (2), then calculate ƒ_q(p_i,h_pi) and ƒ_c(p_i,h_pi) available at different moments for each process node in this domain using Equation (4), and record the local feasible solution path.

Step 6

If the cumulative production parameters fail to meet the requirements when executed to detect node dp, a feedback correction is made, and the scheduling results are adjusted according to the dp, and some scheduling strategies are eliminated.

Step 7

Using Equation (3) with manufacturing expectation ME as a constraint, fully process the set of marked model process nodes P and deposit the feasible scheduling sequence into the Pareto solution set X according to the stage accumulation rule.

Step 8

Determine the dominance relationship based on the manufacturing attributes of each scheduling sequence in the Pareto solution set X and output the optimal scheduling route x*.

5. Algorithm Analysis

5.1. Experimental Environment and Data

The server operating system used for the experiments was Windows Server 2016, with a memory configuration of 4G. The VWF model and the optimized scheduling algorithm VWFA code were constructed in Java. To verify the performance of the optimization algorithm, a sheet metal shop production example described in Section 3.3 was chosen for analysis, setting the manufacturing expectation as ME = (Q, C, H) = (0.93, 50, 51) and a batch of sheet metal production parameters, as shown in

Table 2. Nodes and corresponding resource set.

Node	Resources Set S	Node	Resources Set S
t1	(2, 0.94, 0.2), (3, 0.96, 0.3)	t2	(1, 0.98, 1.2)
t3	(1, 0.97, 0.3), (2, 0.99, 5)	t4	(1, 0.97,1.4)
t5	(2, 0.97, 0.5), (3, 0.99, 0.7)	t6	(1, 0.99, 0.6)
t7	(2, 0.90, 0.8), (3, 0.92, 1.0), (4, 0.95, 1.3)	t8	(15, 0.93, 19)
t9	(2, 0.94, 0.7), (4, 0.98, 1.2)	t10	(3, 0.92, 0.3), (4, 0.95, 0.5), (6, 0.98, 0.6)
t11	(10, 0.94, 14)	t12	(2, 0.96, 0.7), (3, 0.98, 1)

.

After inputting the process model shown in Figure 1 and the data shown in Table 2 into the algorithm VWFA, the optimal solution x* that conforms to the Pareto dominance rule is formed by the process shown in Figure 2.

5.2. Model Optimization Process

The scheduling process in Figure 1 is divided into three optimization segments based on the characteristics of this sheet metal manufacturing process. ME.H = 51 day, and the time constraints of the three optimized sections are set as follows: the first section is equipment commissioning ME.H₁ = 6, the second segment is primary process machining ME.H₂ = 25, and the third segment is finish machining ME.H₃ = 20. According to the requirements set in the detection set DP = {dp₁, dp₂, dp₃}, in the first segment to detect the equipment commissioning situation, set the detection point dp₁ = (σ_1q, σ_1c), where σ_1q = 0.980, σ_1c = 2.5; when converting to this detection node, if ƒ_q(p_i,h_pi) < 0.980, it is necessary to feedback to the precursor node to recommission the equipment with a delay of 1 day, increasing the cost by 0.5 thousand dollars and eliminating the current scheduling scheme if ƒ_c(p_i,h_pi) ≥ 3 thousand dollars. In the second segment, to detect the primary process machining, set the detection node dp₂, where σ_2q =0.950; when converting to this detection node, if ƒ_q(p_i,h_pi) < 0.950, it needs to be fed back to the precursor node to adjust the processing, delaying 2 days and costing an additional 5 thousand dollars with no cost constraint. Set the detection node dp₃ in the third segment, where σ_3q =0.970; when converting to this node, if ƒ_q(p_i,h_pi) < 0.970, feedback to the predecessor node to adjust the processing, which delays 2 days and costs an additional 3 thousand dollars with no cost constraint. The selected scheduling scheme should satisfy ƒ_c(p_i,h_pi) ≤ 50 thousand dollars. The manufacturing attributes of each segment can be passed cumulatively, and the calculation procedure to obtain the Pareto solution set X according to the VWFA algorithm is as follows.

In the first optimization segment, according to Equation (1), the active intervals of process nodes p₁, p₂ and p₃ are respectively TF₁ = [0,2], TF₂ = [2,4] and TF₃ = [3,5]. The ƒ_q and ƒ_c of process nodes p₁, p₂ and p₃ that start at the different moments h_pi are as follows.

