Optimizing task scheduling in cloud computing: a hybrid artificial intelligence approach

Abstract

The artificial intelligence-based allocation of resources can substantially reduce resource wastage and cost. Cloud resource allocation and management have appeared to be the central research direction. Computing through the cloud looks like improvements in network, parallel, and distributed computing. This presentation explains how scheduling works in a cloud environment by comparing it to natural selection, a concept that describes biological evolution. A strategy to reduce the time needed to find a solution by proposing an optimal solution. algorithm adds flexibility, adaptability, parallel processing, and global optimization. Furthermore, the cost function is used to find a beneficial solution by studying the operational completion time, cost of resources, and load balancing. benchmark problems tested through the algorithm show that it performs better than the other algorithms. The algorithm simultaneously solves the scheduling tasks of virtual machines and self-guided vehicles in a cloud computing environment. We Also tested for significant differences among algorithms and the number of jobs using an analysis of variance. Finally, task scheduling using the suggested algorithm indicates an improvement.

Keywords:

• Optimizing artificial intelligence
• cloud computing
• natural selection
• biological evolution task scheduling
• heterogeneous integer programming

1. Introduction

With the rapid development of internet technology, cloud computing has attracted People’s attention as an innovative model (Panda and Jana, 2015). Open service hacks and deploys technology in the cloud to generate revenue by selling computing and storage numbers to users (Yiqiu et al., 2019), meaning that users can easily access the network, pay for usage, and pay by model. Based on a previous study (Mesbahi et al., 2016). Firms and users can rent resources to service providers to save costs, and service providers can earn money by providing services quickly (Lin et al., 2014). Cloud computing has the advantages of excellent access, data security, unlimited storage, mobility, and enhanced collaboration by using virtual resources (Garg et al., 2013). Unlike rental services in cloud computing, only cloud vendors serve cloud users (Tsai et al., 2013). Owing to the need for air services, service providers have increased the quantity and quality of the services they provide (Agarwal and Jain, 2014). Currently, the focus has shifted to a more efficient and effective plan to improve resource use. These computational problems are NP-hard (Dordaie and Navimipour, 2018). Hence, researchers have shifted their focus toward cloud computing to address tasks such as pandas and similar endeavors (Panda and Jana, 2019) provided scheduling methods by considering time and resource usage for assessment. Hussain et al. (2021) proposed an efficient job scheduling technique along with energy savings. In addition, researchers have gained extra attention through heuristic algorithms with issues such as scheduling tasks and balancing of load. placement of the virtual machines and vehicle systems. This algorithm is a hybrid algorithm based on biological evolution and natural selection (Zhou et al., 2020) which effectively illustrates the concept by drawing parallels between cloud resource allocation and biological evolution, emphasizing adaptability and optimization. This strategy yields the solution as an initial and current state, along with crossover and mutation (Song et al., 2020). The solution space problem is a trend in the population (Pan et al., 2021). The hybrid technique switches the closest optimal solutions to optimal global solutions using an algorithm procedure (Zhu et al., 2022). From the perspective of this work, the hill-climbing algorithm was hybridized with a genetic algorithm (GA) to enhance the speed of the algorithm in the global search space (Pan et al., 2020). The contributions of this study are as follows.

• It creates an assessment function using the operational completion time, energy, and reliability.

• It suggests advanced hill-climbing. It merges with a new evolutionary trend.

• Comparison of the anticipated strategy with another algorithm employing 40 benchmark sets for testing with simulation. benchmark problems are utilized to evaluate the algorithm’s performance, demonstrating its superiority over other algorithms in resource allocation tasks within cloud computing environments.

• Utilizing analysis of variance (ANOVA) or evaluating the algorithm’s performance.

The design of this essay’s design is as follows, related work in the second part, the third part of the system model, the proposed algorithm in the fourth part, the fifth section introduces ANOVA, and finally the sixth section introduces analysis of results.

2. Related work

Cloud computing is a pool of resources that can be provisioned and deployed instantly upon user request (Keshanchi et al., 2017). If the number of user requests increases, the cloud computing workload decreases (Liu et al., 2014). Therefore, the algorithm is responsible for allocating the work needed by the user to the service to shorten the time to complete the work, improve the service, and reduce the cost, infrastructure balance, and load (Alhaidari et al., 2019; Geng et al., 2019; Halim and Hajamydeen, 2019). In cloud environments, scheduling strategies encompass various areas. Scheduling issues within the cloud can be categorized into distinct domains, and Quality of Service (QoS) based scheduling tasks can be further divided into different methods (Arunarani et al., 2019). He et al. (2003) proposed a service quality-based method to perform the next scheduling with the min- min heuristic. Wu et al. (2013) allocated resources to the analysis strategy tested it by loading weights on a specific criterion and found its performance to be good. Potluri and Rao (2020) discussed cost reduction and time lag issues and improved resource utilization. Ali et al. (2017) created a rule based on the behavior of the tasks and assigned them to the servers the algorithm groups work by specification and the tasks need to be queued for distribution. Hanini et al. (2019) developed a strategy to determine the number of virtual machines in conjunction with the management of virtual machine demands to manage energy consumption.

Ant colony optimization (ACO) is efficient for solving NP-complete problems, particularly dynamic scheduling task problems (Gupta and Garg, 2017). Liu et al. (2014) developed a model to minimize energy consumption by reducing the number of substantial clouds using ACO algorithms to solve the virtual machine placement problem. Xin et al. (2016) proposed a model to reduce energy consumption and operation time by advancing the scheduling system through a scheduling strategy based on ACO. Delavar et al. (2012) studied grid computing in terms of QoS for scheduling task problems through the ACO approach with an objective function to minimize the operational completion time and cost. Wu et al. (2021) developed a model to reduce resource waste using an assessment model to monitor changes in energy consumption. Kumar and Venkatesan (2019) designated hybrid algorithms with ACO and GA for multi-objective task scheduling systems without considering the security and load of virtual machines. Ragmani et al. (2020) considered an algorithm with alternating parameters to efficiently optimize the operation time and allocation of resources. Sun et al. (2013) developed an advanced ACO called the period ant colony optimization (PACO) algorithm to solve scheduling problems. Pandey et al. (2010) suggested a PSO algorithm to solve scheduling task problems to minimize the execution and transport costs as the objective functions and were then compared with the best resource selection (BRS) algorithm. Juan et al. (2012) addressed the cost vector and potential solutions were found using the PSO algorithm. Alsaidy et al. (2020) considered an objective function to maximize the effectiveness of the cost and consumption of resources using an advanced PSO algorithm. Wen et al. (2012) developed a hybrid model to enhance the convergence speed of an algorithm by using ACO and PSO.

