1. Introduction
The laboratory is an essential and supplementary part of learning to bridge the gap between theory and practice. These days, new technologies provide laboratories with a different form of observation, experimentation, and investigation whereby distance learning garnered the attention of researchers. Hence, the remote lab has been designed to perform the ubiquitous platform from faraway places for learners. The most important feature of remote laboratories is to share expensive resources across the world (Ma and Nickerson, 2006; Abdulwahed et al., 2008). In this regard, some benefits such as flexible access [3], sharing resources (Harward et al., 2008; Lowe et al., 2012; Richter et al., 2011), shared architecture (e.g. Sahara labs (Lowe et al., 2009) and MIT’s iLab (Harward et al., 2008)), security of users, data and devices (Gravier et al., 2008) among many other benefits have been shown to increase attention of users. Regarding the shared platform, user allocations have been a significant issue. Thus, proper scheduling can respond to the demand for having access to the remote laboratory.
Scheduling problems are formulated as a nonlinear optimization problem. Heuristic optimization algorithms are carried out to address this problem. For example, Simulated Annealing (SA) (Gravier et al., 2008), Tabu search (Hertz, 1991), Genetic Algorithm (GA) (Nuntasen and Innet, 2007), Ant Colony Optimization (Nothegger et al., 2012), etc., have been proposed as a methodology to deal with the timetabling problem. Ref (Shiau, 2011) and (Tassopoulos and Beligiannis, 2012) solved course scheduling by applying Hybrid Particle Optimization and Hybrid Harmony Search, respectively. High school timetabling has been proposed as one of the hard nonlinear problems because of considering resources and events (Ahmed et al., 2015). Ref (Akpinar, 2016) proposed a hybrid algorithm with a combination of ant colony optimization and Large Neighborhood Search (LNS). As an example, the LNS approach carries out the roulette wheel method based on each wealthy neighborhood to address hybrid algorithms disadvantages (Li et al., 2015; Sze et al., 2016). SA and Particle Swarm Optimization (PSO) are combined to schedule different types of trains on a single railway track (Jamili et al., 2012). Further, regarding education setting, university timetable planning has been proposed as constraintsatisfaction problems optimizing by GA (Deris et al., 1999; Zandavi and Chung, 2019).
Metaheuristic optimization algorithms fail to respond the online scheduling problem because not only are they designed to solve most of the optimization problems, but also they do not have enough efficiency in timeconsuming (Yen et al., 1998; Fan et al., 2004). On this account, NelderMead simplex has been successfully hybridized with the metaheuristics to increase the rate of convergence. As an example, NelderMead simplex Particle Swarm Optimization (NMPSO) (Fan et al., 2006; Zandavi, 2017), NelderMead simplex Genetic Algorithm (NMGA) (Fan et al., 2006), Stochastic Dual Simplex Algorithm (SDSA) (Zandavi et al., 2019) and Simplex Nondominated Sorting Genetic AlgorithmII (simplexNSGAII) (Pourtakdoust and Zandavi, 2016; Zandavi and Pourtakdoust, 2018) utilize the simplex algorithm as a supplementary part of the main algorithm to improve the exploitation. The main reason is to reach a better compromise between computational efforts and accuracy.
Developing an efficient optimization algorithm is investigated to make better organization when many requests occupy the shared platform. The proposed algorithm is generated to solve scheduling problems which are modeled as multimodal functions. The proposed algorithm consists of two parts; the simplex approach for increasing exploration and NSGA as finding promising areas. When NSGA sorts local optimum points with consideration of a potential area that probably contains a global minimum, the simplex techniques provide a whole searching domain with a diverse exploration. Hence, NSGA and the simplex part have enough potential to detect promising area and increase the diversity of individuals, respectively.
The organization of this paper is as follows: remote lab timetabling problem is explained in section 2. The Neldermead simplex algorithm is detailed in section 3. The nondominated sorting genetic algorithm is represented in section 4. Section 5 provides the proposed algorithm. Numerical results are made in section 6. The conclusion is drawn in section 7.
2. Remote lab Timetabling Problem
The remote lab has been organized as a ubiquitous education technology to expose a set of lab equipment kinds to the world via the internet. Accessing to the appropriate device can play a supplementary role for both education and research activities in a variety of science and engineering disciplines. To enable better usage of available expensive equipment, a remote lab service provider can host multiple users at the same time to amortize the cost of ownership as much as possible. In many situations, allocating the available lab facilities to meet the users’ demand is not a straightforward decision. The main constraint is the limited number of available resources versus the possibility of abundant users’ request to employ a set of lab resources. To begin with, handling such shared access among users with different preferences, employing an optimization technique for resource allocation and scheduling strategy to satisfy a set of predefined constraints is inevitable. The remote lab timetabling problem involves assigning a set of a limited number of available resources to users’ requests to optimize an objective function that reflects unused resources. Also, the constraints to be met are as follows.