Calculation process of process node p₃:

• ƒ_q(p₃,5) = max{0.97} = 0.970; ƒ_c(p₃,5) = 0.3;
• ƒ_q(p₃,4) = max{0.97} = 0.970; ƒ_c(p₃,4) = 0.5;
• ƒ_q(p₃,3) = max{0.97,0.99} = 0.990; ƒ_c(p₃,3) = 0.5.

Calculation process of process node p₂:

• ƒ_q(p₂,4) = max{0.98 × ƒ_q(p₃,5)} = 0.951; ƒ_c(p₂,4) = 1.2;
• ƒ_q(p₂,3) = max{0.98 × ƒ_q(p₃,4)} = 0.970; ƒ_c(p₂,26) = 1.2;
• ƒ_q(p₂,2) = max{0.98 × ƒ_q(p₃,3)} = 0.970; ƒ_c(p₂,25) = 1.2.

Calculation process of process node p₁:

• ƒ_q(p₁,2) = max{0.94 × ƒ_q(p₂,4)} = 0.894; ƒ_c(p₁,2) = 0.2;
• ƒ_q(p₁,1) = max{0.94 × ƒ_q(p₂,3),0.96׃_q (p₂,4)} = 0.913; ƒ_c(p₁,1) = 0.3;
• ƒ_q(p₁,0) = max{0.94 × ƒ_q(p₂,2),0.96׃_q (p₂,3)} = 0.931; ƒ_c(p₁,0) = 0.3.

The above process shows that the reverse iteration to p₁ in this domain produces three local scheduling paths with a cumulative production quality of ƒ_q(p₁,0) = 0.931, ƒ_q(p₁,1) = 0.913 and ƒ_q(p₁,2) = 0.894; the cumulative production times are 6, 5 and 5, and the cumulative production costs are 2, 1.8 and 1.7, respectively. According to the rules described in Definition 8, it is reconstructed as a virtual process node p′_[1–3], and the execution domain of p′_[1–3] is obtained according to Equation (2): θ₁ = [3,5]. The scheduling process and results of p′_[1–3] is in

Table 3. Scheduling process and results of p′_[1–3] under different deadlines.

Deadline	Process
5	Production quality ƒq = 0.931, production time ƒh = 6 and cost ƒc = 2; because ƒq < σ1q, execute correction processing, then ƒ′q = 0.931 + 0.931 × (1 − 0.931) = 0.995. ƒ′h = 6 + 1 = 7, ƒ′c = 2 + 0.5 = 2.5. Due to ƒ′h exceeds ME.H1, it does not meet the requirements and is discarded.
4	The maximum production quality is equivalent to the maximum production accuracy of p1 when it begins at time 1, that ƒq = 0.913, production time ƒh = 5 and cost ƒc = 1.8; because ƒq < σ1q, execute correction processing, and ƒ′q = 0.913 + 0.913 × (1 − 0.913) = 0.992, ƒ′h = 5 + 1 = 6, ƒ′c = 1.8 + 0.5 = 2.3. Meets the requirements of dp1.
3	The maximum production quality is equivalent to the maximum production accuracy of p1 when it begins at time 2, that ƒq = 0.894, production time ƒh = 5 and cost ƒc = 1.7; because ƒq < σ1q, execute correction processing, and ƒ′q(p1,2) = 0.894 + 0.894 × (1 − 0.894) = 0.989, ƒ′h = 5 + 1 = 6, ƒ′c = 1.7 + 0.5 = 2.2. Meets the requirements of dp1.

.

Record the feasible solutions path_1a1 and path_1a2 for the first optimization segment according to Table 3. The serial path consisting of p₄ and p₅ in the first optimization segment without detection nodes produces two locally feasible solutions: path_1b1 and path_1b2, reverse scheduling to p₄ with cumulative production quality and cost of ƒ_q(p₄,0) = 0.940 and ƒ_c(p₄,0) = 1.9, ƒ_q(p₄,0) = 0.960 and ƒ_c(p₄,0) = 2.1, respectively. Cross-combining the locally feasible solutions, four locally feasible solutions, namely, path₁₁, path₁₂, path₁₃ and path₁₄, are obtained.