Genetic Algorithms (GA) can solve NP-hard problems, particularly task scheduling. Kumar and Sharma (2020) proposed an enhanced genetic algorithm methodology for optimizing task selection by selecting the correct VM. Kumar and Verma (2012) suggested a hybrid genetic algorithm and contrasted its outcomes against a standard genetic algorithm utilizing a time-based objective function across various environments. Nagar et al. (2018) proposed a task scheduling model to reduce the time to completion by estimating the earliest completion time for some small projects using the predicted earliest time to finish (PEFT) genetic algorithm. Virangii et al. (2021) proposed a design to optimize the balance of product stability and processing time through hybrid electric search using a genetic algorithm. Manasrah and Ba Ali (2018) worked on a hybrid GA-PSO algorithm for resource allocation in a cloud environmentto minimize the workload and cost performance. Pirozmand et al. (2021) used cloud simulation software to solve the hybrid genetic algorithm problem to optimize virtual machine allocation and energy usage.

Fuzzy Algorithms (FA) are useful for solving NP problems, especially task scheduling. Fahmy (2010) developed a nonperiodic task scheduling model using a fuzzy time-based algorithm and minimizing all tasks over time. Lv et al. (2021) proposed a multi-objective algorithm to optimize queue processing. Zhou et al. (2019) introduced an Infrastructure as a Service (IaaS) model aimed at enhancing the operational speed of intelligent machines through elevated performance standards. Sujana et al. (2020) proposed a cost estimation matrix (CPM) based scheduling heuristic optimization algorithm that merges optimal scheduling with a virtual budget allocation technique. Rezaeipanah et al. (2022) developed a management system for virtual machines in cloud environments. the algorithm tackles scheduling tasks for both virtual machines and self-guided vehicles concurrently, showcasing its versatility and applicability in diverse cloud computing scenarios.

Based on the above kind of literature, it is evident that subsequent research gaps have been identified. Many research activities have been performed using heuristic algorithms or meta-heuristic algorithms, and the simultaneous exploration of task scheduling for machines and automated guided vehicles (AGVs) has not yet been explored using hybrid algorithms. In this study task scheduling was performed using a hybrid hill-climbing algorithm to investigate the task completion time. Numerous researchers have deliberated on this topic and heuristic strategies for cloud computing that have been documented in publications. However, most of them are executed separately not in a hybrid manner, and task scheduling strategies in hybridization have not been examined. Therefore, this study concentrates on scheduling tasks using hybrid cloud computing strategies.

3. System design

Scheduling tasks can be divided into job scheduling and task scheduling in cloud computing. Task scheduling is the most frequently used scheduling method (Padillo et al., 2017; Xu et al., 2016). An important issue in scheduling tasks is determining how to assign jobs to processors to achieve a minimized completion time and cost with high system performance and efficiency (Selvarani and Sadhasivam, 2010).

3.1. Cloud computing system design

In cloud computing, all tasks are independent and are not prioritized, as shown in Figure 1. When a user submits a task from the cloud, the task is queued by master task management and scheduling mechanisms to self-deploy tasks. Driving and using virtual machines. A virtual machine has characteristics such as the number of machines, processing environment, memory, and bandwidth task and possesses two defining attributes: the number of tasks and their respective lengths.

Figure 1. Task scheduling model through cloud computing.

3.2. Evaluation model

This study considers the scheduling task performance in terms of completion time, cost, and degree of imbalance. The performance indicators are explained in detail below: task scheduling performance mainly includes makespan, cost, profit, completion time, and waiting time through the indicators of cloud computing.

Notations

Table

j = job index	u = preceding operation
i = operation index	RTk j = Ij ∩ Ark, $I\in k,j$
J = job set	xj = job j ready time
nj = operation number in job j	xij = job j on machine i steady time.
n = $\sum _{j\in J}{n}_{j,}$ job set J total operations in	yk = ready time of vehicle k
I = {1, 2, 3……n} operation number	R(g) = machine ready time denoted by g
Ij = {Nj + 1, Nj + 2, ……Nj + nj}, I linked with j	R(i) = operation i machine ready time
IDi = I − {h; h ≥ i}, i without operation	mi = machine for operation
IDh = I − {i, i ≤ h} h without operation	S = {max[R(g), bk]}
TT = { } tool to carry out the job’s	OCT = {m1, m2, m4, m3} machines for operations
Tk = { } indices set in I linked with tool k in TT, $k\in \mathit{\text{TT}}$	Pti = operation i processing time
v = operation 1st	Cti = operation i completion time.

Variables $$p_{\mathit{\text{rs}}}=\left\{\begin{array}{c}1\mathit{\text{ if Cr}}<\mathit{\text{Cs}} \\ 0\mathit{\text{ if Cr}}\ge \mathit{\text{Cs}}\end{array}\right\}\mathrm{\text{jobs}}$$ $$q_{\mathit{\text{rs}}}=\left\{\begin{array}{c}1 \mathit{\text{ifCr}}<\mathit{\text{Cs}} \\ 0\mathit{\text{ if Cr}}\ge \mathit{\text{Cs}}\end{array}\right\}\mathrm{\text{machines}}$$ $$Z_{\mathit{\text{hi}}}=\left\{\begin{array}{c}1 \mathit{\text{tool without load}}\\ 0 \mathit{\text{otherwise}} \end{array}\right\}\mathrm{\text{AGVs}}$$ $$Z_{\mathit{\text{oi}}}=\left\{\begin{array}{c}1 \mathit{\text{tool starts from}} L/U\\ 0 \mathit{\text{otherwise}} \end{array}\right\}\mathrm{\text{AGVs}}$$ $$Z_{\mathit{\text{ho}}}=\left\{\begin{array}{c}1 \mathit{\text{tool returns from}} L/U\\ 0 \mathit{\text{otherwise}} \end{array}\right\}\mathrm{\text{AGVs}}$$ $$Z_{\mathit{\text{ijk}}}=\left\{\begin{array}{c}1 \mathit{\text{job}} j \mathit{\text{precedes job}} k\\ 0 \mathit{\text{otherwise}} \end{array}\right\}\mathrm{\text{Machine}}$$

