An improved salp swarm algorithm for permutation flow shop vehicle routing problem

Abstract

Permutation flow shop is a typical production method in discrete manufacturing. In reality, in order to reduce the inventory cost, enterprises need to deliver the produced products to customers in time. Therefore, enterprises need to consider the logistics transportation scheme when making production plans, and minimize the total cost of production and transportation through the collaborative optimization of production scheduling and logistics transportation scheduling. Permutation flow shop vehicle routing problem is studied in this paper. Aiming at the requirements of collaborative optimization of production scheduling and logistics transportation scheduling, a mathematical model of the problem is established, and an improved salp swarm algorithm is proposed to solve it. In order to improve the performance of the algorithm, the proposed algorithm incorporates local search operation to enhance the exploration of the population space. Simulation results show that compared with simulated annealing, genetic algorithm and particle swarm optimization algorithm, the proposed algorithm has better optimization ability. The example application shows that the proposed algorithm can effectively solve permutation flow shop vehicle routing problem.

Introduction

In the current competitive global market, the efficiency of production and distribution processes is crucial for the profitability and sustainability of manufacturing and logistics enterprises. Permutation flow shop vehicle routing problem (PFSVRP) is a complex scheduling and routing issue that is prevalent in industries such as automotive, electronics, and pharmaceuticals, where just-in-time production and delivery are essential. The optimization of PFSVRP can lead to reduced operational costs, minimized delivery times, and enhanced customer service levels, thereby providing a competitive edge in the market. This paper focuses on PFSVRP, aiming to minimize the total cost of production and transportation through the collaborative optimization of production scheduling and logistics transportation scheduling.

In this paper, PFSVRP is studied. The problem can be briefly described as follows: the orders are processed in the factory, the orders are produced according to the permutation flow shop scheduling problem (PFSP), and then the completed orders are transported to each customer by the vehicles. The objective function is to minimize the sum of production cost and transportation cost. The problem includes discrete manufacturing process and logistics transportation scheduling process. Among them, the discrete manufacturing process considers permutation flow shop scheduling problem; the logistics transportation scheduling process considers vehicle routing problem with time windows (VRPTW).

PFSVRP integrates the complexities of production scheduling with the logistical challenges of vehicle routing. Recent studies have explored innovative optimization techniques, focusing on sustainability, dynamic environments, and advanced algorithmic approaches. Here, we review recent studies that significantly contribute to the field: Abreu et al.¹ developed a genetic algorithm for scheduling open shops with sequence-dependent setup times, offering a robust method for handling complex scheduling scenarios. Al-Behadili et al.² focused on a multi-objective biased randomised iterated greedy for the robust permutation flow shop scheduling problem under disturbances, providing valuable insights into handling uncertainties in PFSVRP. Ali et al.³ addressed a multi-objective closed-loop supply chain under uncertainty, presenting an efficient Lagrangian relaxation reformulation using a neighborhood-based algorithm, which could be applied to PFSVRP for better optimization. Bargaoui et al.⁴ explored a novel chemical reaction optimization for the distributed permutation flowshop scheduling problem with makespan criterion, introducing a new heuristic approach that could be beneficial for PFSVRP. Bellio et al.⁵ implemented a two-stage multi-neighborhood simulated annealing for uncapacitated examination timetabling, demonstrating the effectiveness of simulated annealing in solving complex routing and scheduling problems. Fathollahi-Fard et al.⁶ presented a distributed permutation flow-shop considering sustainability criteria and real-time scheduling, aligning with the growing focus on sustainability in PFSVRP. Fu et al.⁷ tackled two-objective stochastic flow-shop scheduling with deteriorating and learning effect in an Industry 4.0-based manufacturing system, offering a forward-looking perspective on the integration of advanced manufacturing technologies with PFSVRP. Ghaleb et al.⁸ focused on real-time production scheduling in the Industry-4.0 context, addressing uncertainties in job arrivals and machines breakdowns, which is crucial for the dynamic nature of PFSVRP. Huang and Gu⁹ utilized a novel biogeography-based optimization algorithm for the distributed assembly permutation flow-shop scheduling problem, providing a innovative heuristic method that could be adapted to PFSVRP. Jing et al.¹⁰ developed local search-based metaheuristics for the robust distributed permutation flowshop problem, emphasizing the importance of robustness in scheduling and routing, a key aspect of PFSVRP.

In recent years, the issue of production and transportation scheduling has attracted the interest of many experts and scholars. Wang et al.¹¹ studied the scheduling problem of a three-stage hybrid flow shop with distribution based on a practical pick-and-distribution system. In order to minimize the maximum delivery time, a mixed integer linear programming model was established. Marandi et al.¹² introduced a new integrated scheduling problem of multi-plant production and distribution in supply chain management. This supply chain consisted of many factories that were connected together in the form of a network. The plant produced intermediate or finished goods and supplied them to other plants or end customers distributed in different geographic areas. The problem involved finding a production plan and vehicle routing solution at the same time to minimize the sum of delay costs and transportation costs. Ganii et al.¹³ studied the integrated scheduling problem of supply chain. A mixed integer nonlinear programming model was introduced, and three multi-objective meta-heuristic algorithms, namely multi-objective particle swarm optimization algorithm, non-dominated sorting genetic algorithm and multi-objective ant colony algorithm, were used to solve the problem. Govindan et al.¹⁴ developed a distribution network model in which the routing problem of multi-product vehicles with time Windows was combined as an operational decision with a strategic decision related to network design. In order to solve the model, particle swarm optimization, electromagnetic mechanism algorithm and artificial bee swarm were proposed and combined with variable neighborhood search. Martins et al.¹⁵ studied a combination problem of hybrid flow shop and vehicle routing. Aiming at minimizing the service time of the last customer, a partial random variable neighborhood decreasing algorithm was proposed.

Moons et al.¹⁶ investigated the job-level scheduling problem of integrated production and distribution, which explicitly took into account vehicle routing decisions in the distribution process. The existing literature on integrated production scheduling and vehicle routing was reviewed and classified. The problem characteristics and corresponding solving methods of the mathematical model were discussed, and the direction of further research was determined. Bahmani et al.¹⁷ established an integration model for the two-stage assembly flow shop scheduling problem and vehicle routing distribution problem under the soft time windows, and proposed an improved meta-heuristic algorithm based on whale optimization algorithm. Hou et al.¹⁸ proposed an integrated distributed production and distribution problem considering the time windows. An enhanced brainstorming optimization algorithm with a specific strategy was designed to deal with the problem under consideration. Yağmur et al.¹⁹ studied a joint production scheduling and outbound distribution planning problem. The goal was to determine the minimum total travel time for vehicles and the delays that might result from delayed delivery. A mixed integer programming formula was proposed. Qiu et al.²⁰ proposed a consideration of hybrid flow workshop production and mixed integer linear programming model for multi-vehicle delivery with travel. An improved meme algorithm was proposed.

Zaied et al.²¹ reviewed PFSP with completion times and their mathematical models, and also reviewed and discussed most of the methods used to solve PFSP. Bhatt²² used two heuristics, either alone or in combination, to find a solution to PFSP. Wei et al.²³ proposed a hybrid genetic simulated annealing algorithm to solve the flow shop scheduling problem. Zhang et al.²⁴ proposed an adaptive step size cuckoo search algorithm based on dynamic balance factors. Firstly, iteration ratio parameters and adaptive ratio parameters were introduced. Then, dynamic balance factor parameters were introduced to adjust the weights of iteration ratio parameters and adaptive ratio parameters. Finally, combining with dynamic balance factors, the calculation method of parameter skewness value and adaptive step strategy were proposed.

Pan et al.²⁵ modeled the multi-travel time-dependent vehicle routing problem with time windows. An iterative algorithm was developed to derive them efficiently. A hybrid meta-heuristic algorithm was designed, which used adaptive large neighborhood search for guided exploration and variable neighborhood descent for intensive development. Fan et al.²⁶ proposed an integer programming model with minimum total cost for multi-depot vehicle routing under time-varying road network. To solve this problem, a hybrid genetic algorithm for variable neighborhood search was proposed. An adaptive neighborhood search frequency strategy and simulated annealing inferior solution acceptance mechanism were used to balance the diversity and exploitability of the iterative algorithm. Chen et al.²⁷ studied the vehicle routing problem with time windows and delivery robot. An adaptive heuristic algorithm for large neighborhood search was proposed. Gmira et al.²⁸ proposed a method for solving time-varying vehicle routing problems with time windows, in which the traveling speed was related to the section in the road network. This solution approach involved a tabu search heuristic.