The second optimization segment according to Equation (1) yields the active intervals of process nodes p₆, p₇, p₈ and p₉ as TF₆ = [0,5], TF₇ = [1,6], TF₈ = [3,8] and TF₉ = [18,23], respectively. The ƒ_q of process nodes p₆, p₇, p₈ and p₉ at the different start moments h_pi are calculated with the same strategy in the first segment. Six local scheduling paths are generated when the reverse iteration is to p₆. With the cumulative production quality of ƒ_q(p₆,0) = 0.857, ƒ_q(p₆,1) = 0.857, ƒ_q(p₆,2) = 0.830, ƒ_q(p₆,3) = 0.822, ƒ_q(p₆,4) = 0.796 and ƒ_q(p₆,5) = 0.779, the cumulative production times used are 24, 24, 23, 22, 21 and 20, respectively. According to the rules described in Definition 8, the path meets the virtual reconstruction requirements, and there is no SDP, so it is reconstructed as a virtual process node p′_[6–9], and the execution domain of p′_[6–9] is obtained according to Equation (2): θ₂ = [18,23]. The scheduling process and results of p′_[6–9] are shown in

Table 4. Scheduling process and results of p′_[6–9] under different deadlines.

Deadline	Process
23	Production quality ƒq = 0.857 and production time ƒh = 24; because ƒq < σ2q, execute correction processing, then ƒ′q = 0.857 + 0.857 × (1 − 0.857) = 0.980. ƒ′h = 24 + 2 = 26. Due to ƒ′h exceeds ME.H2, it does not meet the requirements and is discarded.
…	…
19	The maximum production accuracy is equivalent to the maximum production accuracy of p1 when it begins at time 4, that ƒq = 0.796 and production time ƒh = 21; because ƒq < σ2q, execute correction processing, and ƒ′q = 0.796 + 0.796 × (1 − 0.796) = 0.958, ƒ′h = 21 + 2 = 23. Meets the requirements of dp2.
18	The maximum production accuracy is equivalent to the maximum production accuracy of p1 when it begins at time 5, that ƒq = 0.779 and production time ƒh = 20; because ƒq < σ2q, execute correction processing, and ƒ′q = 0.779 + 0.779 × (1 − 0.779) = 0.951, ƒ′h = 20 + 2 = 22. Meets the requirements of dp2.

.

The corrected cumulative production quality is ƒ′_q(p₆,2) = 0.971, ƒ′_q(p₆,3) = 0.968, ƒ′_q(p₆,4) = 0.958 and ƒ′_q(p₆,5) = 0.951, and 4 locally feasible solutions, namely, path₂₁, path₂₂, path₂₃, and path₂₄, are recorded.

The third optimization segment according to Equation (1) yields the active intervals of process nodes p₁₀, p₁₁ and p₁₂ as TF₁₀ = [0,4], TF₁₁ = [3,7] and TF₁₂ = [13,17], respectively. The ƒ_q and ƒ_c of process nodes p₁₀, p₁₁ and p₁₂ at different start moments h_pi are calculated with the same strategy in the first segment, and 5 local scheduling paths are generated, with ƒ_q(p₁₀,0) = 0.903, ƒ_q(p₁₀,1) = 0.883, ƒ_q(p₁₀,2) = 0.875, ƒ_q(p₁₀,3) = 0.860 and ƒ_q(p₁₀,4) = 0.823, and the cumulative production times are 21, 20, 19, 18 and 17, respectively. According to the rules described in Definition 8, the path meets the virtual reconstruction requirements, and it is reconstructed as a virtual process node p′_[10–12], and the execution domain of p′_[10–12] is obtained according to Equation (2): θ₃ = [13,17]. The scheduling process and results of p′_[10–12] are shown in

Table 5. Scheduling process and result of p′_[10–12] under different deadlines.