Mixed integer linear programming (MILP) model (1) $${\text{minimum}C}_{\mathrm{\text{max}}}.$$(1)

Subjected to (2) $$X\mathrm{\text{ij}}\ge 0,\mathrm{Ɐ}j\in J,i\in M,$$(2) (3) $$x_{{\gamma }_h^i},j\ge x_{{\gamma }_{h-1}^i},j+p_{{\gamma }_{h-1}^i},j,\text{Ɐj}\in J,h=2,\dots \dots ,m,$$(3) (4) $$x_{\mathit{\text{ij}}}\ge t_{\mathit{\text{ik}}}+p_{\mathit{\text{ik}}}-C.z_{\mathit{\text{ijk}}},J,k\in J,j<k,i\in M,$$(4) (5) $$t_{\mathit{\text{ik}}}=x_{\mathit{\text{ij}}}+p_{\mathit{\text{ij}}}-C.(1-z_{\mathit{\text{ijk}}}),J,k\in J,j<k,i\in M,$$(5) (6) $${\mathit{\text{ct}}}_r\ge {\mathit{\text{ct}}}_s+{\mathit{\text{pt}}}_r-H{q}_{\mathit{\text{rs}}}\text{for all}r,s\in I,$$(6) (7) $${\mathit{\text{ct}}}_s\ge {\mathit{\text{ct}}}_r+{\mathit{\text{pt}}}_s-H{(1-q}_{\mathit{\text{rs}}})\text{for all}r,s\in \mathit{\text{Ij}},j,l\in J,j<\mathrm{l},$$(7) (8) $$c_{\mathrm{\text{max}}}=x_{{\gamma }_m^j},j+p_{{\gamma }_m^j},j\text{for all}j\in J,$$(8) (9) $${(1+\mathit{\text{Hq}}}_{\mathit{\text{rs}}}){\mathit{\text{ct}}}_r\ge {\mathit{\text{ct}}}_s+{\mathit{\text{pt}}}_r-H{q}_{\mathit{\text{rs}}}\text{Ɐ r}\in \mathit{\text{Ij}},$$(9) (10) $${(1+\mathit{\text{Hq}}}_{\mathit{\text{rs}}}){\mathit{\text{ct}}}_s\ge {\mathit{\text{ct}}}_r+{\mathit{\text{pt}}}_{\mathit{\text{rs}}}-H(1-q_{\mathit{\text{rs}}})\text{for all}r,s\in \mathit{\text{Ij}},j,l\in J,j<\mathrm{l},$$(10) (11) $$z_{\mathit{\text{ijk}}}\in \left\{0,1\right\},\text{for all}j,k\in J,i\in M,$$(11) $$C_{\mathrm{\text{ti}}}\ge 0\text{for all i}\in I$$ $$P_{\mathrm{\text{rs}}}=\left\{0.1\right\}\text{for all}r\in \mathit{\text{Ij}},S\in \mathit{\text{Ij}},\text{where}j,l,\in J,j<\mathrm{l},$$ $${\mathrm{q}}_{\mathrm{\text{rs}}}=\left\{0.1\right\}\text{for all}r\in \mathrm{\text{Ij}},S\in \mathit{\text{Ij}},\text{where}j,l,\in J,j<\mathrm{l}$$ constraint 1st is stated as an objective function. 2nd constraint is indicated as a job start time of ≥0. Limitation 3rd is a precedence constraint. Constraints 4th, 5th, 6th, and 7th Indicate that the two jobs cannot be programmed simultaneously on the same machine. Constraints 8, 9, 10, and 12 mandate that the operational conclusion time should not be any less than the maximum completion time of the final operation among all jobs. Finally, we evaluated the efficacy of the constraints in the experimental section. The indexing of machines and vehicles is unclear because each job routing is known in the design of the smart manufacturing industry through artificial intelligence. The aim is to minimize the completion time, formulated nonlinearly using mixed integer programming.

4. Schedule model

A suitable heuristic algorithm is crucial for obtaining the optimum solution for scheduling problems with minimum time. Attaining specific goals through heuristic algorithms with a set of rules and multiple options to establish the evolution function. This method enriches the efficiency of a search process by yielding the claims of the organization and the extensiveness of the best.

4.1. Hill climbing

To attain the peak of the mountain or problem a local search method, such as a hill-climbing algorithm, continuously transports in the direction of rising height or value. Through the algorithm, the problem reaches the highest value, and it terminates when no neighbor has a higher value. The various stages of hill climbing algorithms are represented graphically in Figure 2, and the pseudocode is presented in Algorithm 1. Evaluate the cost or objective function of both the current and previous states of the problem and then choose the superior one as the current state.

Figure 2. State-space vs objective function.

Algorithm 1.

Hill climbing algorithm.

Step 1: Initial solution = i

Step 2: f(s) ≤ f(i) S €Neighbours (i) do

Step 3: S €Neighbours (i), Generates;

Step 4: if cost (s) > cost (i) then

Step 5: Restore (s) with the I;

Step 6: Exit

In Algorithm 1, (i) the existing state within the solution space, denoted a represents the neighboring state in the hill climbing algorithm. After calculating the cost function, if the cost function of the neighboring state is greater than the existing state, then restore the neighboring state to the existing state or else keep the existing state as an optimum state for the next search. The above algorithm was integrated with the vehicle heuristic algorithm shown in Algorithm 2.

Algorithm 2.

Vehicle heuristic algorithm.

Step 1: parameters initialize

Step 2: generate initial solution ();

Step 3: machine number identification; M1–M4; ∀j $\in$ J; ∀ i $\in$ I

Step 4: V(1) = i(1); V(2) = i(2); V{1, 2}; i{1, 2…. n}

Step 5: POMN; previous operation machine number; M1–M4; ∀j $\in$ J; ∀ i $\in$I

Step 6: VRT; vehicle ready time; V(1, 2)

Step 7: VETT; vehicle empty travel time; V(1, 2); ∀j $\in$ J; ∀ i $\in$ I.