Commonly used optimization algorithms include simulated annealing²⁹, genetic algorithm³⁰, particle swarm optimization³¹, ant colony optimization algorithm³² and bat algorithm^33,34 Based on the basic principle of the salp swarm algorithm (SSA)^35,36, this paper proposes an improved salp swarm algorithm (ISSA) to solve the PFSVRP problem. For the discrete manufacturing process, local search operation is introduced; for the logistics transportation scheduling process, the salp swarm algorithm is applied to the discrete domain, and local search strategy is adopted to enhance the search effect of the algorithm. The algorithm is compared with simulated annealing, genetic algorithm and particle swarm algorithm, and the experimental results are given.

The remainder of this paper is organized as follows: Section "Problem description and mathematical model" introduces the problem description and mathematical model. Section "Algorithm design" gives the algorithm to solve the testing problem. The simulation analysis results are presented in Section "Simulation analysis". Section “Conclusion” draws the conclusion.

Problem description and mathematical model

Problem description

This paper studies the PFSVRP problem. The problem can be described as follows: the orders are processed in the factory, orders are processed on machines, each order has processes, each order is processed on different machines, the order is processed in the same order on all machines, each order has its due date, and then the completed orders are delivered to each customer by the vehicles. The objective function is to minimize the sum of production cost and transportation cost. The problem includes discrete manufacturing process and logistics transportation scheduling process. Among them, the discrete manufacturing process considers permutation flow shop scheduling problem³⁷; the logistics transportation scheduling process takes into account vehicle routing problem with time windows³⁸.

Figure 1 shows the schematic diagram of PFSVRP problem, including 6 orders, 1 factory and 6 customers with 3 processing machines and 3 vehicles.

Fig. 1.

The schematic diagram of PFSVRP problem.

Table 1 presents an example of PFSP. In total, 4 orders and 3 processing machines are included. Among them, each order has 3 processes. As shown in Table 1, the number corresponding to the order represents the processing time of the order on the machine. The unit of processing time is second(s).

Table 1.

An example for PFSP.

Machines	Orders
	1 s	2 s	3 s	4 s
1	1 s	7 s	8 s	4 s
2	5 s	3 s	9 s	5 s
3	7 s	3 s	4 s	3 s

The PFSVRP problem makes the following assumptions.

1. All orders are available at zero time.

2. The machine configuration constitutes a flow shop.

3. The machine is always available, there is no breakdown time and maintenance time.

4. The machine setup time can be ignored.

5. The order is not allowed to be preempted, that is, once the order processing is started, it cannot be interrupted.

6. Assume that the order start time is ignored.

7. Each order belongs to a different customer.

8. The due date of each order is greater than or equal to its completion time.

9. The capacity of each vehicle is the same.

10. Each customer has its own time window.

Mathematical model

Mathematical symbols

The mathematical symbols and meanings of the PFSVRP problem are shown in Table 2.

Table 2.

Mathematical symbols for PFSVRP.

Symbol	Meaning
	Number of orders (here, the number of orders equals the number of customers)
,,	The index of the order (here, the index of the customer is the same as the index of the order)
	Number of machines
	Index of the machine
	Number of vehicles
	Index of the vehicle
	The processing time of order on machine
	Unit time cost of order processing
	A large positive number
	The completion time of the order on the machine
	The completion time of the order
	Maximum completion time of the orders
	The due date of the order
	Unit time cost of order delay
	Unit distance cost of the vehicle
	Maximum load capacity of factory vehicle
	Distance between the depot and customer
	Distance from customer to the customer
	Customer collection for the vehicle
	The requirements of the customer
	Time window of the customer ; is the allowed service start time and is the allowed service end time
	Actual service start time of the customer
	A sequence of orders in a certain processing order
	If is order of the sequence , ; otherwise, . Here,
	If vehicle has customer as its first customer, ; otherwise,
	If vehicle serves customer and transports directly to customer , ; otherwise,
	If the last customer of the vehicle is customer , ; otherwise,

Objective function

The objective function is to minimize the sum of production cost and transportation cost.

1

Among them, the first item in formula (1) is the delay cost; the second item is the production cost; the third and fourth items are the transportation cost.

Constraints

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Here, formula (2) ensures that the maximum completion time is greater than or equal to the completion time of the order on the last machine; formula (3) and (4) ensure that each order in a sequence can occur only once; formula (5) represents the completion time of the first order processed on the first machine; formula (6) ensures that one order cannot be processed by multiple machines at the same time; formula (7) ensures that one machine can only process one order at the same time; formula (8) ensures that the completion time of all processes is greater than zero; formula (9) ensures the value range of decision variables; formula (10) ensures that each vehicle can only transport one order at most; formula (11) ensures that the vehicle is continuous during transportation; formula (12) ensures that the order of vehicle transportation does not exceed its maximum of its capacity; formula (13) ensures that the completion time of each order shall not exceed the maximum completion time; formula (14) ensures that the completion time of each order is less than or equal to its due date; formula (15) ensures that the vehicle does not generate sub-loops during transportation; formula (16) ensures the constraints of time windows for each customer. Here, (actual service start time) is a decision variable representing the service start time for customer . is not directly calculated from the preceding equations but is determined during the algorithm’s solution process based on vehicle scheduling and route arrangement. The algorithm will consider all customer service time window constraints (as shown in Eq. (16)) and other relevant constraints to determine the for each customer, thereby optimizing the entire logistics delivery plan.

Algorithm design

Coding and decoding of Solutions

Coding of solutions

This section mainly introduces the coding of the solution of PFSP and the solution of VRPTW. The coding of the solution of PFSP consists of two parts, operations sequencing (OS) and machines selection (MS). Here, . The coding of the solution of VRPTW contains vehicle routing (VR). The coding of the solution is shown in Fig. 2. Here, the dimension of MS is , the dimension of OS is and the dimension of VR is .

• 1. Operations sequence (OS)

Fig. 2.

Coding of the solutions.

The value on each bit of operations sequencing is represented by a number. For example, the number 1 indicates the first order, and the number 4 indicates the fourth order. The number of occurrences of each order number is equal to the number of machines . For the example in Table 1, operations sequence can be encoded as OS = [1 4 3 2 1 4 3 2 1 4 3 2], as shown in Fig. 3.

• 2. Machines Selection (MS)

Fig. 3.

Coding of solutions for the example of Table 1.

The value on each bit of machines selection is represented by the machine number. Based on the selected operation, determine the machine number to which the order is to be processed. The number of occurrences of each machine number is equal to the number of orders . For the example in Table 1, machines selection can be encoded as MS = [1 1 1 1 2 2 2 2 3 3 3 3], as shown in Fig. 3.

• 3. Vehicle Routing (VR)

The vertex number is (including customers and 1 factory), and the vehicle number is , and the dimension is . Vehicle routing is coded as a substitution of . As shown in Fig. 1, , , the vehicle routing can be expressed as , where element and elements greater than are considered to be the factory .

Decoding of solutions

This section introduces the decoding of the solution of VRPTW.

The part of vehicle routing is decoded. The element whose value is and the elements whose values are greater than are both considered to be the factory . The front and end of vehicle routing are added respectively to form a complete VRPTW path. The points passing between the two constitutes the access path of a vehicle, which is the decoding of vehicle routing. As shown in Fig. 1, , , vehicle routing is , and its corresponding access path is . Therefore, the access path of vehicle 1 is , and the access path of vehicle 2 is , and the access path of vehicle 3 is .

SSA

Salp swarm algorithm is a bio-heuristic optimization algorithm that simulates the swarm behavior of salp swarm. Salps are Marine organisms that navigate and forage in the ocean in groups, a behavior known as a salp chain. The design of salp swarm algorithm is inspired by this group behavior of salps, in particular how they achieve efficient movement and foraging through rapid and coordinated changes.

SSA can be expressed by the following formulas.

17

18

19

Here, represents the position of the first salp (the leader) in the dimension . is the position of the food source in the dimension . shows the upper bound of the dimension , and shows the lower bound of the dimension . and are the random numbers with the range from zero to one. is the number of the current iteration, and is the maximum number of iterations. represents the position of the salp (the follower) in the dimension .

ISSA

ISSA to solve PFSP and ISSA to solve VRPTW are detailed below.

ISSA to solve PFSP

This section describes in detail ISSA to solve PFSP. On the basis of the SSA , local search operation is incorporated, forming an ISSA. Below is a detailed description of the incorporated local search operation.

Select individuals from the population whose fitness values are in the top 20% for local search operation.

For each selected individual, follow these steps.

• (1) Initialize local search operation parameters.

Set the iteration number to start from 1, the maximum number of jobs to be removed as , the number of jobs to be deleted as , the selection probability for each neighborhood as . The current temperature is , where is the initial temperature. The maximum number of iterations is . The initial solution is .

• (2) Local search process.