Deadline	Process
17	Production quality ƒq = 0.903, production time ƒh = 21 and cost ƒc = 15.6; because ƒq < σ3q, execute correction processing, then ƒ′q = 0.903 + 0.903 × (1 − 0.903) = 0.990. ƒ′h = 21 + 2 = 23, ƒ’c = 18.6. Due to ƒ′h exceeds ME.H3, it does not meet the requirements and is discarded.
…	…
14	The maximum production accuracy is equivalent to the maximum production accuracy of p1 when it begins at time 3, that ƒq = 0.986 ƒh = 18 and ƒc = 15.6; because ƒq < σ3q, execute correction processing, and ƒ′q = 0.860 + 0.860 × (1 − 0.860) = 0.980, ƒ′h = 18 + 2 = 20 and ƒ’c = 18.6. Meets the requirements of dp3.
13	The maximum production accuracy is equivalent to the maximum production accuracy of p1 when it begins at time 4, that ƒq = 0.823 ƒh = 17 and ƒc = 15.6; because ƒq < σ3q, execute correction processing, and ƒ′q = 0.823 + 0.823 × (1 − 0.823) = 0.969, ƒ′h = 17 + 2 = 19, ƒ’c = 18.6. Meets the requirements of dp3.

.

The paths whose cumulative production quality does not meet the requirements of σ_3q are fed back to the corresponding nodes for correction. The paths starting from moments 0 and 1 are eliminated due to the corrected cumulative production time exceeding ME.H₃, and the corrected cumulative production quality is ƒ′_q(p₁₀,2) = 0.984, ƒ′_q(p₁₀,3) = 0.980 and ƒ′_q(p₁₀,4) = 0.969; the cumulative production cost is 18.5, 18.5, 18.6 and 18.6, which meets the requirement of σ_3c; and the three local feasible solutions path₃₁, path₃₂ and path₃₃ are recorded.

6. Algorithm Process and Results Analysis

6.1. VWFA Reconstruction Process

The VWFA algorithm divides the set of process nodes P in the VWF model into three optimization segments according to the process characteristics, none of the three optimized segments has SDP, and the process nodes of the three optimized segments are fictitiously formed into p′_[1–5], p′_[6–9] and p′_[10–12] according to the rules. The virtual reconstruction process of the VWFA algorithm is shown in Figure 3.

Table 6. Local feasible solutions for each optimization section.

Segment	Local Solution	Path
First segment	path11	pB/tB→p1/t1S2→p2/t2S1→p3/t3S1, B0→p4/t4S1→p5/t5S1
	path12	pB/tB→p1/t1S2→p2/t2S1→p3/t3S1, B0→p4/t4S1→p5/t5S2
	path13	pB/tB→p1/t1S1→p2/t2S1→p3/t3S2, B0→p4/t4S1→p5/t5S1
	path14	pB/tB→p1/t1S1→p2/t2S1→p3/t3S2, B0→p4/t4S1→p5/t5S2
Second segment	path21	p6/t6S1→p7/t7S2→p8/t3S1→p9/t9S2
	path22	p6/t6S1→p7/t7S3→p8/t3S1→p9/t9S1
	path23	p6/t6S1→p7/t7S2→p8/t3S1→p9/t9S1
	path24	p6/t6S1→p7/t7S1→p8/t3S1→p9/t9S2
Third segment	path31	p10/t10S2→p11/t11S1→p12/t12S2→pE/tE
	path32	p10/t10S2→p11/t11S1→p12/t12S1→pE/tE
	path33	p10/t10S1→p11/t11S1→p12/t12S1→pE/tE

shows the local feasible solutions generated by the three optimization segments. On the basis of the characteristics of the accumulative transferability of the manufacturing attributes in each segment, the scheduling solutions of the three optimization segments are cross-integrated to obtain a Pareto solution set X.

Take the ƒ_q, ƒ_c and ƒ_h as the object function. According to the domination rule in Definition 11: determine the dominant relation of each feasible solution; the solution without Pareto improvement is the optimal solution; the Pareto optimal solution x* = path₁₁→path₂₁→path₃₁ is obtained; the final cumulative ƒ_q = 0.947, ƒ_c = 50.5 and ƒ_h = 50. Under this resource allocation and scheduling strategy, one can get the optimal equilibrium property.

6.2. Results Analyses

The data from the example described in Section 3.3 and Table 2 are input into the minimum critical path algorithm (CPM) and the production accuracy maximization algorithm (PAM). To ensure the validity and fairness of the comparison results, all three algorithms use the experimental environment described in Section 5.1 and set the placement manufacturing expectation ME = (Q, C, H) = (0.93,50,51). Three different scheduling paths path were obtained, as shown in Figure 4.

