Journal of Applied MathematicsVolume 2011 (2011), Article ID 218078, 11 pagesdoi:10.1155/2011/218078

Research Article

A Heuristic Algorithm for Resource Allocation/Reallocation Problem

S. Raja Balachandar and K. Kannan

Department of Mathematics, School of Humanities & Sciences, SASTRA University, Tamil Nadu, Thanjavur 613401, India

Received 3 February 2011; Revised 16 June 2011; Accepted 19 July 2011

Academic Editor: Yuri Sotskov

Copyright © 2011 S. Raja Balachandar and K. Kannan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents a 1-opt heuristic approach to solve resource allocation/reallocation problem which is known as 0/1 multichoice multidimensional knapsack problem (MMKP). The intercept matrix of the constraints is employed to find optimal or near-optimal solution of the MMKP. This heuristic approach is tested for 33 benchmark problems taken from OR library of sizes upto 7000, and the results have been compared with optimum solutions. Computational complexity is proved to be of solving heuristically MMKP using this approach. The performance of our heuristic is compared with the best state-of-art heuristic algorithms with respect to the quality of the solutions found. The encouraging results especially for relatively large-size test problems indicate that this heuristic approach can successfully be used for finding good solutions for highly constrained NP-hard problems.

1. Introduction

A cellular network is a mobile network in which resources are managed in cells. Each cell, an abounded area, is served by an antenna or base station. Cell size and shape depend on signal strength, the presence of obstacles to signal propagation, customer capacity, and cost constraints. Allocation/reallocation problem is formulated as 0/1 multichoice multidimensional knapsack problem (MMKP), which is NP-hard combinatorial optimization problem and detailed study of this problem is presented in [1]. Literally MMKP is defined as follows: Given a set of groups of variables, one tries to select the best variable in each group. Each variable in a group has a value in an objective function and consumes a certain amount of resources as well. The problem is to select the variables, subject to resource constraints, so that the objective function is maximized.

The 0/1 multichoice multidimensional knapsack problem may thus be mathematically formulated as follows: given groups of items to pack in some knapsack of capacity . Each item has a profit and a weight , and the problem is to choose one item from each group such that the profit sum is maximized without having the weight sum to exceed : All coefficients , , and are positive integers, and the classes are mutually disjoint. Totally there are variables, and they are divided into groups.

In the resource allocation/reallocation problem a group represents a terminal that needs to be reallocated. For terminals subjected to reallocation the constraints contain the power estimation constraints, and the constraints express that a channel cannot allocate more than one terminal from a particular cell at a time. Equation (1.3) expresses that the terminals must be reallocated to a channel. For terminals to be allocated the constraints contain the power estimation constraints, the constraints that express that a terminal cannot be allocated to more than one channel from a particular cell at time.

The other applications of MMKP are quality of service degradation model, utility model, multisession adaptive multimedia system, problem of allocation of resources on a packet network, and nursing personnel scheduling problem. MMKP has been solved by several algorithms, and they have been cited and compared as benchmarks many times in the literature.

This paper is organized as follows. A brief survey of various researchers’ works pertaining to this problem is elucidated in Section 2. The dominant principles of intercept matrix, dominance principle-based Heuristic (DPH) approach for solving MMKP, and computational complexity of DPH are explained in Section 3. We have furnished the results obtained by DPH for all the benchmark problems in Section 4. This section also includes the extensive comparative study of results of our heuristic with known optimum or best solutions of MMKP. Salient features of this algorithm are also enumerated in Section 4, and finally concluding remarks and future direction are also given in Section 5.

2. Previous Work

Depending on the nature of the solution, the existing algorithm for MMKP can be divided into two groups, namely, exact algorithm striving for exact solutions and heuristic algorithms producing near-optimal solution. The Exact algorithm includes branch and bound and Lagrange multipliers technique [2, 3].

Khan et al. [2, 4] presented an exact algorithm for the MMKP based on branch and bound with linear programming technique. A solution is explored in each iteration of this algorithm by picking the items of a particular group. All the possible alternative picks of a group are estimated by applying linear programming. The solution with highest estimated total value is selected for further exploration. A solution is termed as the optimal solution if it gives the highest total value among all the explored solutions and an item from each group is already picked. Khan et al. [4] had applied the concept of aggregate resource consumption [5] to pick a new candidate item in a group to solve the MMKP which is resulted in a heuristic named HEU. In HEU, a new item of group is selected for possible upgrade if it gives the highest change in earned value per unit change of aggregate resource consumption. Akbar et al. [6] presented a modified version of HUE, namely, M-HEU. In M-HEU, a preprocessing step to find a feasible solution and a postprocessing step to improve the total value of the solution with one upgrade followed by one or more downgrades were added.