Each user can only employ one rig across any requests.

The amount of time that a user can employ a rig is limited. Such a limitation needs to be determined based on the nature of the experiment as well as the number of other users waiting in queue to access the same rig.

None of the students can have inactive time (also referred to as idle time) during the possession of a rig.
The primary aim of this timetable problem is to minimize the number of unused rigs of each rig type. Such a minimization leads to increase in the total number of satisfied users (about the constraints mentioned above). Notably, the objective function of this optimization system to be minimized can be modeled as follows:
(1) 
where is the number of rig types, and is defined as a rig set in each type. and are the capacity of rig and the total number of users who are employing such a rig, respectively. is a matrix that is set to one if the associated rig is applied for an experiment in each timeslot and rig type , otherwise it is equal to zero.
As an example, let us assume that four users want to use a remote lab system consists of rig types (i.e., ). It is assumed that each rig type has one rig (i.e., ), while each of them has three capacity levels (i.e., ). While each of the first three users uses the first rig type that includes the first rig, then the associated is set to 1. So, the value of is set to . While the requests of forth user to access any other rig types can be delivered without any delay, any request of this user to access rig must be queued by the system as other users occupy it. In such a circumstance, remains to one while the value of (C  P) is equal to . By terminating any experiment of the first three users, rig is ready to be used by the fourth user. This can be inferred from the above equation by noting that (C  P) is not equal to zero anymore. Figure 1 depicts a schema for the scheduling process of the case mentioned above.
In Figure 1, represents a situation when all users select the Rig in Rig Type . Also, Rig is not allocated to the fourth user as others occupy it. When an experiment terminates at the defined procedure, the fourth user can be assigned to use Rig after elapsing the threshold time ( minutes in this example). Thus, it is imperative to use an efficient search method to produce the optimal timetable to satisfy all constraints.
2.1. Simplex Method
NelderMead Simplex Algorithm method is classified as a heuristic optimization algorithm and direct research method because the objective function is directly utilized to achieve the best optimal point without derivation. A simplex is a geometrical object produced by
points in dimension space (Chelouah and Siarry, 2003; Rao, 2009; Nobahari et al., 2016). For example, the triangle is a simplex in twodimension space. The basic idea in the simplex method is to compare the value of the objective function at the vertices of a simplex and move the simplex gradually toward the optimum point through an iterative process. The vertices of a regular simplex (equilateral triangle in two dimensions) of size a, within dimensional space, are generated by Eq 2 (Rao, 2009).(3) 
(4) 
For reaching the optimal solution, operational tools (reflection, contraction, and expansion) are utilized, deforming the simplex scheme geometrically (see Figure 2). After each transformation, the current worst vertex is swapped by a better one. Therefore, simplex gradually moves toward to the optimum point. The reflected, expanded and contracted points are given by: , and , respectively.
(5) 
(6) 
(7) 
During these transformations, is the centroid of all vertices except ; where is the index of the worst point. The parameters , and are called reflection, expansion and contraction coefficients, respectively.
Reflection is acting as reflecting the worst vertex, named high, concerning the centroid . If the reflected point is better than all other points, expansion operates to expand the simplex in the reflection direction; otherwise, if it is at least better than the worst, the algorithm performs the reflection with the new worst point again (Rao, 2009). The contraction is operation because of which the worst point is at least as good as the reflected point.
2.2. Nondominated Sorting Genetic Algorithm
NSGA method is a direct optimization algorithm, like simplex. This algorithm is known as a fast approach (Srinivas and Deb, 1994). In this approach, to identify the solution of the first nondominated front in a population of size , each solution can be compared with every other solution in the population to find out if it is dominated. This process is continued to find all members of the first nondominated level in the population. This algorithm is utilized Selection, Crossover, and Mutation to find optimal points.
3. Simplex Nondominated Sorting Genetic Algorithm
This algorithm utilizes simplex optimization as exploration and NSGA as exploitation to find the optimal solution. The flowchart of SNSGA is shown in Figure 3. Also, the parameters of SNSGA are tuned and shown in Table 1.