From iteration number to the maximum number of iterations t_max, based on the current selection probability for each neighborhood, use the roulette method to choose a neighborhood type, determine the size of , and the neighborhood scale size is . Select a position from the current solution , and then start from this position, continuously remove jobs. If , then, continuously remove jobs ; otherwise, continuously remove jobs , and then continuously remove jobs . Starting from the job , insert it into the best position in the remaining part of the solution, and repeat this process until all jobs are inserted, and thus obtain a new solution . If the objective function value of the new solution is better than the current solution , then update the current solution to , and update the neighborhood selection probability, i.e., .

• (3) Acceptance criterion.

Update the current temperature, i.e., . If the objective function value of the solution is better than the initial solution , the acceptance criterion is . Where is the maximum completion time of the solution , is the maximum completion time of the solution , and is a random number, . If the acceptance criterion is met, then update the solution to .

• (4) Termination condition.

After completing the local search and acceptance criterion for the individual, the entire optimization process ends.

The pseudocode of local search operation applied in ISSA to solve PFSP is shown in Pseudocode 1.

Pseudocode 1.

Local search operation applied in ISSA to solve PFSP.

The steps of ISSA to solve PFSP are shown in Algorithm 1.

Algorithm 1.

ISSA to solve PFSP.

ISSA to solve VRPTW

According to the formulas (17), (18) and (19), we redefine the formulas in the discrete domain as follows.

Formula (17) can be transformed in the following. Define the food source as which refers to the best solution. is the leader which is the first salp. Suppose as , where . Here, represents , where . However, formula (18) in the continuous domain is unchanged. We directly use formula (18) in the discrete domain. If , . Otherwise, remains unchanged.

Formula (19) can be described as follows. is the th follower salp where . Randomly generate , where . If , . Here, refers to the last salp. Otherwise, stays the same.

Local search method adopts the large neighborhood search algorithm.

The steps of ISSA used to solve VRPTW are shown in Algorithm 2.

Algorithm 2 .

ISSA to solve VRPTW.

Simulation analysis

The implementation is programed in Matlab and run on an Intel i7 12700H computer (2.70 GHz CPU) with 16.00 GB RAM.

Performance test of algorithms

The examples of PFSP uses 120 benchmarks³⁹ of Taillard’s instances of permutation flow shop scheduling problem. The goal of PFSP is to minimize the makepan. ISSA is compared with simulated annealing algorithm (SA)⁴⁰, genetic algorithm (GA)⁴¹ and particle swarm algorithm (PSO)⁴², and the comparison of the algorithms for PFSP is shown in Table 3, where the deviation is defined as . It can be concluded that 57 of the 120 results are obtained by ISSA in precision. Compared with SA, 76 optimal solutions can be obtained by ISSA. Comparing ISSA with GA, 88 optimal solutions can be obtained. Comparing ISSA with PSO, 106 optimal solutions can be obtained. There are 85 results with a deviation of less than 1%.

Table 3.

Comparison of the algorithms for PFSP.

Instance	BKS	SA	GA	PSO	ISSA	Deviation (%)
ta001	1278	1278	1278	1278	1278	0.00
ta002	1359	1359	1359	1359	1359	0.00
ta003	1081	1081	1081	1081	1081	0.00
ta004	1293	1293	1293	1293	1293	0.00
ta005	1235	1235	1235	1235	1235	0.00
ta006	1195	1195	1195	1195	1195	0.00
ta007	1234	1239	1239	1239	1234	0.00
ta008	1206	1206	1206	1206	1206	0.00
ta009	1230	1230	1230	1230	1230	0.00
ta010	1108	1108	1108	1108	1108	0.00
ta011	1582	1582	1582	1582	1582	0.00
ta012	1659	1659	1659	1659	1659	0.00
ta013	1496	1496	1496	1496	1496	0.00
ta014	1377	1377	1377	1377	1377	0.00
ta015	1419	1419	1419	1419	1419	0.00
ta016	1397	1397	1397	1397	1397	0.00
ta017	1484	1484	1484	1484	1484	0.00
ta018	1538	1538	1538	1543	1538	0.00
ta019	1593	1593	1593	1593	1593	0.00
ta020	1591	1591	1591	1598	1591	0.00
ta021	2297	2297	2297	2297	2297	0.00
ta022	2099	2099	2099	2100	2099	0.00
ta023	2326	2326	2326	2326	2326	0.00
ta024	2223	2223	2223	2223	2223	0.00
ta025	2291	2291	2291	2294	2291	0.00
ta026	2226	2226	2226	2228	2226	0.00
ta027	2273	2273	2273	2273	2273	0.00
ta028	2200	2200	2200	2200	2200	0.00
ta029	2237	2237	2237	2237	2237	0.00
ta030	2178	2178	2178	2178	2178	0.00
ta031	2724	2724	2724	2724	2724	0.00
ta032	2834	2836	2834	2836	2834	0.00
ta033	2621	2621	2621	2621	2621	0.00
ta034	2751	2751	2751	2751	2751	0.00
ta035	2863	2863	2863	2863	2863	0.00
ta036	2829	2829	2829	2829	2829	0.00
ta037	2725	2725	2725	2725	2725	0.00
ta038	2683	2683	2683	2683	2683	0.00
ta039	2552	2552	2552	2555	2552	0.00
ta040	2782	2781	2781	2782	2782	0.00
ta041	2991	3024	3021	3051	3023	1.07
ta042	2867	2882	2902	2915	2873	0.21
ta043	2839	2852	2871	2889	2852	0.46
ta044	3063	3063	3070	3071	3063	0.00
ta045	2976	2982	2998	3024	2984	0.27
ta046	3006	3006	3024	3036	3006	0.00
ta047	3093	3122	3122	3133	3107	0.45
ta048	3037	3042	3063	3049	3039	0.07
ta049	2897	2911	2914	2923	2902	0.17
ta050	3065	3077	3076	3131	3078	0.42
ta051	3771	3889	3874	3950	3883	2.97
ta052	3668	3714	3734	3761	3715	1.28
ta053	3591	3667	3688	3741	3672	2.26
ta054	3635	3754	3759	3806	3748	3.11
ta055	3553	3644	3644	3688	3636	2.34
ta056	3667	3708	3717	3758	3693	0.71
ta057	3672	3754	3728	3763	3722	1.36
ta058	3627	3711	3730	3788	3721	2.59
ta059	3645	3772	3779	3831	3761	3.18
ta060	3696	3778	3801	3830	3777	2.19
ta061	5493	5493	5493	5493	5493	0.00
ta062	5268	5268	5268	5268	5268	0.00
ta063	5175	5175	5175	5175	5175	0.00
ta064	5014	5014	5014	5014	5014	0.00
ta065	5250	5250	5250	5250	5250	0.00
ta066	5135	5135	5135	5135	5135	0.00
ta067	5246	5246	5246	5246	5246	0.00
ta068	5094	5094	5094	5094	5094	0.00
ta069	5448	5448	5448	5448	5448	0.00
ta070	5322	5322	5322	5322	5322	0.00
ta071	5770	5776	5770	5790	5770	0.00
ta072	5349	5360	5358	5377	5349	0.00
ta073	5676	5677	5676	5679	5679	0.05
ta074	5781	5792	5792	5849	5792	0.19
ta075	5467	5467	5467	5514	5467	0.00
ta076	5303	5311	5311	5308	5303	0.00
ta077	5595	5596	5605	5602	5599	0.07
ta078	5617	5625	5617	5664	5626	0.16
ta079	5871	5891	5877	5907	5875	0.07
ta080	5845	5845	5845	5858	5848	0.05
ta081	6106	6257	6303	6414	6278	2.82
ta082	6183	6226	6266	6383	6245	1.00
ta083	6252	6342	6351	6437	6330	1.25
ta084	6254	6303	6360	6407	6303	0.78
ta085	6262	6380	6408	6509	6378	1.85
ta086	6302	6427	6453	6551	6458	2.48
ta087	6184	6306	6332	6476	6314	2.10
ta088	6315	6472	6482	6640	6464	2.36
ta089	6204	6380	6343	6462	6327	1.98
ta090	6404	6485	6506	6593	6488	1.31
ta091	10862	10872	10885	10872	10885	0.21
ta092	10480	10487	10495	10556	10523	0.41
ta093	10922	10941	10941	10950	10963	0.38
ta094	10889	10889	10889	10893	10939	0.46
ta095	10524	10524	10524	10537	10537	0.12
ta096	10329	10346	10346	10378	10337	0.08
ta097	10854	10868	10866	10882	10882	0.26
ta098	10730	10741	10741	10777	10753	0.21
ta099	10438	10451	10451	10450	10438	0.00
ta100	10675	10680	10684	10727	10676	0.01
ta101	11152	11287	11339	11535	11373	1.98
ta102	11143	11277	11344	11596	11402	2.32
ta103	11281	11418	11445	11676	11537	2.27
ta104	11275	11376	11434	11665	11445	1.51
ta105	11259	11365	11369	11548	11385	1.12
ta106	11176	11330	11292	11546	11356	1.61
ta107	11337	11398	11481	11702	11517	1.59
ta108	11301	11433	11442	11675	11500	1.76
ta109	11145	11356	11313	11554	11391	2.21
ta110	11284	11446	11424	11683	11458	1.54
ta111	26040	26187	26228	26656	26379	1.30
ta112	26500	26799	26688	27153	26962	1.74
ta113	26371	26496	26522	26923	26656	1.08
ta114	26456	26612	26586	26894	26738	1.07
ta115	26334	26514	26541	26768	26542	0.79
ta116	26469	26661	26582	26965	26711	0.91
ta117	26389	26529	26660	26799	26596	0.78
ta118	26560	26750	26711	27066	26867	1.16
ta119	26005	26223	26148	26488	26298	1.13
ta120	26457	26619	26611	26923	26721	1.00

The iteration convergence times of ISSA solving PFSP to obtain the optimal solution are shown in Table 4. The iteration convergence times usually refers to the number of iterative steps required for an iterative algorithm to achieve a predetermined accuracy or satisfy convergence conditions.