Moser et al. [3] used the concept of graceful degradation from the most valuable items based on Lagrange multipliers to solve the MMKP. The selected highest valued item might be infeasible because of high resource consumption. That is why graceful degradation is done in multiple iterations towards selection of a feasible suboptional solution. Hifi et al. [7, 8] proposed two different approximate approaches. The first approach is a guided local search-based heuristic in which the trajectories of the solutions were oriented by increasing the cost function with a penalty term; it penalizes bad features of preciously visited solutions.

The second approach is a reactive local search where an explicit check for the repetition of configuration is added to the local search. The algorithm starts by an initial solution and improves by using a fast literature procedure. Later both deblocking and degrading procedure are introduced in order to (i) escape to local optimal and to (ii) introduce diversification in the search space. Finally, a memory list is applied in order to forbid the repetition of configuration. Recently Cherfi and Hifi [9, 10] have developed two hybrid algorithms, namely, HLB and AHLB, and compared the solutions with another procedure called column generation method (ALGO) [10]. HLB is the combination of local branching and column generation solution procedure, and AHLB is known as augmented HLB. These two algorithms are extended version of Hifi et al.’s [7, 8] previous work. The computational time of these algorithms was fixed from 300 to 1200 sec. The overall best solutions of these two algorithms are presented and compared with our heuristic in Section 4.

Drexl [11] presented a simulated annealing approach to solve a slightly different variant of MMKP (without choice constraints (1.3)), namely, the multidimensional knapsack problem (MDKP). Genetic algorithm approaches are not suitable to solve for real-time admission control, as they require long time to find a suboptional solution.

Parra-Hernández and Dimopoulos [12] proposed another heuristic HMMKP, called linear programming relaxations of the MDKP reduced from the MMKP problem. A PRAM model approximation algorithm was devised by Newton et al. [13] for solving the MMKP in time using operations. Shahriar et al. [14] proposed a multiprocessors-based heuristic algorithm (MP-HEU) for MMKP which runs time, where is the time requested by the algorithm using single processors, is the number of processors, and is a function of , the synchronization overhead. Sbihi [15] has presented a best first search exact algorithm for the MMKP. The main principle of the algorithm is twofold: (i) to generate an initial feasible solution as a starting lower bound and (ii) at different levels of the search tree to determine an intermediate upper bound obtained by solving an auxiliary problem and perform the strategy of fixing items during the exploration.

In this paper, we propose a 1-opt heuristic algorithm based on dominance principle of intercept matrix to solve MMKP. The dominance principle-based heuristic algorithm has been implemented successfully to solve 0-1 multiconstrained knapsack problem [16]. The main principle of the algorithm is twofold: (i) to generate an initial feasible solution as a starting lower bound and (ii) to improve the initial feasible solution to optimal or near optimal by using this heuristic iteratively.

3. Dominance Principle-Based Heuristic (DPH)

Many researchers have included all the possible constraints and variables in linear programming (LP) model of the real-life optimization problems, but some of the constraints and variables may not involve in the optimality and it may consume additional computational complexity. Such variables and constraints are known as redundant constraints and variables.

The preprocessing techniques are necessary to remove such redundant constraints and variables. Researchers [4, 17–25] have proposed many algorithms for LP models; in particular, Paulraj et al. [26] used the intercept matrix of the constraints to identify redundant constraints prior to the start of the solution process in his heuristic approach. Here we used the intercept matrix to identify redundant variables (0 valued variable) as well as selected variables (1 valued variable) for solving MMKP, since MMKP is a well-known 0-1 integer programming problem and many variables have zero values.

The intercept matrix of the constraints (1.2) is used to identify the variables of value 1 and 0. The algorithm starts by selecting the lowest cost-valued item of each group as (initial) feasible solution (Step  1) and improves the objective value by using dominance principle of intercept matrix (Step  3 to Step  5). This sequence of operations is performed ten times.