Parameter  Description  Value 
Population Size  30  
Maximum Generation  60  
Crossover Ratio  1.2  
Scale Parameter  0.1  
Shrink Parameter  0.5  
a  Side of Simplex  2 
Reflection Coefficient  1  
Expansion Coefficient  4  
Contraction Coefficient  0.2  
Maximum Iteration  30  
3.1. General Setting out of the Algorithm
SNSGA contains two heuristic algorithms, simplex optimization and NSGA, to reach the optimum global point. In this regard, simplex provides a wide range of variety across search space and acts as exploration; also, NSGA plays a vital role in exploiting the potential area which likely consists of global optimization solutions. Moreover, simplex performs to generate the new population with enough diversity at each generation whereby selection, crossover, and mutation propagate vertices of the simplex. Therefore, simplex operations (reflection, expansion, and contraction) are carried out to deform and move simplex toward likelihood regions of the search space until a maximum number of iterations () is reached. SNSGA has ten parameters that must be set before the execution of the algorithm.
3.2. Initial Population
In this proposed scheme, the primary parent population is generated randomly with uniform distribution. Not only is a population of each generation sorted based on nondomination scheme, but also each solution is assigned the fitness equal to its nondomination level. Furthermore, recombination based on binary tournament selection is carried out generating a first offspring having the same size as the parent population.
3.3. Population Update
In each generation, the population is sorted based on fast nondomination strategy. Simplex part of proposed algorithm organizes the new parent population. Simplex performs to generate the new population with enough diversity, updating population through reflection, expansion and contraction operators for the next generation. Further, any individual that is generated by operational genetic tools is used as vertices of the simplex to achieve new population.
3.4. Stop Condition
Simplex part of the proposed algorithm performed in the inner loop terminates at the maximum number of iterations. NSGA part of the algorithm operated in outer loop satisfies termination by the maximum number of generation. Figure 3 illustrates that the parameters of SNSGA, listed in Table 1, must be set before running the algorithm. Then, initial populations that are appropriated for NSGA are produced by simplex method optimization. Therefore, the simplex method helps to increase exploration. Having created a population, NSGA carries out crossover and mutation to find the optimum global point. Afterward, the SNSGA will be stopped if stopping condition satisfies.
4. Numerical Results
SNSGA is applied on the benchmarks, listed in (Jamili et al., 2012). According to Ref (Jamili et al., 2012), SNSGA is executed times to measure the rate of successful minimization, the average of the objective function evaluation numbers, and the average error on the objective function. Once either one of the termination criteria is first reached, the algorithm stops and returns the coordinates of a final point as well as the final optimal objective function value ( (algorithm)). Analytical minimum objective value () is compared with , and thus the solution is said to be ”successful” if the following inequality holds:
(8) 
where is an average of the objective function.
The average of the objective function evaluation numbers is only accounted for the ”successful minimization”. The average error is defined as the average of FOBJ deviation between the best successful point and the known global optimum, where only the ”successful minimization” achieved by the algorithm. The results of SNGSA tests performed over ten benchmarks are shown in Table 2.
Test Function  Rate of successful minimization  Average of objective function numbers  Average gap between the best successful point and the known global optimum 
RC  100  109  
GP  100  124  
B2  100  94  
SH  100  206  
100  189  
100  227  
100  185  
98  345  
100  105  
100  148  
The performance of SNGSA is compared with other algorithms such as Continuous Hybrid Algorithm (CHA) (Chelouah and Siarry, 2003), Enhanced Continuous Tabu Search (ECTS) (Chelouah and Siarry, 2000b), Continuous Genetic Algorithm (CGA) (Chelouah and Siarry, 2000a), Enhanced Simulated Annealing (ESA) (Siarry et al., 1997), Continuous Reactive Tabu Search minimum (CRTS min) (Battiti and Tecchiolli, 1996), Continuous Reactive Tabu Search average (CRTS ave) (Battiti and Tecchiolli, 1996), Tabu search (TS) (Cvijović and Klinowski, 1995), INTEROPT (Cvijović and Klinowski, 1995), Hybrid NelderMead simplex method and Genetic Algorithm (NMGA) (Bilbro and Snyder, 1991) and Hybrid NelderMead simplex method and Particle Swarm Optimization (NMPSO) (Fan et al., 2006). Table 3 illustrates the average numbers of function evaluation over 100 simulation runs for each benchmark and of the optimization algorithm.