Table 4.

The iteration convergence times of ISSA solving PFSP.

Instance	BKS	ISSA	Iteration convergence times
ta001	1278	1278	2
ta002	1359	1359	2
ta003	1081	1081	1
ta004	1293	1293	8
ta005	1235	1235	14
ta006	1195	1195	1
ta007	1234	1234	46
ta008	1206	1206	2
ta009	1230	1230	1
ta010	1108	1108	2
ta011	1582	1582	11
ta012	1659	1659	5
ta013	1496	1496	50
ta014	1377	1377	23
ta015	1419	1419	11
ta016	1397	1397	4
ta017	1484	1484	4
ta018	1538	1538	23
ta019	1593	1593	7
ta020	1591	1591	27
ta021	2297	2297	3
ta022	2099	2099	23
ta023	2326	2326	1
ta024	2223	2223	7
ta025	2291	2291	27
ta026	2226	2226	58
ta027	2273	2273	11
ta028	2200	2200	11
ta029	2237	2237	31
ta030	2178	2178	10
ta031	2724	2724	1
ta032	2834	2834	22
ta033	2621	2621	1
ta034	2751	2751	13
ta035	2863	2863	6
ta036	2829	2829	3
ta037	2725	2725	2
ta038	2683	2683	1
ta039	2552	2552	17
ta040	2782	2782	3
ta044	3063	3063	89
ta046	3006	3006	205
ta061	5493	5493	1
ta062	5268	5268	9
ta063	5175	5175	1
ta064	5014	5014	87
ta065	5250	5250	8
ta066	5135	5135	1
ta067	5246	5246	6
ta068	5094	5094	3
ta069	5448	5448	24
ta070	5322	5322	7
ta071	5770	5770	205
ta072	5349	5349	242
ta075	5467	5467	176
ta076	5303	5303	66
ta099	10,438	10,438	1

Based on the iteration convergence times in Table 4, Convergence curve graph is shown in Fig. 4. It can more intuitively display the convergence times.

Fig. 4.

Convergence curve graph based on Table 4.

Examples of VRPTW can be described as follows: V001 to V090 from Solomon benchmark, V091 to V120 from Gehring & Homberger benchmark, as shown in Table 5.

Table 5.

Examples of VRPTW.

Instance	Benchmark	Instance	Benchmark
V001	The first 20 customers of C101	V061	C101
V002	The first 20 customers of C102	V062	C102
V003	The first 20 customers of C103	V063	C103
V004	The first 20 customers of C104	V064	C104
V005	The first 20 customers of C105	V065	C105
V006	The first 20 customers of C106	V066	C106
V007	The first 20 customers of C107	V067	C107
V008	The first 20 customers of C108	V068	C108
V009	The first 20 customers of C109	V069	C109
V010	The first 20 customers of C201	V070	C201
V011	The first 20 customers of C202	V071	C202
V012	The first 20 customers of C203	V072	C203
V013	The first 20 customers of C204	V073	C204
V014	The first 20 customers of C205	V074	C205
V015	The first 20 customers of C206	V075	C206
V016	The first 20 customers of C207	V076	C207
V017	The first 20 customers of C208	V077	C208
V018	The first 20 customers of R101	V078	R101
V019	The first 20 customers of R102	V079	R102
V020	The first 20 customers of R103	V080	R103
V021	The first 20 customers of R104	V081	R104
V022	The first 20 customers of R105	V082	R105
V023	The first 20 customers of R106	V083	R106
V024	The first 20 customers of R107	V084	R107
V025	The first 20 customers of R108	V085	R108
V026	The first 20 customers of R109	V086	R109
V027	The first 20 customers of R110	V087	R110
V028	The first 20 customers of R111	V088	R111
V029	The first 20 customers of R112	V089	R112
V030	The first 20 customers of R201	V090	R201
V031	The first 50 customers of C101	V091	C1_2_1
V032	The first 50 customers of C102	V092	C1_2_2
V033	The first 50 customers of C103	V093	C1_2_3
V034	The first 50 customers of C104	V094	C1_2_4
V035	The first 50 customers of C105	V095	C1_2_5
V036	The first 50 customers of C106	V096	C1_2_6
V037	The first 50 customers of C107	V097	C1_2_7
V038	The first 50 customers of C108	V098	C1_2_8
V039	The first 50 customers of C109	V099	C1_2_9
V040	The first 50 customers of C201	V100	C1_2_10
V041	The first 50 customers of C202	V101	C2_2_1
V042	The first 50 customers of C203	V102	C2_2_2
V043	The first 50 customers of C204	V103	C2_2_3
V044	The first 50 customers of C205	V104	C2_2_4
V045	The first 50 customers of C206	V105	C2_2_5
V046	The first 50 customers of C207	V106	C2_2_6
V047	The first 50 customers of C208	V107	C2_2_7
V048	The first 50 customers of R101	V108	C2_2_8
V049	The first 50 customers of R102	V109	C2_2_9
V050	The first 50 customers of R103	V110	C2_2_10
V051	The first 50 customers of R104	V111	The first 500 customers of C1_6_1
V052	The first 50 customers of R105	V112	The first 500 customers of C1_6_2
V053	The first 50 customers of R106	V113	The first 500 customers of C1_6_3
V054	The first 50 customers of R107	V114	The first 500 customers of C1_6_4
V055	The first 50 customers of R108	V115	The first 500 customers of C1_6_5
V056	The first 50 customers of R109	V116	The first 500 customers of C1_6_6
V057	The first 50 customers of R110	V117	The first 500 customers of C1_6_7
V058	The first 50 customers of R111	V118	The first 500 customers of C1_6_8
V059	The first 50 customers of	V119	The first 500 customers of C1_6_9
V060	The first 50 customers of R201	V120	The first 500 customers of C1_6_10

VRPTW aims to minimize the number of vehicles needed (NV) and the total distance (TD). ISSA is compared with SA, GA and PSO and the comparison of the algorithms for VRPTW is shown in Table 6. It can be concluded that among 120 results, 91 better results are obtained by ISSA on NV and TD in precision. On NV and TD, ISSA compared with SA, 113 better results are obtained. 94 better results are obtained by ISSA compared to GA. 104 better results are obtained by ISSA compared to PSO.

Table 6.

Comparison of the algorithms for VRPTW.