The construction of the intercept matrix (by dividing values by coefficients of (1.2)) is explored in Step  3. The elements of intercept matrix are used to find and which are arranged in decreasing order, and the leading element is the dominant variable (Step  5). We use this dominant variable to improve the current feasible solution, and this procedure provides optimum or near-optimum solution of MMKP. The dominant principle focuses at the resource matrix with lower requirement come forward to maximize the profit. The intercept matrix of the constraints (1.2) plays a vital role in achieving the goal, in a heuristic manner. Step  4 is used to identify the 0-value variables, that is, redundant variables. The various stages of DPH are presented in Algorithm 1.

Notation:

            : current objective function value

            : new objective function value

            : current solution vector

            : new solution vector

initial value:

            : zero valued vectors

            : zero

            k: 1 (iteration)

Step  1: For to do begin

             ,

            ,

            end for

do begin

Step  2: For our convenience convert

            ,

            ,

For to do begin

Step  3: Intercept matrix

Step  4: Redundant variables

            ,

            ,

Step  5: Dominant variable s

            ,

            if

            then

            ,

            ,

            else

            ,

            end if

end for

while

Display and .

Algorithm 1: DPH algorithm for MMKP.

3.1. Example

Consider an MMKP with 3 groups, 8 items, and 2 resources, that is, , , and : (we convert two-dimensional problem into one-dimensional notation for our convenience), Initial feasible solution: by using Step  1 of the algorithm, we have found the initial feasible solution of MMKP (001, 01,010) and objective value is 29. Next we update this solution by using DPH algorithm iteratively (from Step  3 to Step  5). Table 1 illustrates the first iteration reports of DPH algorithm. This heuristic updates the solution vector and objective function value.

Table 1: First Iteration reports of DPH.

Group 1
Group 2
Group 3

Const 1
3.4
4.25
3.4
2.125
2.833333
2.428571
3.4
1.7

Const 2
2.142857
2.142857
3
7.5
3.75
2.142857
5
1.875

Total =
5.542857
6.392857
6.4
9.625
6.583333
4.571429
8.4
3.575

Total *  
55.42857
89.5
57.6
105.875
85.58333
54.85714
58.8
60.775

Group 1
Group 2
Group 3

Const 1
—
—
—
1.625
2.166667
2.428571
2.6
*

Const 2
—
—
—
2
2
2
2.666667
*

Total =
—
—
—
3.625
4.166667
4.428571
5.266667
*

Total *  
—
—
—
39.875
54.16667
53.14286
36.86667
*

Group 1
Group 2
Group 3

Const 1
—
—
—
—
—
2.428571
2.6
*

Const 2
—
—
—
—
—
2
2.666667
*

Total =
—
—
—
—
—
4.428571
5.266667
*

Total *  
—
—
—
—
—
53.14286
36.86667
*

*: less than 1 in any one of the entry in that column, —: group is omitted.

There is no improvement in the 2nd iteration.

Iteration. Consider .2nd item in group 1 dominates other items, and it improves the objective function value from 29 to 34. Thus, the new solution vector is (010, 01,010) and objective function value is 34.Consider .There is no dominated variable other than the 2nd item in group 2. In this iteration no feasible upgrade is possible.Consider .2nd item in group 3 dominates other items, but it is available already. In this iteration no feasible upgrade is possible. Thus, the final solution vector is (010, 01,010) and corresponding objective function value is 34.

Iteration. For , and 3, there is no change in the objective function value. We terminate the iteration process and display the final solution, 45 which is optimum.

Theorem 3.1. DPH can be realized in time, polynomial in the number of groups (), item types (), constraints (), and number of iterations ().

Proof. The computational complexity of finding the heuristic solution of MMKP using DPH can be obtained as follows. For simplicity, let us assume that the number of variables per group is a constant (in case of different numbers of items per group, assume that is the maximum number of elements in a group) and there are resources and groups. It is easy to verify that the procedure for construction of intercept matrix takes operations. Step  3 and Step  4 take and operations, respectively. One iteration performs times to complete the groupwise selection. So the overall running time of the procedure DPH for one iteration can be deduced as follows: For iteration algorithm, the overall time complexity is .