Algorithm  SH  

CHA (Chelouah and Siarry, 2003)  295  259  132  345  459  215  492  598 
ECTS (Chelouah and Siarry, 2000b)  245  231  210  370  480  195  548  825 
CGA (Chelouah and Siarry, 2000a)  620  410  320  575  960  620  582  610 
ESA (Pradhan and Panda, 2012)    783      796  15820  698  1137 
CRTS min (Velazquez et al., 2014)  41  171          609  664 
CRTSave (Velazquez et al., 2014)  38  248          513  812 
TS (Hemmatian et al., 2014)  492  486    727      508   
INTEROPT (Shi and Kong, 2015)  4172  6375          1113  3700 
NMGA (Hancer et al., 2015)  356  422  529  1009  738  339  688  2366 
NMPSO (Hancer et al., 2015)  230  304  325  753  440  186  436  850 
SNSGA  109  124  94  206  189  227  185  345 
The numerical results demonstrate that SNSGA has a significant performance improvement in comparison with other hybrid heuristic algorithms. Overall, there is a considerable decrease in the average number of objective function evaluation for almost all benchmarks except for . Some of the test functions listed in Table 3 are utilized to describe the performance of the proposed algorithm in finding of optimum global point efficiently. Note that the objective function is normalized. The normalized objective functions are formulated as follow:
(9) 
where NOF is the normalized objective function in each iteration. OF is the real value of the objective function, while max(OF) and min(OF) are maximum and minimum values of the objective functions respectively, throughout the iteration.
Figure 4 represents the convergence performance of SNSGA starting from an initial random point versus the number of iteration. As seen, the SNSGA has precisely reached to global optimums during the iterations. Thus, SNSGA is competitive and even better than other metaheuristic optimization schemes.
SNSGA is utilized to reach the optimum scheme of a scheduling problem for remote labs. Figure 5 shows the results of the optimization for the scheduling problem. Figure 5 represents the periodical form of timetable problem. This periodical item is completely depended on the number of requests using the specific rig type. When the scheduling process starts with one user, the value of the objective function begins from 2 decreasing gradually in zero based on the request and capacity. Therefore, this trend is periodically occurred to make room for those who cannot use rigs.
5. Conclusion
A new hybrid heuristic optimization algorithm was proposed to respond to the demand for having access to the remote laboratories. The proposed hybrid algorithm formulates the scheduling problem as a nonlinear optimization problem. The hybridization of NelderMead simplex and nondominated sorting genetic algorithm reached a good compromise between timeconsuming and accuracy. The numerical results show that the proposed algorithm has a competitive performance in solving scheduling problems.
References
 Beyond the classroom walls: remote labs, authentic experimentation with theory lectures. Cited by: §1.
 Solving high school timetabling problems worldwide using selection hyperheuristics. Expert Systems with Applications 42 (13), pp. 5463–5471. Cited by: §1.
 Hybrid large neighbourhood search algorithm for capacitated vehicle routing problem. Expert Systems with Applications 61, pp. 28–38. Cited by: §1.

The continuous reactive tabu search: blending combinatorial optimization and stochastic search for global optimization
. Annals of Operations Research 63 (2), pp. 151–188. Cited by: §4.  Optimization of functions with many minima. IEEE Transactions on Systems, Man, and Cybernetics 21 (4), pp. 840–849. Cited by: §4.
 A continuous genetic algorithm designed for the global optimization of multimodal functions. Journal of Heuristics 6 (2), pp. 191–213. Cited by: Table 3, §4.
 Tabu search applied to global optimization. European journal of operational research 123 (2), pp. 256–270. Cited by: Table 3, §4.
 Genetic and nelder–mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions. European Journal of Operational Research 148 (2), pp. 335–348. Cited by: §2.1, Table 3, §4.
 Taboo search: an approach to the multiple minima problem. Science 267 (5198), pp. 664–666. Cited by: §4.

Incorporating constraint propagation in genetic algorithm for university timetable planning.
Engineering applications of artificial intelligence
12 (3), pp. 241–253. Cited by: §1.  Hybrid simplex search and particle swarm optimization for the global optimization of multimodal functions. Engineering optimization 36 (4), pp. 401–418. Cited by: §1.
 A genetic algorithm and a particle swarm optimizer hybridized with nelder–mead simplex search. Computers & industrial engineering 50 (4), pp. 401–425. Cited by: §1, §4.
 State of the art about remote laboratories paradigmsfoundations of ongoing mutations. International Journal of Online Engineering 4 (1), pp. http–www. Cited by: §1, §1.

