Figure shows an example of MIRCPSP with activities

Figure 1 shows an example of MIRCPSP with 7 activities where 1 and 7 are dummy activities.
An example of the mode identity structure for this network is presented in Table 1. The set of activities is partitioned into four disjoint subsets. The second subset, for example, consists of activities 2, 3 and 5 for which four possible modes are specified. All three activities, however, must be executed in the same mode. If mode 2 is selected, the three activities, 2, 3 and 5 are executed in the second mode.

Proposed GA
In this section, we describe the Genetic Algorithm (GA), which is a well-known meta-heuristic that has been successfully applied to a noticeable number of project scheduling problems  [5–10], to solve MIRCPSP. In order to increase the quality of the proposed GA, we implement an efficient local search. We also consider an exact method based on the branch and bound algorithm  [3] to provide comparable computational efforts for the GA.

Parameter tuning
Full factorial experimentation is an exhaustive approach to designing experimental investigations and calibrating the parameters of algorithms  [22,23]. When the number of parameters is high, a Fractional Factorial Experiment (FFE) is applied to reduce the number of required experiments  [24]. Taguchi  [25] developed a method based on FFE and offered a set of orthogonal arrays for designing experiments of quality improvement in manufacturing. A comprehensive study of orthogonal arrays can be found in Hedayat et al.  [26].
Taguchi divides factors into controllable and noise factors. Although there is no direct control of noise factors, the Taguchi method, using the concept of robustness, minimizes the effect of noise and determines the optimal level of controllable factors  [25]. The aim is to maximize the Signal-to-Noise (S/N) ratio, where the term signal denotes a desirable value (response variable) and noise denotes an undesirable value (standard deviation).
There are various formulas for the S/N ratio and it order Cisplatin depends on the experimental objective. In general, there are three basic quality characteristics: smaller is better, larger is better and nominal is best. Since we are trying to minimize the makespan, we use the smaller is better type. The corresponding S/N ratio is: where is the response variable and is the number of experiments. Each level that has a higher value of is selected as the optimum level.
In the proposed GA, the factors that should be tuned are PS, , NU and NL, where PS is the population size, and are the reproduction and mutation probabilities, NU is the number of best chromosomes used for updating Memory, and UL is the number of iterations in the local search. Different levels of these factors are shown in Table 2. Using MINITAB 16, L27 is selected as the most fit orthogonal array design to fulfill all our requirements. This design is shown in Table 3.
Figure 7 shows the average S/N ratio obtained at each level. According to this table, the optimal levels of PS, , NU and NL are 30, 0.2, 0.15, 3 and 30, respectively.
Using MINITAB16, a variance analysis (ANOVA) for SN ratios is performed and the results are shown in Table 4. This Table indicates the relative significance of individual factors in terms of their main effect on the objective function. Table 5 is response table for signal to noise ratios. The last row in these tables ranks the factors according to their impact degree, where lower rank means higher impact.

Experimental results
In order to validate the proposed genetic algorithm for the MIRCPSP, a problem set, consisting of 180 problems, was generated by the project generator, , which was developed by Drexl et al.  [27], using the parameters given in Table 6.
The indication means that the value is randomly generated in the interval . Resource availability is assumed to be constant over time. For each combination of parameters (number of activities and job subset strength), 10 problem instances were generated. The resource factor, RF, reflects the average portion of resource required per activity. The resource strength, RS, reflects the scarceness of the resource. The job subset strength, JSS, introduced by Drexl et al.  [27], is an index which determines the number of disjoint subsets of activities, , depending on the number of project activities, , according to: If , then activity subsets with one activity per subset are created. If , then activity subsets are created with (dummy start activity), (dummy finish activity).