Step 8: VLTT; vehicle loaded travel time; V(1, 2); ∀j $\in$ J; ∀ i $\in$ I.

Step 9: Current location of the vehicle; Vi = Mi; ∀v $\in$

Step 10: Vehicle availability; If V(1) TT ≤ V(2)TT; V1; otherwise V(2)

Step 11: termination criterion

Step 12: Exit

In Algorithm 2, the operation sequence is identified through the algorithm: according to the sequence identifying the machine number the 1st AGV assigns to the first job after the 2nd AGV assigns to the second job. From 3rd job onwards, the vehicle heuristic algorithm is incorporated to assign tasks to vehicles depending on their availability. After obtaining the optimal sequence through an integrated hill climbing algorithm, a neighbor sequence is generated for crossover and mutation operations, which a shown in Algorithm 3. The algorithm incorporates flexibility, adaptability, parallel processing, and global optimization techniques to speed up finding optimal resource allocation solutions.

Algorithm 3.

Genetic Algorithm.

Step 1: Initiation

Step 2: Population generation (); P_g; g{1, 2 … n}

Step 3: Fitness function; OCT; ∀j $\in$ J; ∀ i $\in$I

Step 4: Repeat; i{1, 2 … n}

Step 5: Selection; ∀j $\in$ J; ∀ i $\in$ I; M1–M4;

Step 6: Crossover; Pg; and generate OCTCg

Step 7: Mutation; Pg; and generate OCTMg

Step 8: Fitness function; OCT; ∀j $\in$ J; ∀ i $\in$I

Step 9: termination criterion

Step 10: Exit

In Algorithm 3, after population generation from Algorithms 1 and 2 crossover and mutation operations are implemented to obtain an enhanced optimum solution. The hybrid algorithm is an alternative to the regular metaheuristic that we demonstrate in Figure 3.

Figure 3. Hybrid hill climbing algorithm.

4.2. Experiments

Hill Climbing (HC) heuristic search implementation job number 1 out of 10 numbers and layout numbers 1, out of four layouts deliberated as an example (Ulusoy et al., 1997). A coding system is utilized to label the example problems found in the initial column of Table 5. The digits following “1.1” signify the specific job set and layout, considering both process time and travel time. Following this, the digit succeeding “1.10” indicates the job set and layout, with the processing time doubled and the travel time halved, resulting in an additional set of 40 problems. Similarly, the digit after “1.11” denotes the job set and layout, with the processing time tripled and the travel time halved, thus generating another set of 40 problems (Alla et al., 2023). The algorithm evaluates the CT for ten jobs and the sequence is accomplished based on the neighbor.

Hill climbing algorithm enlightened for job number 1:

Step 1: Select the job number from 10 jobs.

Job number 1, layout number 1, number of jobs 5, number of operations 13.

Step 2: Current state.

4-5-6-1-2-3-12-13-10-11-7-8-9

According to the search, the technique evaluates the cost function as shown in Table 1. The cost function evaluates operational completion time (OCT), cost of resources, and load balancing to determine the most advantageous resource allocation solutions.

Table 1. Cost function calculation for J1 and L1 benchmark problems.

O_No	M_No	V_No	T_T	J_R	JRH	OCT
O4	M1	V1	0	6	6	26
O5	M3	V2	26	34	34	44
O6	M2	V1	44	50	50	68
O1	M1	V2	42	48	48	56
O2	M2	V2	56	62	68	84
O3	M4	V1	84	92	92	104
O12	M3	V2	72	82	82	92
O13	M1	V2	92	100	100	115
O10	M4	V1	98	110	110	124
O11	M2	V1	124	132	132	150
O7	M3	V2	112	122	122	134
O8	M4	V2	134	140	140	148
O9	M1	V1	148	158	158	173

Step 3: Generate a neighbor sequence of the current state.

4-5-6-12-13-1-2-3-10-11-7-8-9—fitness function-190

Step 4: Compare these two solutions and select the best one

Sequence 1: 4-5-6-1-2-3-12-13-10-11-7-8-9: fitness function: 173

Sequence 2: 4-5-6-12-13-1-2-3-10-11-7-8-9: fitness function: 190

The best sequence was 4-5-6-1-2-3-12-13-10-11-7-8-9 with an OCT of 173.

Step 5: Generate another neighbor sequence from the best one.

4-5-6-1-2-3-10-11-12-13-7-8-9 -fitness function-179

Step 6: Compare the OCT for the new neighbor solution and the best one from the previous comparison and select the finest one.

The best sequence is sequence 1.

Create an additional neighboring sequence to incorporate into the Genetic Algorithm

Step 7: Combine the genetic algorithm and hill climbing algorithm

Holland et al. (1975) developed genetic algorithms (GAs) which are applied to different fields of engineering problems very effectively. It is very popular among all the algorithms. The algorithm mechanism is based on the natural evolutionary process simplifications shown in Figure 4.

Figure 4. Population, chromosome, and genes in the genetic algorithm.

4.2.1 Genetic operators

Crossover and mutations are accomplished to produce new chromosomes.

4.2.2 Crossover operators

Based on the literature survey, single- and double-point crossovers were deemed as shown in Figure 5.

Figure 5. The crossover position within the genetic algorithm.

(i) Single-point crossover

parents are swapped with the genes.

Before:

    Random sequence 1: 2-1|-4-5-3

    neighbor sequence 2: 2-4|-1-3-5

After:

    Random sequence 1: 2-1-1-3-5

    neighbor sequence 2: 2-4-4-5-3

After repair and replacement:

    Random sequence 1: 2-1-4-3-5

    neighbor sequence 2: 2-4-1-5-3

According to job order operations, the sequences are: 4-5-6-1-2-3-10-11-7-8- 9-12-13-OCT-.179

4-5-6-10-11-1-2-3-12-13-7-8-9-OCT-185

(ii) Two-point crossover

genes in between two cut points are exchanged.

