The steps of TSMRAP can be briefly expressed as follows:

Step 0: Generate random initial feasible solutions.

Step 1: Explore all possibilities of the neighborhood through the defined move methods.

Step 2: If the best solution of the step 1 is in the tabu list and is not better than the current best, then repeat Step 1 to find the next best. Otherwise, select the solution as the next best move.

Step 3: If the stopping criterion is satisfied, then stop. Otherwise, go to Step 1.

The initial solutions of TSMRAP are randomly generated, and the scheme of randomly constructing the initial solutions is nearly identical to the general scheme, except that it allows either the module or the component for redundancy, due to the characteristics of our problem. In our experiments, TSMRAP was applied 5 times with different initial solutions for each problem.

The tabu list is one of the mechanism to prevent cycling and guide the search toward unexplored regions of the solution space. Kultruel-Konak et al. [8] showed that the dynamic size of a tabu list plays an important role in finding the better solutions for RAP. For the long-term memory, the size of a tabu list is reset every 20 iterations to the value of between [n, 3n] uniformly distributed. Once the list is full, the oldest element of the tabu list is removed as a new one is added.

The TS generally adopts the penalty function to allow to explore the search towardthe promising infeasible regions. The penalty function of our TSMRAP is employed as the same that of TS [8] . The penalty function for only one cost constraint is as follows:

Where R_all is the unpenalized (feasible or infeasible) system reliability of the best solution found so far, R_feas is the system reliability of the best feasible solution found so far, and ∆c represents the magnitude of the constraint violation. The initial value of NFT_c is set to 1% of the cost constraints limit C and k is set to 1, though computational results are not sensitive to this value.

In TSMRAP, neighborhood solutions are generated by two moves, that is, the first and second move. The first move is similar to the TS [8] for the series-parallel system, that is, to change the allocated redundant number of the module or component by adding or subtracting one. The second move is somewhat more complicated than the first move due to the characteristics of the problem. The scheme of the second move is to replace the current solution with the redundant value of the module or component. That is, if the current solution within a subsystem is assigned to the module, then it is replaced with a series of the component. Reversely, if the current solution within a subsystem is assigned to the component, then it is replaced by the module.

For example, let us consider the following system structure with 2 modules and 5 components in Table 1. The cost function is as follows.

The cost limit is set to 95. Suppose that the current solution during the iterations of TSMRAP to obtain the optimal solution for our problem to be given in Table 2, that is, the encoding status of (2, 0, 0, 0, 0, 2, 1, 0, 0). From the current solution, all the neighborhood solutions generated by the two moves are shown in Table 2. The number of the neighborhood solutions by the first move (adding or subtracting one) are 5 cases shown in Table 2. In the first move, notice that when the redundant value of component (122) equals to 1, it is not allowed to subtract one from the component (122).

The scheme of the second move is to replace the current solution with the redundant value of the module or component. The number of the neighborhood solutions generated by the second move is 4 cases. As shown in Table 2, when the value of module (11) equals to 2, it is replaced with two cases of components (111, 112, 113), that is, (1, 1, 1) and (2, 2, 2). Namely, it is allowed to assign a series of the redundant value of components up to the value of the redundant number of module (11), that is, 2. Reversely, when the value of component (121, 122) equals to (2, 1), it is replaced with two cases of module (12), that is, 1 and 2. We assign the redundant value of module up to the maximum value of the redundant number among components (121) and (122), that is, 2.

In Table 2, the system reliability for each neighborhood solution is evaluated by the penalty function of Eq. (2). For example, (2, 0, 0, 0, 0, 2, 2) has the penalty value of 0.796362 instead of 0.895469. Among 9 candidates of neighborhood solutions, (2, 2, 0, 0, 0, 0, 0), which has the maximum value of the system reliability (0.852996), is selected for the next best move of TSMRAP.

We compare the traditional RAP of considering only the component redundancy for the lowest-level with MRAP of allowing the modular and the component redundancy. We consider the 3-level system. Table 3 gives the values of input variables for reliability, price, and additive cost parameter. Test problems are generated by varying the cost limit from 150 to 340. A comparison between the traditional RAP and MRAP is shown in Table 4. As the results in Table 4 indicate, MRAP provides the superior solutions to the traditional RAP for 19 cases of 20 test problems. For only one case (cost limit of 190), the traditional RAP and MRAP have the same result of 0.8878.

Specifically, for the cost limit of 170, both of the system structuresare shown in Figure 1. In this case, the optimal solution of the traditional RAP and MRAP are given by 0.8455 and 0.8511, respectively. This result indicates that MRAP considering the modular redundancy as well as the component redundancy at the lowest-level provides the possibility of having the better system reliability than the traditional RAP of considering only the component redundancy.

Figure 1: 
The system structure

To additionally evaluate the performance of the previous GA[16] and the proposed TSMRAP, the new test problems for moderate size of system structure are designed. They are composed of 4 sets of 10 test problems for each level, that is, totally 120 test problems. For the case of 3-level, we replicated the data in Table 3 by h-fold, where h ranges from 4, 6, 8, to 10, that is, the number of subsystems of each problem became n=3 × h (h=4, 6, 8, and 10). Similarly, for the cases of 4-level, the number of subsystems of each problem became n=2 × h (h =6, 9, 12, and 15). For each set, 10 test problems were generated according to increase the cost limit of the constraint by 100. We increase the cost limit(C) of the constraints appropriately for each problem according to the size of problems in order to assign the reasonable value (not to be very low) of system reliability for each problem. In our numerical experiments, the stopping criterion of TSMRAP was defined as 200 iterations without finding an improvement in the best feasible solution. TSMRAP was applied 5 times with different starting initial solutions for each test problem. The computational results for evaluating the performance between GA and TSMRAP are summarized in Table 5. Two algorithms are coded in C/C++, and experiments are performed on a Pentium IV 3.0 GHz PC. Performances of GA and TSMRAP are assessed in terms of average relative error(A), maximum relative error(M), optimality rate(O) and average execution time(sec.) of 10 problems(T) defined as follows.

O = the number of cases in which each method yields the best solution among 10 problems.

R_j = the system reliability obtained by each method for test problem j.

R_j^*= the best system reliability obtained by both of GA and TSMRAP.

As the results in Table 5 indicate, TSMRAP outstandingly outperformed the GA. TSMRAP obtained a higher system reliability in 115 of 120 test cases. TSMRAP failed to obtain the best solution for only two cases in the case of n=30 for 4-level. Even for these cases, the average relative error (A) of GA is very poor, scoring the higher value of 0.0356 than 0.0016 of TSMRAP. GA obtained totally the best solution in only 5 cases, in which two cases are the higher system reliability than TSMRAP and 3 cases are the same system reliability of TSMRAP.

In terms of computing time, GA and TSMRAP are nearly equal for the case n=12 and 18 of 3-level and 4-level. For the cases (n=24 and 30) of 3-level and 4-level, TSMRAP is generally faster than GA. Specifically, when n=30 of 3-level and 4-level, the computing time of GA is almost 3 times of TSMRAP’s.