A multiobjective artificial bee colony approach to feature selection using fuzzy mutual information
. In2015 IEEE Congress on Evolutionary Computation (CEC)
, pp. 2420–2427. Cited by: Table 3.  The ilab shared architecture: a web services infrastructure to build communities of internet accessible laboratories. Proceedings of the IEEE 96 (6), pp. 931–950. Cited by: §1.
 Optimization of hybrid laminated composites using the multiobjective gravitational search algorithm (mogsa). Engineering Optimization 46 (9), pp. 1169–1182. Cited by: Table 3.
 Tabu search for large scale timetabling problems. European journal of operational research 54 (1), pp. 39–47. Cited by: §1.
 Solving a periodic singletrack train timetabling problem by an efficient hybrid algorithm. Engineering Applications of Artificial Intelligence 25 (4), pp. 793–800. Cited by: §1, §4.
 Iterated local search embedded adaptive neighborhood selection approach for the multidepot vehicle routing problem with simultaneous deliveries and pickups. Expert Systems with Applications 42 (7), pp. 3551–3561. Cited by: §1.
 LabShare: towards crossinstitutional laboratory sharing. In Internet accessible remote laboratories: scalable elearning tools for engineering and science disciplines, pp. 453–467. Cited by: §1.
 LabShare: towards a national approach to laboratory sharing. In Proceedings of the 20th Annual Conference for the Australasian Association for Engineering Education, pp. 458–463. Cited by: §1.
 Handson, simulated, and remote laboratories: a comparative literature review. ACM Computing Surveys (CSUR) 38 (3), pp. 7. Cited by: §1.

Simplex filter: a novel heuristic filter for nonlinear systems state estimation
. Applied Soft Computing 49, pp. 474–484. Cited by: Figure 2, §2.1.  Solving the post enrolment course timetabling problem by ant colony optimization. Annals of Operations Research 194 (1), pp. 325–339. Cited by: §1.
 Application of genetic algorithm for solving university timetabling problems: a case study of thai universities. Cited by: §1.
 A hybrid simplex nondominated sorting genetic algorithm for multiobjective optimization. International Journal of Swarm Intelligence & Evolutionary Computation 5 (3), pp. 1–11. Cited by: §1, Figure 2.
 Solving multiobjective problems using cat swarm optimization. Expert Systems with Applications 39 (3), pp. 2956–2964. Cited by: Table 3.
 Engineering optimization: theory and practice. John Wiley & Sons. Cited by: §2.1, §2.1.
 Lila: a european project on networked experiments. In Automation, Communication and Cybernetics in Science and Engineering 2009/2010, pp. 307–317. Cited by: §1.
 A multiobjective ant colony optimization algorithm based on elitist selection strategy.. Metallurgical & Mining Industry (6). Cited by: Table 3.
 A hybrid particle swarm optimization for a university course scheduling problem with flexible preferences. Expert Systems with Applications 38 (1), pp. 235–248. Cited by: §1.
 Enhanced simulated annealing for globally minimizing functions of manycontinuous variables. ACM Transactions on Mathematical Software (TOMS) 23 (2), pp. 209–228. Cited by: §4.
 Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary computation 2 (3), pp. 221–248. Cited by: §2.2.
 A hybridisation of adaptive variable neighbourhood search and large neighbourhood search: application to the vehicle routing problem. Expert Systems with Applications 65, pp. 383–397. Cited by: §1.
 A hybrid particle swarm optimization based algorithm for high school timetabling problems. applied soft computing 12 (11), pp. 3472–3489. Cited by: §1.
 Multiobjective compact differential evolution. In 2014 IEEE Symposium on Differential Evolution (SDE), pp. 1–8. Cited by: Table 3.
 A hybrid approach to modeling metabolic systems using a genetic algorithm and simplex method. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 28 (2), pp. 173–191. Cited by: §1.
 Stochastic dual simplex algorithm: a novel heuristic optimization algorithm. IEEE Transactions on Cybernetics. Cited by: §1.
 State estimation of nonlinear dynamic system using novel heuristic filter based on genetic algorithm. Soft Computing 23 (14), pp. 5559–5570. Cited by: §1.
 Multidisciplinary design of a guided flying vehicle using simplex nondominated sorting genetic algorithm ii. Structural and Multidisciplinary Optimization 57 (2), pp. 705–720. Cited by: §1.
 Surfacetoair missile path planning using genetic and pso algorithms. Journal of Theoretical and Applied Mechanics 55 (3), pp. 801–812. Cited by: §1.
Comments
There are no comments yet.