Random sequence 1: 2|-1-4-3|-5

neighbor sequence 2: 2-4-1-5-3

After Repair and replacement:

Random sequence 1: 2|-4-1-3|-5neighbor sequence 2: 2-|1-4-5|-3

According to job order operations sequences are:

4-5-6-10-11-1-2-3-7-8-9-12-13- OCT- 178

4-5-6-1-2-3-10-11-12-13-7-8-9 - OCT- 197

From crossover operations select the best one for the next operation:

4-5-6-10-11-1-2-3-7-8-9-12-13- OCT- 178

4.2.3 Mutation operators

Mutation operations performed on chromosomes to efficiently search for the search space are shown in Figure 6.

Figure 6. The mutation process within a GA.

4.2.4 Random mutation

Randomly selecting chromosomes is repaired, if necessary, after swapping two genes randomly.

Before: 2-4-1-3-5, After: 3-4-1-2-5

Operation sequence is: 7-8-9-10-11-1-2-3-4-5-6-12-13-OCT-173

4.2.5 Adjacent mutation

Exchange the genes that are adjacent to each other.

Before 3-4-1-2-5, After 3-4-2-1-5

The operation sequence was 7-8-9-10-11-4-5-6-1-2-3-12-13-OCT-173, shown in Table 2.

Table 2. Benchmark problems calculation for neighbor sequence.

O_No	M_No	V_No	T_T	J_R	JR_H	OC_T
O7	M3	V1	0	10	10	22
O8	M4	V2	22	28	28	36
O9	M1	V1	36	46	46	61
O10	M4	V2	34	46	46	60
O11	M2	V2	60	68	68	86
O4	M1	V1	58	64	64	84
O5	M3	V1	84	92	92	102
O6	M2	V2	102	108	108	126
O1	M1	V1	100	106	106	114
O2	M2	V1	114	120	126	142
O3	M4	V2	142	150	150	162
O12	M3	V1	130	140	140	150
O13	M1	V1	150	158	158	173

Step 7: Repeat the procedure until execution.

To get the effective performance of the algorithm at least 10 to 20 runs are required to note down.

The different functions and layouts used to determine the completion time through HCA are presented in Table 3.

Table 3. Cost function through hill-climbing algorithm.

J.No	t/p	HCA	J.No	t/p	HCA	J.No	t/p	HCA	J.No	t/p	HCA
1.1	0.59	107	1.2	0.47	92	1.3	0.52	92	1.4	0.74	118
2.1	0.61	116	2.2	0.49	94	2.3	0.54	96	2.4	0.77	126
3.1	0.59	121	3.2	0.47	101	3.3	0.51	98	3.4	0.74	135
4.1	0.91	127	4.2	0.73	107	4.3	0.8	106	4.4	1.14	144
5.1	0.85	95	5.2	0.68	76	5.3	0.74	81	5.4	1.06	103
6.1	0.78	129	6.2	0.54	111	6.3	0.54	112	6.4	0.78	140
7.1	0.78	136	7.2	0.62	100	7.3	0.68	108	7.4	0.97	151
8.1	0.58	170	8.2	0.46	151	8.3	0.5	155	8.4	0.72	190
9.1	0.61	133	9.2	0.49	124	9.3	0.53	127	9.4	0.76	138
10.1	0.55	166	10.2	0.44	157	10.3	0.49	167	10.4	0.69	194

The above table gives the test results for the 40 problems using the hill climbing algorithm for the travel time vs process time ratio (t/p > 0.25). The OCT for job sets and layouts for the genetic hill climbing algorithm is presented in Table 4.

Table 4. Cost function through the hybrid algorithm.

F(xi)	ti/pi	GAHCA	F(xi)	ti/pi	HCA	F(xi)	ti/pi	HCA	F(xi)	ti/pi	HCA
1.1	0.59	102	1.2	0.47	84	1.3	0.52	88	1.4	0.74	111
2.1	0.61	112	2.2	0.49	89	2.3	0.54	94	2.4	0.77	122
3.1	0.59	116	3.2	0.47	95	3.3	0.51	97	3.4	0.74	127
4.1	0.91	125	4.2	0.73	100	4.3	0.8	102	4.4	1.14	141
5.1	0.85	89	5.2	0.68	73	5.3	0.74	76	5.4	1.06	101
6.1	0.78	125	6.2	0.54	110	6.3	0.54	108	6.4	0.78	139
7.1	0.78	130	7.2	0.62	93	7.3	0.68	100	7.4	0.97	135
8.1	0.58	165	8.2	0.46	151	8.3	0.5	153	8.4	0.72	175
9.1	0.61	131	9.2	0.49	112	9.3	0.53	117	9.4	0.76	133
10.1	0.55	167	10.2	0.44	153	10.3	0.49	154	10.4	0.69	191

Table 5. Percentage improvement between CDS and HCS.

Job	t/p	CDS (Prasad et al., 2023)	HCA	% of Improvement
4.3	0.8	234	106	120.75
9.4	0.76	295	138	113.77
4.2	0.73	225	107	110.28
4.4	1.14	299	144	107.64
1.2	0.47	190	92	106.52
4.1	0.91	260	127	104.72
9.2	0.49	249	124	100.81
9.1	0.61	263	133	97.74
9.3	0.53	251	127	97.64
5.2	0.68	149	76	96.05
10.1	0.55	312	166	87.95
10.2	0.44	284	157	80.89
10.4	0.69	348	194	79.38
5.3	0.74	145	81	79.01
5.4	1.06	182	103	76.70
3.3	0.51	173	98	76.53
3.2	0.47	178	101	76.24
1.3	0.52	162	92	76.09
6.1	0.78	225	129	74.42
3.1	0.59	211	121	74.38
6.4	0.78	237	140	69.29
10.3	0.49	279	167	67.07
3.4	0.74	224	135	65.93
1.1	0.59	173	107	61.68
6.2	0.54	179	111	61.26
1.4	0.74	189	118	60.17
5.1	0.85	147	95	54.74
8.1	0.58	261	170	53.53
7.4	0.97	227	151	50.33
8.4	0.72	285	190	50.00
2.1	0.61	172	116	48.28
2.2	0.49	137	94	45.74
7.1	0.78	194	136	42.65
6.3	0.54	156	112	39.29
7.2	0.62	139	100	39.00
2.4	0.77	174	126	38.10
2.3	0.54	130	96	35.42
7.3	0.68	142	108	31.48
8.2	0.46	181	151	19.87
8.3	0.5	183	155	18.06

The above table gives the test results for the 40 problems using the genetic hill climbing algorithm, which includes crossover, again with the mutations for the travel time vs process time ratio (t/p > 0.25). Table 5 illustrates the performance of CDS (Prasad et al., 2023) and HCA, showing the percentage improvement between the algorithms.