Instance	SA		GA		PSO		ISSA
	NV	TD	NV	TD	NV	TD	NV	TD
V001	3	175.37	3	175.37	3	175.37	3	175.37
V002	2	167.16	2	167.16	2	167.16	2	167.16
V003	2	161.57	2	161.57	2	161.57	2	161.57
V004	2	159.48	2	159.48	2	159.48	2	159.48
V005	3	175.37	3	175.37	3	175.37	3	175.37
V006	3	175.37	3	175.37	3	175.37	3	175.37
V007	3	175.37	3	175.37	3	175.37	3	175.37
V008	3	175.37	3	175.37	3	175.37	3	175.37
V009	2	169.45	2	160.82	2	160.82	2	160.82
V010	2	211.57	2	199.01	2	199.01	2	199.01
V011	2	199.01	2	199.01	2	199.01	2	199.01
V012	1	193.09	1	193.09	1	193.09	1	193.09
V013	1	187.21	1	187.21	1	187.21	1	187.21
V014	2	199.01	2	199.01	2	199.01	1	265.52
V015	2	211.57	2	199.01	2	199.01	1	247.08
V016	3	222.85	2	198.06	2	198.06	2	198.98
V017	2	198.84	2	198.84	2	198.84	1	204.49
V018	6	516.91	7	513.66	7	511.25	7	514.50
V019	6	439.41	6	434.91	6	434.91	6	434.91
V020	5	376.77	5	371.58	5	371.58	4	380.67
V021	4	351.00	3	331.88	3	331.88	3	331.88
V022	5	434.88	5	432.48	5	432.48	5	432.48
V023	5	395.65	4	384.65	4	384.65	4	384.65
V024	4	356.62	4	352.91	4	350.18	4	356.62
V025	3	314.90	3	314.90	3	314.90	3	314.90
V026	4	373.54	4	379.07	4	373.54	4	376.87
V027	4	364.91	4	364.63	4	361.78	4	364.63
V028	4	352.30	4	360.63	4	353.30	4	353.30
V029	3	310.70	3	310.70	3	310.70	3	310.70
V030	4	389.52	3	388.12	3	388.12	3	388.12
V031	6	428.82	5	363.25	6	422.74	5	363.25
V032	5	392.24	5	362.17	5	362.17	5	362.17
V033	6	424.00	5	362.17	5	398.78	5	362.17
V034	5	390.03	5	358.88	5	382.69	5	358.88
V035	6	428.23	5	363.25	5	363.25	5	363.25
V036	6	433.65	5	363.25	5	363.25	5	363.25
V037	6	429.32	5	363.25	6	429.85	5	363.25
V038	5	363.25	5	363.25	5	363.25	5	363.25
V039	5	363.25	5	363.25	5	397.18	5	363.25
V040	4	417.13	3	361.80	3	361.80	3	361.80
V041	4	417.13	3	361.80	3	361.80	3	361.80
V042	4	466.62	3	361.41	3	361.41	3	361.41
V043	3	383.66	2	351.72	2	351.72	2	351.72
V044	4	446.74	3	361.41	3	361.41	3	361.41
V045	5	470.79	3	361.41	3	361.41	3	361.41
V046	4	392.98	3	361.21	3	378.45	3	361.21
V047	4	405.95	2	352.12	2	352.12	2	352.12
V048	13	1055.48	12	1051.98	13	1135.13	12	1046.70
V049	11	911.44	12	917.89	11	928.72	11	913.60
V050	10	816.30	9	782.09	9	780.77	8	784.56
V051	6	644.25	6	637.05	6	638.31	6	638.31
V052	11	948.63	9	924.02	9	934.75	10	921.60
V053	10	850.75	8	813.96	8	826.96	8	806.77
V054	8	754.19	7	713.50	8	733.48	7	727.15
V055	7	646.24	6	635.92	7	633.60	6	631.40
V056	9	815.07	8	796.02	8	812.70	8	795.70
V057	8	739.90	7	699.38	7	742.77	7	699.38
V058	8	728.95	7	710.03	8	738.61	7	718.98
V059	7	667.52	7	654.63	6	647.21	6	641.31
V060	8	855.55	5	801.77	5	802.07	4	829.91
V061	10	828.94	10	828.94	10	828.94	10	828.94
V062	12	900.77	10	828.94	10	850.99	10	828.94
V063	11	942.79	10	853.83	10	886.58	10	828.06
V064	10	902.95	10	824.78	10	957.25	10	824.78
V065	13	1043.82	10	828.94	10	828.94	10	828.94
V066	11	861.70	10	828.94	10	828.94	10	828.94
V067	11	887.15	10	828.94	10	828.94	10	828.94
V068	12	1007.06	10	828.94	11	929.68	10	828.94
V069	10	857.70	10	828.94	10	1004.01	10	828.94
V070	8	841.73	3	591.56	3	591.56	3	591.56
V071	7	770.52	3	591.56	3	591.56	3	591.56
V072	7	882.13	4	604.87	3	607.55	3	591.17
V073	4	691.14	4	619.72	4	619.72	3	590.60
V074	7	778.00	3	588.88	3	588.88	3	588.88
V075	6	723.07	3	588.49	3	588.49	3	588.49
V076	7	735.96	3	588.29	3	588.29	3	588.29
V077	6	747.06	3	588.32	3	588.32	3	588.32
V078	22	1739.39	20	1661.62	20	1666.38	19	1651.10
V079	20	1552.53	18	1479.89	18	1490.95	18	1473.62
V080	16	1292.23	14	1228.13	14	1240.78	14	1220.83
V081	13	1098.94	11	1004.13	11	1050.90	10	998.55
V082	18	1490.12	16	1415.67	15	1430.77	15	1387.27
V083	15	1289.28	14	1299.46	13	1300.65	13	1254.35
V084	14	1175.85	12	1109.57	12	1157.85	11	1085.49
V085	12	1028.78	11	1023.39	11	1022.22	10	959.41
V086	16	1263.15	14	1236.92	13	1250.02	13	1170.82
V087	14	1198.12	12	1166.70	12	1141.15	12	1098.27
V088	13	1116.39	12	1102.00	12	1108.97	11	1067.20
V089	12	1070.40	11	1040.84	11	1036.70	10	996.43
V090	10	1198.86	7	1189.34	7	1175.23	5	1203.62
V091	24	3272.67	20	2704.57	20	2806.19	20	2704.57
V092	24	3288.13	20	2700.65	20	2809.79	20	2700.65
V093	24	3387.46	20	2683.28	20	3289.11	19	2748.07
V094	23	3279.25	19	2682.45	19	3650.64	18	2752.97
V095	24	3298.38	20	2702.05	21	3099.08	20	2702.05
V096	26	3527.01	20	2701.04	23	3421.09	20	2701.04
V097	24	3345.68	20	2731.25	21	3594.87	20	2709.37
V098	24	3254.30	20	2690.27	21	3700.51	20	2690.28
V099	23	3214.45	20	2747.30	20	3822.84	19	2715.71
V100	23	3497.34	20	2740.26	20	4040.52	19	2762.32
V101	14	2029.16	7	1984.01	7	2015.82	7	1985.24
V102	15	2455.46	8	1959.74	7	1948.31	7	1858.65
V103	14	2519.10	8	1835.97	7	1919.23	8	1846.10
V104	11	2046.21	8	1801.53	7	1887.02	7	1820.95
V105	15	2346.39	6	1904.36	7	1965.48	7	1998.41
V106	12	2177.58	8	1990.58	7	1935.20	7	1868.78
V107	13	2374.65	7	1870.65	7	2052.80	7	1993.23
V108	11	2128.22	9	1956.92	7	1883.45	7	1958.15
V109	13	2244.06	8	1864.45	8	1922.70	7	1847.28
V110	12	2158.28	7	1805.33	8	1910.66	7	1828.30
V111	73	17,876.57	59	14,198.43	66	20,173.71	60	13,940.77
V112	65	16,260.82	56	13,572.86	60	19,185.98	55	13,373.04
V113	61	15,845.44	50	13,812.26	52	18,992.69	49	12,776.23
V114	53	13,961.63	47	13,221.43	48	19,841.14	47	12,698.50
V115	68	17,416.29	57	13,815.20	65	20,864.66	57	13,753.12
V116	66	16,562.71	58	13,907.92	66	22,066.91	56	13,650.83
V117	67	17,074.79	55	13,820.07	62	22,653.31	55	13,702.45
V118	66	16,950.98	54	13,881.50	63	23,500.70	53	13,668.19
V119	61	16,180.71	51	13,469.27	56	20,551.42	50	13,102.00
V120	56	14,896.15	49	13,846.39	56	21,457.53	49	13,607.89

Optimal solution for V091 can be shown as follows.

Optimal Solution:

Number of Vehicles: 20, Total Distances: 2704.5678, Number of Violated Constraint Routes: 0, Number of Violated Constraint Customers: 0

Route1: 0- > 190- > 5- > 10- > 193- > 46- > 128- > 106- > 167- > 34- > 95- > 158- > 0.

Route2: 0- > 177- > 3- > 88- > 8- > 186- > 127- > 98- > 157- > 137- > 183- > 0.

Route3: 0- > 32- > 171- > 65- > 86- > 115- > 94- > 51- > 174- > 136- > 189- > 0.

Route4: 0- > 114- > 159- > 38- > 150- > 22- > 151- > 16- > 140- > 187- > 142- > 111- > 63- > 56- > 0.

Route5: 0- > 170- > 134- > 50- > 156- > 112- > 168- > 79- > 29- > 87- > 42- > 123- > 0.

Route6: 0- > 30- > 120- > 19- > 192- > 196- > 97- > 14- > 96- > 130- > 28- > 74- > 149- > 0.

Route7: 0- > 21- > 23- > 182- > 75- > 163- > 194- > 145- > 195- > 52- > 92- > 0.

Route8: 0- > 62- > 131- > 44- > 102- > 146- > 68- > 76- > 0.

Route9: 0- > 57- > 118- > 83- > 143- > 176- > 36- > 33- > 121- > 165- > 188- > 108- > 0.

Route10: 0- > 148- > 103- > 197- > 124- > 141- > 69- > 200- > 0.

Route11: 0- > 113- > 155- > 78- > 175- > 13- > 43- > 2- > 90- > 67- > 39- > 107- > 0.