According to the scheduling path shown in Figure 4, the CPM or PAM algorithm has the deficiency of the single optimization goal or the lack of prediction in service selection. The VWFA algorithm determines the executable range of each task by dividing the work-flow into scheduling segments and calculating the execution domain of the virtual tasks. In this domain, the service selection scheme of each task under different conditions is simulated to expand the solution space and avoid the imbalance of overall resource allocation caused by the optimization of the local tasks. Finally, the optimal solution of time, cost and quality balance is obtained.

Table 7. Scheduling results of different algorithms.

Algorithm and Parameter	CPM			PAM			VWFA
	ƒq	ƒc	ƒh	ƒq	ƒc	ƒh	ƒq	ƒc	ƒh
First segment	0.929	5.0	5	0.995	7.8	6	0.989	6.2	6
Second segment	0.951	26.1	22	0.969	32.1	28	0.971	25.8	25
Third segment	0.970	18.0	17	0.990	18.6	21	0.986	18.5	19
Cumulative	0.857	50.1	44	0.955	58.5	55	0.947	50.5	50

compares the scheduling results of the three algorithms.

Compared with CPM, the optimized production quality improvement of VWFA under the constraint of expectations is Δ = (ƒ_q − ƒ_q1)/ƒ_q1 × 100% = 10.5%. Compared with PAM, VWFA reduces the time by 9.1% to meet the time constraint and saves 13.7% of the cost. This result demonstrates that the algorithm VWFA has scheduling advantages under a limited range of EC constraints.

7. Impact of Other Parameters on the Performance of VWFA

7.1. Impact of the Number of Process Nodes

Since the algorithm is influenced by many factors during execution, this paper only investigates two important influencing factors: the number of process nodes and the performance impact of the finite range of engineering expectations EC.H on the algorithm.

The cumulative production quality of the work-flow model is directly influenced by the production quality of each process node. A random number containing {10, 15, 20, 25} process nodes set P, corresponding to the number of {1, 2, 3, 4, 5} in the transfer node set T, is randomly generated. The effect of the change in the number of process nodes on the performance of the CPM and VWFA algorithms is shown in Figure 5.

Figure 5 shows that the number of process nodes is inversely proportional to the cumulative production quality ƒ_q. Compared with the CPM algorithm, the algorithm VWFA proposed in this paper improves on the cumulative production quality ƒ_q by 12.6%, 17.1%, 20.7% and 26.8%, respectively.

7.2. Impact of Time Constraints on Algorithm Performance

The deadline is the latest completion time of the business process, and usually the production quality increases with the increase of the time constraint. Randomly generating {10,15,20} process nodes, the number of service pools S_i is taken from the interval [2,5] for any integer, increasing the deadline EC.H proportionally to 10%, 15%, 20%, 25% and 30% as the deadline and analyzing the impact on the algorithm VWFA. The results are shown in Figure 6.

In Figure 6, the experimental results show an increase in production quality as the deadline is delayed. The reason is that with the increase of the deadline, so does the execution domain increase, which means that some tasks can choose services with production accuracy. Therefore, the deadline can be increased appropriately to improve the efficiency of the algorithm.

8. Conclusions

Because of the complexity of a multi-objective nonlinear process, it is difficult to obtain high-quality solution results by a single-objective algorithm. This paper proposes a virtual work-flow model, VWF, and a staged scheduling optimization algorithm, VWFA, to address this challenge. The complex paths in the nonlinear work-flow are simplified and combined by virtualization, and the work-flow is divided into multiple optimization segments. To ensure the constraints can be satisfied and the utilization of resources can be maximized, the execution domain of the task is calculated to expand the solution space and avoid insufficient local resource scheduling. The feasible solution is obtained by combining backward and forward scheduling, and the optimal solutions satisfying the multi-objective optimization are selected according to the Pareto domination relation. The experimental results analysis shows that our approach performed better than the CPM and PAM algorithms. Using the proposed algorithm can provide a scheduling scheme to the users in an efficient manner, but there is still room for improvement. For example, when task parameters change due to proficiency or workpiece wastage, the effect of parameter changes on the algorithm should also be considered. In the future, we plan to include an adjustment strategy to improve the robustness of the algorithm and to provide better QoS.