The highest percentage improvement is 120, and the lowest percentage improvement is 18.06 when comparing HCA with the CDS (Prasad et al., 2023) algorithm while observing from the above table. Table 6 lists the performance metrics of the HCS and GHCA, including the percentage improvement between the HCA and GHCA.

Table 6. Percentage improvement between HCS and GHCS.

Job	t/p	HCA	GHCA	% Improvement
7.4	0.97	151	135	11.85
9.2	0.49	124	112	10.71
1.2	0.47	92	84	9.52
8.4	0.72	190	175	8.57
9.3	0.53	127	117	8.55
10.3	0.49	167	154	8.44
7.3	0.68	108	100	8.00
7.2	0.62	100	93	7.53
4.2	0.73	107	100	7.00
5.1	0.85	95	89	6.74
5.3	0.74	81	76	6.58
3.2	0.47	101	95	6.32
1.4	0.74	118	111	6.31
3.4	0.74	135	127	6.30
2.2	0.49	94	89	5.62
1.1	0.59	107	102	4.90
7.1	0.78	136	130	4.62
1.3	0.52	92	88	4.55
3.1	0.59	121	116	4.31
5.2	0.68	76	73	4.11
4.3	0.8	106	102	3.92
9.4	0.76	138	133	3.76
6.3	0.54	112	108	3.70
2.1	0.61	116	112	3.57
2.4	0.77	126	122	3.28
6.1	0.78	129	125	3.20
8.1	0.58	170	165	3.03
10.2	0.44	157	153	2.61
2.3	0.54	96	94	2.13
4.4	1.14	144	141	2.13
5.4	1.06	103	101	1.98
4.1	0.91	127	125	1.60
10.4	0.69	194	191	1.57
9.1	0.61	133	131	1.53
8.3	0.5	155	153	1.31
3.3	0.51	98	97	1.03
6.2	0.54	111	110	0.91
6.4	0.78	140	139	0.72
8.2	0.46	151	151	0.00
10.1	0.55	166	167	−0.60

The highest percentage improvement of the genetic hill climbing algorithm was 11.85 and the lowest percentage improvement of the genetic hill climbing algorithm was −0.60 when compared with the HCA.

4.3. Cloud environment

Following the simulation, two distinct cloud computing environments were established to assess the effectiveness of the algorithms. The results are presented in Tables 7 and 8.

Table 7. Simulation cloud environment 1.

Entity	Constraint	Value	Entity	Constraint	Value
Virtual machine	No of VM	6	Job	No of jobs	40–120
	RAM	512 MB		Length	50–5000
	Bandwidth	1000 MB	Host	No of Hosts	2
	MIPS	100–1000		Bandwidth	10000
	Type of policy	Time share		RAM	2048 MB
	size	10,000		Storage	1,000,000
	OS	Linux	Datacenter		2
	CPU	1
	VVM	Xen

Table 8. Simulation cloud environment 2.

Entity	Constraint	Value	Entity	Constraint	Value
Virtual machine	No Virtual Machines	6	Job	No of jobs	40–120
	RAM	128–15,360 MB		Length	100–10,000
	Bandwidth	128–15,360 MB	Host	No of Hosts	2
	MIPS	256–30,720		Bandwidth	10 GB
	Type of policy	Time share		RAM	20 GB
	size	10 GB		Storage	1TB
	OS	Linux	Datacenter		2
	CPU	1
	VVM	Xen

Tables 7 and 8 illustrate how the two meteorological conditions aid in demonstrating the stability and performance of the algorithm. The Wilcoxon signed-rank test scores for the CDS, HCA, and GHCA algorithms are displayed in Table 9 when the correlation was set to 0.05. The (+) sign indicates that GHCA performance is the best at the level of significance 0.05. Current operation and indicator. This shows that the GHCA performs worse than the algorithm.

Table 9. Comparison with the results of CDS, HCA, and GAHCA at a significant level a = 0.05.

F(x)	CDS (Alla et al., 2023)	HCA	GAHCA	F(x)	CDS	HCA	GAHCA
F1	234	106 +	135−	F21	237	140−	102+
F2	295	138−	112+	F22	279	167−	133+
F3	225	107−	84+	F23	224	135	108+
F4	299	144+	175−	F24	173	107+	112−
F5	190	92+	117−	F25	179	111+	122−
F6	260	127+	154−	F26	189	118+	125−
F7	249	124−	100+	F27	147+	95+	165−
F8	263	133−	93+	F28	261	170−	153+
F9	251	127−	100+	F29	227	151−	94+
F10	149	76+	89−	F30	285	190−	141+
F11	312	166−	76+	F31	172	116-	101+
F12	284	157−	95+	F32	137	94+	125−
F13	348	194−	111+	F33	194	136+	191−
F14	145	81+	127−	F34	156	112+	131−
F15	182	103−	89+	F35	139+	100+	153−
F16	173	98+	102−	F36	174	126−	97+
F17	178	101+	130−	F37	130	96+	110−
F18	162	92−	88+	F38	142	108+	139−
F19	225	129−	116+	F39	181	151≈	151≈
F20	211	121−	73+	F40	183	155+	167−

Table 9 shows that the GAHCA algorithm outperforms the CDS and HCA in terms of overall performance.

5. Significant differences among algorithms and number of jobs

The Friedman test is a non-parametric test developed by Friedman. As with ANOVA, differences between the treatments were detected. Analysis of variance (ANOVA) is a tool used to test the equality of two or more individual instruments. According to R.A Fisher, ANOVA is “the separation of the variable in one group due to the variable’s association with the other group.” Statistics can be categorized as univariate (univariate) or multivariate. Two levels (two-way). The significance of differences was assessed using an analysis of variance (ANOVA), which allowed for a statistical comparison of algorithm performances and their scalability with varying job loads.