Route12: 0- > 93- > 55- > 135- > 58- > 184- > 199- > 37- > 81- > 138- > 0.

Route13: 0- > 60- > 82- > 180- > 84- > 191- > 125- > 4- > 72- > 17- > 0.

Route14: 0- > 45- > 178- > 27- > 173- > 154- > 24- > 61- > 100- > 64- > 179- > 109- > 0.

Route15: 0- > 164- > 66- > 147- > 160- > 47- > 91- > 70- > 0.

Route16: 0- > 73- > 116- > 12- > 129- > 11- > 6- > 122- > 139- > 0.

Route17: 0- > 101- > 144- > 119- > 166- > 35- > 126- > 71- > 9- > 1- > 99- > 53- > 0.

Route18: 0- > 161- > 104- > 18- > 54- > 185- > 132- > 7- > 181- > 117- > 49- > 0.

Route19: 0- > 133- > 48- > 26- > 152- > 40- > 153- > 169- > 89- > 105- > 15- > 59- > 198- > 0.

Route20: 0- > 20- > 41- > 85- > 80- > 31- > 25- > 172- > 77- > 110- > 162- > 0.

Optimal delivery route solved by ISSA for V091 is shown as Fig. 5.

Fig. 5.

Optimal delivery route solved by ISSA for V091.

Optimal solution for V096 can be shown as follows.

Optimal Solution:

Number of Vehicles: 20, Total Distances: 2701.0354, Number of Violated Constraint Routes: 0, Number of Violated Constraint Customers: 0

Route1: 0- > 73- > 116- > 12- > 129- > 11- > 6- > 122- > 139- > 0.

Route2: 0- > 20- > 41- > 85- > 80- > 31- > 25- > 172- > 77- > 110- > 162- > 0.

Route3: 0- > 32- > 171- > 65- > 86- > 115- > 94- > 51- > 174- > 136- > 189- > 0.

Route4: 0- > 164- > 66- > 147- > 160- > 47- > 91- > 70- > 0.

Route5: 0- > 60- > 82- > 180- > 84- > 191- > 125- > 4- > 72- > 165- > 188- > 108- > 0.

Route6: 0- > 177- > 3- > 88- > 8- > 186- > 127- > 98- > 157- > 138- > 0.

Route7: 0- > 133- > 48- > 26- > 152- > 40- > 153- > 169- > 89- > 105- > 15- > 59- > 198- > 0.

Route8: 0- > 114- > 159- > 38- > 150- > 22- > 151- > 16- > 140- > 187- > 142- > 111- > 63- > 56- > 0.

Route9: 0- > 170- > 134- > 50- > 156- > 112- > 168- > 79- > 29- > 87- > 42- > 123- > 0.

Route10: 0- > 190- > 5- > 10- > 193- > 46- > 128- > 106- > 167- > 34- > 95- > 158- > 0.

Route11: 0- > 45- > 178- > 27- > 173- > 154- > 24- > 61- > 100- > 64- > 179- > 109- > 0.

Route12: 0- > 113- > 155- > 78- > 175- > 13- > 43- > 2- > 90- > 67- > 39- > 107- > 0.

Route13: 0- > 93- > 55- > 135- > 58- > 184- > 199- > 37- > 81- > 137- > 0.

Route14: 0- > 30- > 120- > 19- > 192- > 196- > 97- > 14- > 96- > 130- > 28- > 74- > 149- > 0.

Route15: 0- > 148- > 103- > 197- > 124- > 141- > 69- > 200- > 0.

Route16: 0- > 101- > 144- > 119- > 166- > 35- > 126- > 71- > 9- > 1- > 99- > 53- > 0.

Route17: 0- > 21- > 23- > 182- > 75- > 163- > 194- > 145- > 195- > 52- > 92- > 0.

Route18: 0- > 57- > 118- > 83- > 143- > 176- > 36- > 33- > 121- > 17- > 0.

Route19: 0- > 62- > 131- > 44- > 102- > 146- > 68- > 76- > 0.

Route20: 0- > 161- > 104- > 18- > 54- > 185- > 132- > 7- > 181- > 117- > 49- > 183- > 0.

Optimal delivery route solved by ISSA for V096 is shown as Fig. 6.

Fig. 6.

Optimal delivery route solved by ISSA for V096.

Experimental results and analysis

Examples of PFSVRP problem contain examples of PFSP and examples of VRPTW, as shown in Table 7.

Table 7.

Examples of PFSVRP problem.

Problem	Instance	Problem	Instance
C001	ta001 + V001	C061	ta061 + V061
C002	ta002 + V002	C062	ta062 + V062
C003	ta003 + V003	C063	ta063 + V063
C004	ta004 + V004	C064	ta064 + V064
C005	ta005 + V005	C065	ta065 + V065
C006	ta006 + V006	C066	ta066 + V066
C007	ta007 + V007	C067	ta067 + V067
C008	ta008 + V008	C068	ta068 + V068
C009	ta009 + V009	C069	ta069 + V069
C010	ta010 + V010	C070	ta070 + V070
C011	ta011 + V011	C071	ta071 + V071
C012	ta012 + V012	C072	ta072 + V072
C013	ta013 + V013	C073	ta073 + V073
C014	ta014 + V014	C074	ta074 + V074
C015	ta015 + V015	C075	ta075 + V075
C016	ta016 + V016	C076	ta076 + V076
C017	ta017 + V017	C077	ta077 + V077
C018	ta018 + V018	C078	ta078 + V078
C019	ta019 + V019	C079	ta079 + V079
C020	ta020 + V020	C080	ta080 + V080
C021	ta021 + V021	C081	ta081 + V081
C022	ta022 + V022	C082	ta082 + V082
C023	ta023 + V023	C083	ta083 + V083
C024	ta024 + V024	C084	ta084 + V084
C025	ta025 + V025	C085	ta085 + V085
C026	ta026 + V026	C086	ta086 + V086
C027	ta027 + V027	C087	ta087 + V087
C028	ta028 + V028	C088	ta088 + V088
C029	ta029 + V029	C089	ta089 + V089
C030	ta030 + V030	C090	ta090 + V090
C031	ta031 + V031	C091	ta091 + V091
C032	ta032 + V032	C092	ta092 + V092
C033	ta033 + V033	C093	ta093 + V093
C034	ta034 + V034	C094	ta094 + V094
C035	ta035 + V035	C095	ta095 + V095
C036	ta036 + V036	C096	ta096 + V096
C037	ta037 + V037	C097	ta097 + V097
C038	ta038 + V038	C098	ta098 + V098
C039	ta039 + V039	C099	ta099 + V099
C040	ta040 + V040	C100	ta100 + V100
C041	ta041 + V041	C101	ta101 + V101
C042	ta042 + V042	C102	ta102 + V102
C043	ta043 + V043	C103	ta103 + V103
C044	ta044 + V044	C104	ta104 + V104
C045	ta045 + V045	C105	ta105 + V105
C046	ta046 + V046	C106	ta106 + V106
C047	ta047 + V047	C107	ta107 + V107
C048	ta048 + V048	C108	ta108 + V108
C049	ta049 + V049	C109	ta109 + V109
C050	ta050 + V050	C110	ta110 + V110
C051	ta051 + V051	C111	ta111 + V111
C052	ta052 + V052	C112	ta112 + V112
C053	ta053 + V053	C113	ta113 + V113
C054	ta054 + V054	C114	ta114 + V114
C055	ta055 + V055	C115	ta115 + V115
C056	ta056 + V056	C116	ta116 + V116
C057	ta057 + V057	C117	ta117 + V117
C058	ta058 + V058	C118	ta118 + V118
C059	ta059 + V059	C119	ta119 + V119
C060	ta060 + V060	C120	ta120 + V120

The comparison of the algorithms for PFSVRP problem is shown in Table 8. Here, , . ISSA is compared with SA, GA and PSO. It can be concluded that among 120 results, 87 better results are obtained by ISSA. 111 better results are obtained for ISSA compared to SA. 90 better results are obtained by ISSA compared with GA. ISSA compared with PSO, 108 better results are obtained. We calculate the performance improvement percentage of ISSA algorithm. The performance improvement percentage of the ISSA algorithm can be calculated using the following formula: . Here, represents compared algorithms (SA, GA, and PSO). PIS represents the percentage performance improvement of ISSA algorithm compared to SA algorithm. PIG represents the percentage performance improvement of ISSA algorithm compared to GA algorithm. PIP represents the percentage performance improvement of ISSA algorithm compared to PSO algorithm. In this way, the performance advantages of ISSA algorithm compared to other algorithms can be visually demonstrated.

Table 8.

Comparison of the algorithms for PFSVRP problem.