5.1. Friedman test

It is an alternative to one-way ANOVA with frequent measures.

5.1.1 Elements

Three or more strategies of measures at investigational conditions. There was only one dependent it can be a ratio interval or ordinal.

5.1.2 Assumptions

A random sample from the population and the sample is not normally distributed.

5.1.3 Hypothesis

Classified as alternate and null hypothesis.

Null hypothesis: Probability distributions for conditions are the same.

Alternate hypothesis: Differing from each other there are at least two of them.

H1: At least two of them show significant differences.

H0: P1 = P2 = P3 = …… Pk; p = Median

5.1.4 Statistics for assessment

n = Number of jobs total, k = measured blocks

R_k = ranks sum for a strategy i.

5.1.5 Decision rules

Decision-making is based on below below-mentioned rules.

Fr > Critical value—reject the hypothesis.

P-value ≤ (alpha)—reject the null hypothesis.

Post hoc analysis was performed if the null hypothesis was rejected to find a given pair difference.

5.1.6 Friedman test

Step 1: Define the hypothesis.

H0 (null hypothesis): All three Strategies have the same probability distribution.

H1 (alternate hypothesis)

Step 2: Significance

Significance level = 0.05

Step 3: DOF calculation

DF = number of strategies to be measured (K) − 1, DF = 3 − 1 = 2.

Step 4: Chi-Square value.

alpha = 0.05 and DF = 2. Critical Chi-square value = 5.991

Step 5: Decision rule

You have the option to verify either of the two rules

FR > 5.991, reject the Null Hypothesis.

P-value ≤ (alpha), reject the null hypothesis.

Step 6: The ranks are in ascending order and are shown in Table 10.

Table 10. Ranks according to Friedman test.

F(x)	RANK	RANK	RANK	F(x)	RANK	RANK	RANK
F1	R3	R2	R1	F21	R3	R1	R2
F2	R3	R1	R2	F22	R3	R1	R2
F3	R3	R1	R2	F23	R3	R1	R2
F4	R3	R2	R1	F24	R3	R2	R1
F5	R3	R2	R1	F25	R3	R2	R1
F6	R3	R2	R1	F26	R3	R2	R1
F7	R3	R1	R2	F27	R2	R3	R1
F8	R3	R1	R2	F28	R3	R1	R2
F9	R3	R1	R2	F29	R3	R1	R2
F10	R3	R2	R1	F30	R3	R1	R2
F11	R3	R1	R2	F31	R3	R1	R2
F12	R3	R1	R2	F32	R3	R2	R1
F13	R3	R1	R2	F33	R3	R2	R1
F14	R3	R2	R1	F34	R3	R2	R1
F15	R3	R1	R2	F35	R2	R3	R1
F16	R3	R2	R1	F36	R3	R1	R2
F17	R3	R2	R1	F37	R3	R2	R1
F18	R3	R1	R2	F38	R3	R2	R1
F19	R3	R1	R2	F39	R3	R1.5	R1.5
F20	R3	R1	R2	F40	R3	R2	R1
					∑=118	∑= 61.5	∑= 60.5

Step 7: Statistics assessment calculation $$\mathrm{Q}=\frac{12}{\mathit{\text{nk}}(k+1)}*\sum R_j^2-3n(k+1),$$ $$\mathrm{Q}=\frac{12}{40*3(3+1)}*21366.5-3*40\left(3+1\right)\mathrm{Q}=534.1625-480,\mathrm{Q}=54.1625$$ where n = number of jobs, k = number of strategies, and R_j² = sum of ranks for the jth group.

Step 8: Result

Rejected the null hypothesis because Q was greater than the Critical Chi-square value. Therefore, we conclude that. There is a notable contrast in the decrease in the average OCT in a minimum of one pair of methods. Finally, the ANOVA test is presented in Section 5.2.

5.2. Two-way ANOVA

Step 1: Hypotheses

Null Hypotheses: H01: μM1= μM2= μM3 (for treatments) (algorithms)

In other words, there was an absence of notable divergence in the average execution time among the three algorithms.

H02: μS1= μS2= μS3 = μS4 = …… = μS40 (for blocks) (number of jobs)

Alternative Hypotheses:

H11: There exists a distinction among the three algorithms.

H12: There is a difference among the means of at least one of the 40 items compared to the other.

Step 2: Data

Information retrieved from Tables 7 and 8

Step 3: significance level α = 5%

Step 4: Statistic test

F0t (treatment) = MST/MSE, F0b (block) = MSB/MSE

Step 5: Calculation of the Test Statistic

Table

S.No	CDS (Prasad et al., 2023)	HCA	GAHCA	Xi	Xi^2
X1	234	106	135	475	225,625
X2	295	138	112	545	297,025
X3	225	107	84	416	173,056
X4	299	144	175	618	381,924
X5	190	92	117	399	159,201
X6	260	127	154	541	292,681
X7	249	124	100	473	223,729
X8	263	133	93	489	239,121
X9	251	127	100	478	228,484
X10	149	76	89	314	98,596
X11	312	166	76	554	306,916
X12	284	157	95	536	287,296
X13	348	194	111	653	426,409
X14	145	81	127	353	124,609
X15	182	103	89	374	139876
X16	173	98	102	373	139,129
X17	178	101	130	409	167,281
X18	162	92	88	342	116,964
X19	225	129	116	470	220,900
X20	211	121	73	405	164,025
X21	237	140	102	479	229,441
X22	279	167	133	579	335,241
X23	224	135	108	467	218,089
X24	173	107	112	392	153,664
X25	179	111	122	412	169,744
X26	189	118	125	432	186,624
X27	147	95	165	407	165,649
X28	261	170	153	584	341,056
X29	227	151	94	472	222,784
X30	285	190	141	616	379,456
X31	172	116	101	389	151,321
X32	137	94	125	356	126,736
X33	194	136	191	521	271,441
X34	156	112	131	399	159,201
X35	139	100	153	392	153,664
X36	174	126	97	397	157,609
X37	130	96	110	336	112,896
X38	142	108	139	389	151,321
X39	181	151	151	483	233,289
X40	183	155	167	505	255,025
Total Xi	8444	4994	4786	18,224	8,587,098
Xi^2	71,301,136	24,940,036	22,905,796