Problem	SA	GA	PSO	ISSA	PIS	PIG	PIP
C001	180411	180411	180411	180411	0.00	0.00	0.00
C002	186048	186048	186048	186048	0.00	0.00	0.00
C003	156571	156571	156571	156571	0.00	0.00	0.00
C004	177144	177144	177144	177144	0.00	0.00	0.00
C005	176111	176111	176111	176111	0.00	0.00	0.00
C006	172111	172111	172111	172111	0.00	0.00	0.00
C007	176511	176511	176511	176011	0.00	0.00	0.00
C008	173211	173211	173211	173211	0.00	0.00	0.00
C009	173835	171246	171246	171246	0.01	0.00	0.00
C010	174271	170503	170503	170503	0.02	0.00	0.00
C011	217903	217903	217903	217903	0.00	0.00	0.00
C012	223827	223827	223827	223827	0.00	0.00	0.00
C013	205763	205763	205763	205763	0.00	0.00	0.00
C014	197403	197403	197403	217356	− 0.10	− 0.10	− 0.10
C015	205371	201603	201603	216024	− 0.05	− 0.07	− 0.07
C016	206555	199118	199118	199394	0.03	0.00	0.00
C017	208052	208052	208052	209747	− 0.01	− 0.01	− 0.01
C018	308873	307898	307675	308150	0.00	0.00	0.00
C019	291123	289773	289773	289773	0.00	0.00	0.00
C020	272131	270574	271274	273301	0.00	− 0.01	− 0.01
C021	335000	329264	329264	329264	0.02	0.00	0.00
C022	340364	339644	339744	339644	0.00	0.00	0.00
C023	351295	347995	377995	347995	0.01	0.00	0.08
C024	329286	328173	327354	329286	0.00	0.00	− 0.01
C025	323570	323570	323870	323570	0.00	0.00	0.00
C026	334662	336321	334862	335661	0.00	0.00	0.00
C027	336773	336689	335834	336689	0.00	0.00	0.00
C028	325690	328189	325990	325990	0.00	0.01	0.00
C029	316910	316910	316910	316910	0.00	0.00	0.00
C030	334656	334236	334236	334236	0.00	0.00	0.00
C031	401046	381375	399222	381375	0.05	0.00	0.04
C032	401272	392051	392251	392051	0.02	0.00	0.00
C033	389300	370751	381734	370751	0.05	0.00	0.03
C034	392109	382764	389907	382764	0.02	0.00	0.02
C035	414769	395275	395275	395275	0.05	0.00	0.00
C036	412995	391875	391875	391875	0.05	0.00	0.00
C037	401296	381475	401455	381475	0.05	0.00	0.05
C038	377275	377275	377275	377275	0.00	0.00	0.00
C039	364175	364175	374654	364175	0.00	0.00	0.03
C040	403239	386640	386740	386740	0.04	0.00	0.00
C041	427539	410640	413640	410840	0.04	0.00	0.01
C042	428186	398623	399923	395723	0.08	0.01	0.01
C043	400298	392616	394416	390716	0.02	0.00	0.01
C044	440322	415423	415523	414723	0.06	0.00	0.00
C045	439437	408223	410823	406823	0.07	0.00	0.01
C046	418494	410763	417135	408963	0.02	0.00	0.02
C047	433985	417836	418936	416336	0.04	0.00	0.01
C048	620844	621894	645439	617910	0.00	0.01	0.04
C049	564532	566767	570916	564280	0.00	0.00	0.01
C050	552590	542227	547331	543168	0.02	0.00	0.01
C051	582175	578515	586493	579793	0.00	0.00	0.01
C052	655989	650606	656525	647980	0.01	0.00	0.01
C053	621925	612988	622188	609231	0.02	0.01	0.02
C054	601657	589950	600644	592945	0.01	− 0.01	0.01
C055	558272	555176	558880	553020	0.01	0.00	0.01
C056	615321	610506	619610	608010	0.01	0.00	0.02
C057	597370	582614	599131	582014	0.03	0.00	0.03
C058	589785	586009	600383	587794	0.00	0.00	0.02
C059	577456	574289	577263	568493	0.02	0.01	0.02
C060	634465	620631	623621	626673	0.01	− 0.01	0.00
C061	797982	797982	797982	797982	0.00	0.00	0.00
C062	797031	775482	782097	775482	0.03	0.00	0.01
C063	800337	773649	783474	765918	0.04	0.01	0.02
C064	772285	748834	788575	748834	0.03	0.00	0.05
C065	838146	773682	773682	773682	0.08	0.00	0.00
C066	772010	762182	762182	762182	0.01	0.00	0.00
C067	790745	773282	773282	773282	0.02	0.00	0.00
C068	811518	758082	788304	758082	0.07	0.00	0.04
C069	802110	793482	846003	793482	0.01	0.00	0.06
C070	784719	709668	709668	709668	0.10	0.00	0.00
C071	808756	754468	756468	754468	0.07	0.00	0.00
C072	800639	717261	719965	712251	0.11	0.01	0.01
C073	775042	753516	753816	745080	0.04	0.01	0.01
C074	812600	755864	761564	755864	0.07	0.00	0.01
C075	763621	723247	727947	723247	0.05	0.00	0.01
C076	751888	707587	707287	706787	0.06	0.00	0.00
C077	783718	736996	736696	736396	0.06	0.00	0.00
C078	1084317	1060186	1066314	1057930	0.02	0.00	0.01
C079	1054859	1031667	1037985	1029586	0.02	0.00	0.01
C080	972169	952939	958034	951049	0.02	0.00	0.01
C081	955382	931539	956670	927365	0.03	0.00	0.03
C082	1069636	1051301	1067531	1040681	0.03	0.01	0.03
C083	1020984	1024938	1033895	1009305	0.01	0.02	0.02
C084	983055	968871	988055	955947	0.03	0.01	0.03
C085	946634	947817	957566	925623	0.02	0.02	0.03
C086	1021645	1016376	1030106	997046	0.02	0.02	0.03
C087	990036	983210	989945	960881	0.03	0.02	0.03
C088	982117	978800	996691	966560	0.02	0.01	0.03
C089	959120	946552	957210	931629	0.03	0.02	0.03
C090	1008158	1007402	1011869	1009886	0.00	0.00	0.00
C091	2069001	1899871	1929057	1899871	0.08	0.00	0.02
C092	2035139	1859695	1898537	1862495	0.08	0.00	0.02
C093	2110338	1899084	2081733	1920721	0.09	− 0.01	0.08
C094	2072675	1893635	2184492	1919791	0.07	− 0.01	0.12
C095	2041914	1863015	1983424	1864315	0.09	0.00	0.06
C096	2092703	1844912	2064127	1844012	0.12	0.00	0.11
C097	2090504	1905975	2166661	1901011	0.09	0.00	0.12
C098	2050390	1881181	2187853	1882384	0.08	0.00	0.14
C099	2009435	1869290	2191852	1858513	0.08	0.01	0.15
C100	2117202	1890478	2284856	1896296	0.10	0.00	0.17
C101	1737448	1729103	1758246	1732872	0.00	0.00	0.01
C102	1864338	1722322	1744093	1697795	0.09	0.01	0.03
C103	1897530	1695291	1743369	1707530	0.10	− 0.01	0.02
C104	1751463	1683859	1732606	1690785	0.03	0.00	0.02
C105	1840417	1708208	1744444	1738023	0.06	− 0.02	0.00
C106	1786274	1726374	1735160	1696234	0.05	0.02	0.02
C107	1852195	1709295	1786040	1749669	0.06	− 0.02	0.02
C108	1781766	1731276	1732535	1737445	0.02	0.00	0.00
C109	1808818	1690635	1732210	1693284	0.06	0.00	0.02
C110	1792084	1683999	1741498	1694290	0.05	− 0.01	0.03
C111	7981671	6882329	8717713	6820131	0.15	0.01	0.22
C112	7558146	6740658	8471094	6708112	0.11	0.00	0.21
C113	7403232	6795878	8390107	6498469	0.12	0.04	0.23
C114	6849689	6625029	8641742	6483350	0.05	0.02	0.25
C115	7876287	6798660	8936198	6780136	0.14	0.00	0.24
C116	7634913	6830576	9316573	6766349	0.11	0.01	0.27
C117	7775337	6812021	9475893	6770335	0.13	0.01	0.29
C118	7760294	6835550	9756810	6787157	0.13	0.01	0.30
C119	7476513	6655581	8814226	6560400	0.12	0.01	0.26
C120	7130745	6815017	9129559	6754467	0.05	0.01	0.26

Conclusion

In this study, we proposed an Improved Salp Swarm Algorithm (ISSA) to address the Permutation Flow Shop Vehicle Routing Problem (PFSVRP). By comparing ISSA with Simulated Annealing (SA), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO), ISSA achieved relatively better results in multiple instances. The practical application of ISSA lies in its direct applicability to real-world industrial problems involving production scheduling and logistics distribution, especially in industries that require just-in-time production and delivery, such as automotive manufacturing and pharmaceuticals.

A direct application of the ISSA algorithm is in the automotive manufacturing industry, where production efficiency and logistics costs directly impact the competitiveness of enterprises. For example, an automobile manufacturing plant may need to schedule the production of multiple orders on several production lines and ensure that the completed orders are delivered to customers on time. The ISSA algorithm can help the factory optimize production scheduling and vehicle delivery routes, reducing the total cost of production and transportation while shortening delivery times. Specifically, the algorithm can determine which production line to process which order and the best delivery sequence for vehicles, thereby achieving collaborative optimization of production and logistics.

For instance, suppose an automobile manufacturing plant has multiple orders that need to be processed on three production lines and the finished components need to be delivered to different customers. The ISSA algorithm can provide the factory with an optimal production and delivery plan that minimizes the total production and logistics costs while meeting customer delivery time requirements. In this way, the ISSA algorithm not only improves production efficiency but also enhances customer satisfaction, providing strong support for the enterprise in fierce market competition.

Future research will explore the integration of ISSA with other intelligent algorithms to solve this problem.

Datasets

Examples of PFSP use 120 benchmarks of Taillard’s instances of permutation flow shop scheduling problem. The numbers are from ta001 to ta120.

Examples of VRPTW can be described as follows: V001 to V090 from Solomon benchmark, V091 to V120 from Gehring & Homberger benchmark, as shown in Table 9.

Table 9.

Examples of VRPTW.

Instance	Benchmark	Instance	Benchmark
V001	The first 20 customers of C101	V061	C101
V002	The first 20 customers of C102	V062	C102
V003	The first 20 customers of C103	V063	C103
V004	The first 20 customers of C104	V064	C104
V005	The first 20 customers of C105	V065	C105
V006	The first 20 customers of C106	V066	C106
V007	The first 20 customers of C107	V067	C107
V008	The first 20 customers of C108	V068	C108
V009	The first 20 customers of C109	V069	C109
V010	The first 20 customers of C201	V070	C201
V011	The first 20 customers of C202	V071	C202
V012	The first 20 customers of C203	V072	C203
V013	The first 20 customers of C204	V073	C204
V014	The first 20 customers of C205	V074	C205
V015	The first 20 customers of C206	V075	C206
V016	The first 20 customers of C207	V076	C207
V017	The first 20 customers of C208	V077	C208
V018	The first 20 customers of R101	V078	R101
V019	The first 20 customers of R102	V079	R102
V020	The first 20 customers of R103	V080	R103
V021	The first 20 customers of R104	V081	R104
V022	The first 20 customers of R105	V082	R105
V023	The first 20 customers of R106	V083	R106
V024	The first 20 customers of R107	V084	R107
V025	The first 20 customers of R108	V085	R108
V026	The first 20 customers of R109	V086	R109
V027	The first 20 customers of R110	V087	R110
V028	The first 20 customers of R111	V088	R111
V029	The first 20 customers of R112	V089	R112
V030	The first 20 customers of R201	V090	R201
V031	The first 50 customers of C101	V091	C1_2_1
V032	The first 50 customers of C102	V092	C1_2_2
V033	The first 50 customers of C103	V093	C1_2_3
V034	The first 50 customers of C104	V094	C1_2_4
V035	The first 50 customers of C105	V095	C1_2_5
V036	The first 50 customers of C106	V096	C1_2_6
V037	The first 50 customers of C107	V097	C1_2_7
V038	The first 50 customers of C108	V098	C1_2_8
V039	The first 50 customers of C109	V099	C1_2_9
V040	The first 50 customers of C201	V100	C1_2_10
V041	The first 50 customers of C202	V101	C2_2_1
V042	The first 50 customers of C203	V102	C2_2_2
V043	The first 50 customers of C204	V103	C2_2_3
V044	The first 50 customers of C205	V104	C2_2_4
V045	The first 50 customers of C206	V105	C2_2_5
V046	The first 50 customers of C207	V106	C2_2_6
V047	The first 50 customers of C208	V107	C2_2_7
V048	The first 50 customers of R101	V108	C2_2_8
V049	The first 50 customers of R102	V109	C2_2_9
V050	The first 50 customers of R103	V110	C2_2_10
V051	The first 50 customers of R104	V111	The first 500 customers of C1_6_1
V052	The first 50 customers of R105	V112	The first 500 customers of C1_6_2
V053	The first 50 customers of R106	V113	The first 500 customers of C1_6_3
V054	The first 50 customers of R107	V114	The first 500 customers of C1_6_4
V055	The first 50 customers of R108	V115	The first 500 customers of C1_6_5
V056	The first 50 customers of R109	V116	The first 500 customers of C1_6_6
V057	The first 50 customers of R110	V117	The first 500 customers of C1_6_7
V058	The first 50 customers of R111	V118	The first 500 customers of C1_6_8
V059	The first 50 customers of	V119	The first 500 customers of C1_6_9
V060	The first 50 customers of R201	V120	The first 500 customers of C1_6_10

Examples of PFSVRP problem contain examples of PFSP and examples of VRPTW, as shown in Table 10.

Table 10.

Examples of PFSVRP problem.

Problem	Instance	Problem	Instance
C001	ta001 + V001	C061	ta061 + V061
C002	ta002 + V002	C062	ta062 + V062
C003	ta003 + V003	C063	ta063 + V063
C004	ta004 + V004	C064	ta064 + V064
C005	ta005 + V005	C065	ta065 + V065
C006	ta006 + V006	C066	ta066 + V066
C007	ta007 + V007	C067	ta067 + V067
C008	ta008 + V008	C068	ta068 + V068
C009	ta009 + V009	C069	ta069 + V069
C010	ta010 + V010	C070	ta070 + V070
C011	ta011 + V011	C071	ta071 + V071
C012	ta012 + V012	C072	ta072 + V072
C013	ta013 + V013	C073	ta073 + V073
C014	ta014 + V014	C074	ta074 + V074
C015	ta015 + V015	C075	ta075 + V075
C016	ta016 + V016	C076	ta076 + V076
C017	ta017 + V017	C077	ta077 + V077
C018	ta018 + V018	C078	ta078 + V078
C019	ta019 + V019	C079	ta079 + V079
C020	ta020 + V020	C080	ta080 + V080
C021	ta021 + V021	C081	ta081 + V081
C022	ta022 + V022	C082	ta082 + V082
C023	ta023 + V023	C083	ta083 + V083
C024	ta024 + V024	C084	ta084 + V084
C025	ta025 + V025	C085	ta085 + V085
C026	ta026 + V026	C086	ta086 + V086
C027	ta027 + V027	C087	ta087 + V087
C028	ta028 + V028	C088	ta088 + V088
C029	ta029 + V029	C089	ta089 + V089
C030	ta030 + V030	C090	ta090 + V090
C031	ta031 + V031	C091	ta091 + V091
C032	ta032 + V032	C092	ta092 + V092
C033	ta033 + V033	C093	ta093 + V093
C034	ta034 + V034	C094	ta094 + V094
C035	ta035 + V035	C095	ta095 + V095
C036	ta036 + V036	C096	ta096 + V096
C037	ta037 + V037	C097	ta097 + V097
C038	ta038 + V038	C098	ta098 + V098
C039	ta039 + V039	C099	ta099 + V099
C040	ta040 + V040	C100	ta100 + V100
C041	ta041 + V041	C101	ta101 + V101
C042	ta042 + V042	C102	ta102 + V102
C043	ta043 + V043	C103	ta103 + V103
C044	ta044 + V044	C104	ta104 + V104
C045	ta045 + V045	C105	ta105 + V105
C046	ta046 + V046	C106	ta106 + V106
C047	ta047 + V047	C107	ta107 + V107
C048	ta048 + V048	C108	ta108 + V108
C049	ta049 + V049	C109	ta109 + V109
C050	ta050 + V050	C110	ta110 + V110
C051	ta051 + V051	C111	ta111 + V111
C052	ta052 + V052	C112	ta112 + V112
C053	ta053 + V053	C113	ta113 + V113
C054	ta054 + V054	C114	ta114 + V114
C055	ta055 + V055	C115	ta115 + V115
C056	ta056 + V056	C116	ta116 + V116
C057	ta057 + V057	C117	ta117 + V117
C058	ta058 + V058	C118	ta118 + V118
C059	ta059 + V059	C119	ta119 + V119
C060	ta060 + V060	C120	ta120 + V120

Author contributions

The contribution of Yanguang Cai is the financial support of the entire article. The contribution of Huajun Chen is the completion of the theoretical and experimental parts of the article.

Data availability

Dataset are represented at the end of the main manuscript.

Declarations

Competing interests

The authors declare no competing interests.