Squares: $$\sum \sum {\mathrm{ X}}_{\mathrm{\text{ij}}}{}^2=\text{3165458}$$

Correction factor: $$\text{C F}=G^2/n={\left(18224\right)}^2/120=332114176/120=2767618$$

Total sum of squares: $$\begin{array}{c}\text{TSS=}\sum \sum \text{Xij2‐CF}\\ \text{=3165458‐2767618 = 397840}\end{array}$$

Treatments due to sum of squares: $$\begin{array}{c}\text{SST}\left(\text{number of jobs}\right)=\frac{\sum _{i=1}^k{X}_j^2}{K}–\text{C F}\\ =8587098/3–2767618=2862366‐2767618=94748\end{array}$$

Blocks due to sum of squares: $$\mathrm{\text{SSB}}\left(\mathrm{\text{algorithms}}\right)=\frac{\sum _{i=1}^k{X}_i^2}{K}–\text{C F}=119146968/40–2767618=211056$$

Error due to sum of squares: $$\text{SSE}=\text{TSS}–\text{SST}–\text{SSB}=\text{39784}0–\text{94748}–\text{211}056=\text{92}03$$

Mean sum of squares for treatment: MST = $\frac{\mathrm{\text{SST}}}{k-1}$ = $\frac{94748}{40-1}$= 2429

Mean sum of squares for blocks: $$\mathrm{\text{MSB}}=\frac{\mathrm{\text{SSB}}}{m-1}=\frac{211056}{3-1}=105528.$$

Mean sum of squares for errors: $$\mathrm{\text{MSE}}=\frac{\mathrm{\text{SSE}}}{(k-1)(m-1)}=\frac{92036}{(40-1)(3-1)}=1180$$

ANOVA table (two-way)

Table

Variation	Squares sum	DOF	Squares mean	F-ratio
Treatments between	94,748	39	2429	F0t = 2429/1180 = 2.05
Blocks between	211,056	2	105528	F0b = 105,528/1180 = 89.43
Error	92,036	78	1180
Total		119

Step 6: Critical value

Values for treatment and blocks

F _{(39,78) 0.05} = 1.55

F _{(02,78) 0.05} = 3.113

Step 7: Decision

1. Calculation F0t = 2.05 > f(39.78)0.05 = 1.55, and by rejecting the null hypothesis, we deduce that there is no substantial difference in OCT among our methods. The conclusion is presented in this section.

2. Calculate F0b = 89.43 > f(2,78)0.05 = 3.113, rejecting the null hypothesis and the conclusion drawn is that there exists no notable difference in the OCT across forty questions.

Figure 2 illustrates the performance of the algorithm for non-equilibrium tasks in two distinct environments, with a range of 50 to 500. The x-axis denotes the function, and the y-axis represents the measurement function value. A comparison between CDS, HCA, and GHCA is shown in the figure below.

Figure 7 shows the performance of the HCA, GAHCA, and CDS algorithms within a smart manufacturing environment, utilizing 10 tasks with layout number 1. The X-axis indicates the number of jobs, while the y-axis represents the performance evaluation value.

Figure 7. Number of jobs vs performance matrix for layout 1.

Figure 8 shows the performance of the HCA, GAHCA, and CDS algorithms in a smart manufacturing setting, employing 10 tasks with layout number 2. The X-axis denotes the number of jobs, and the y-axis represents the performance evaluation value.

Figure 8. Number of jobs vs performance matrix for layout 2.

Figure 9 illustrates the performance of the HCA, GAHCA, and CDS algorithms within a smart manufacturing environment using 10 tasks with layout number 3. The X-axis indicates the number of jobs, while the y-axis represents the value of the performance evaluation.

Figure 9. Number of jobs vs performance matrix for layout 3.

Figure 10 shows the performance of the HCA, GAHCA, and CDS algorithms within a smart manufacturing environment using 10 tasks with layout number 4. The X-axis displays the number of jobs, while the y-axis shows the value of the performance evaluation metric. Genetic Hill Climbing demonstrated superior performance compared to the other algorithms based on observations from the graph.

Figure 10. Number of jobs vs performance matrix for layout 4.

6. Conclusion

This study focuses on addressing task scheduling challenges within cloud computing environments by proposing a hybrid hill-climbing algorithm. Task scheduling using the suggested algorithm leads to a notable improvement in performance, suggesting enhanced efficiency, reduced resource wastage, and cost savings in cloud computing environments. This new algorithm aims to strike a balance between solution optimization and exploration capabilities. The implementation was conducted using Python, wherein the algorithm was tested across 10 different problem sets featuring various layouts. Python code encompassed multiple algorithms, including HCA and GHCA, integrated within the vehicle system.

The results were derived from 20 iterative runs of an evolutionary procedure. The conclusions drawn from this research were primarily based on the application of HCA and GHCA to NP-hard problems. The evaluations encompassed 40 problems categorized across different layouts: layout 1 with 10 job sets, layout 2 with 20 job sets, layout 3 with 30 job sets, and layout 4 with 40 job sets. For instance, “10.1” signifies the 10th job set within layout 1.

The analysis indicated that GHCA outperformed other algorithms based on observations. Hypothesis testing was conducted to ascertain the significant differences between the algorithms and job set problems. Experiments revealed that the hybrid strategy was particularly effective in addressing NP-hard problems, resulting in reduced cost functions and enhanced utilization of virtual machine resources.

Future endeavors include delving into multi-objective problems and exploring dynamic load balancing mechanisms to further mitigate cloud resource waste the study hints at future research directions by demonstrating the effectiveness of the proposed algorithm, encouraging further exploration into refining its capabilities or applying it to different cloud computing scenarios for even greater optimization and efficiency gains

Author contributions

Methodology, MNR and AVSRP. Software, MNR. Validation, MNR and LSP. Formal analysis, MNR. Writing-original draft preparation, MNR and SK. writing-review and editing, VSK. Visualization, MNR. Supervision, ND and AKS. Project administration, MNR.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.