4. Experimental Design

The heuristic was tested on 33 instances corresponding to two groups; the first group contains existing instances, namely, I01 to I13 (13 instances) which was tested by Khan [2], and the second group consists of 20 instances, namely, Ins01 to Ins20 which were randomly generated by Hifi et al. [8]. Our algorithm was coded in C++ and performed on Pentium IV 2.40 GHz computer with 512 MB memory, running under Windows XP Professional. For all the 33 instances DPH has reached the optimum/best/improved solutions. Table 2 illustrates the computational results of DPH, Cplex, ALGO, and HLB, AHLIB solutions with known optimum/best solutions [8] for the test instances (°indicates optimum solution and *indicates best solution). DPH reaches the best solution for the problem I07 in the first iteration itself, and the first iteration objective function values of this problem for to 100 are presented in Figure 1.

Table 2: Computational results of Cplex, ALGO, HLB, AHLB, and DPH with known optimum/best [8].

Ins
Optimum/best
DPH multiple run solutions (maximum 10 iterations)
Cplex
ALGO
HLB
AHLB

Initial/min
Max
Iter

I01
173°
127
173
2
173
173
173
173

I02
364°
304
364
1
364
364
364
364

I03
1602°
1202
1602
3
1602
1602
1602
1602

I04
3597°
2497
3597
2
3597
3597
3597
3597

I05
3905.70°
2907.70
3905.70
2
3905.70
3905.70
3905.70
3905.70

I06
4799.30°
3796.30
4799.30
4
4799.30
4799.30
4799.30
4799.30

I07
24587*
21356
24587*
1
24584
24587*
24587*
24587*

I08
36877
29467
36894*
5
36869
36892
36878
36894*

I09
49167
40457
49179*
6
49155
49176
49171
49179*

I10
61437
53466
61464*
3
61446
61461
61450
61464*

I11
73773
62731
73783*
6
73759
73775
73777
73783*

I12
86071
66045
86080*
6
86071
86078
86078
86080*

I13
98429
72457
98438*
7
98418
98431
98431
98438*

Ins01
10714
4554
10738*
8
10709
10732
10732
10738*

Ins02
13598*
9459
13598*
4
13597
13598*
13598*
13598*

Ins03
10943
7305
10944*
2
10934
10943
10944*
10944*

Ins04
14429
10249
14442*
1
19422
14440
14432
14442*

Ins05
17053*
12034
17053*
2
17041
17053*
17053*
17053*

Ins06
16823
11237
16828*
2
16815
16825
16826
16827

Ins07
16423
12235
16440*
4
16407
16435
16432
16440*

Ins08
17506
11068
17510*
2
17484
17510*
17508
17510*

Ins09
17754
14545
17761*
3
17747
17760
17758
17761*

Ins10
19314
13422
19316*
2
19285
19314
19315
19316*

Ins11
19431
13109
19441*
5
19424
19434
19434
19441*

Ins12
21730
16253
21732*
4
21725
21731
21731
21732*

Ins13
21569
19345
21577*
2
21569
21575
21571
21577*

Ins14
32869
27109
32874*
2
32866
32870
32871
32874*

Ins15
39154
32568
39160*
2
39154
39157
39151
39160*

Ins16
43357
39733
43363*
3
43357
43361
43357
43362

Ins17
54349
34453
54360*
5
54349
54349
53454
54360*

Ins18
60456
40340
60464*
3
60455
60460
60457
60464*

Ins19
64921
59931
64924
2
64919
64923
64924
64925*

Ins20
75603
67890
75611
3
75603
75611
75609
75612*

Figure 1: First iteration objective function value for I07.

The comparative study of DPH with other existing heuristic algorithms (HMS: Hifi et al. algorithm [7]; RLS, MRLSa, MRLSb: Reactive Local Search, Modified Reactive Local Search algorithm [8]; Cplex: Cplex Solver; KLMA: Khan et al.’s algorithm [4]; MOSER: Moser et al.’s algorithm [3]; Opt/best: optimum or best solutions [8]; ALGO: column generation method [10]; HLBCGSP: local branching and column generation [9]; AHLBCGSP: augmented hybrid procedure [9]) has been furnished in Tables 3(a) and 3(b) in terms of the number of optimal or best solutions, the average deviation from optimal/best solution obtained for group 1 problem, and the number of optimum/overall best solutions.

Table 3: (a) Summarized results for the solution quality of group 1 problem. (b) Summarized results forthe solution quality of group 2 problem.

(a)

HMS
RLS
MRLSa
MRLSb
Cplex
MOSER
KLMA
ALGO
HLB
AHLB
DPH

Number of problems
13
13
13
13
13
13
13
13
13
13
13

Number of optimum/best
2
5
4
12
8
0
1
13
13
13
13

Average deviation
1.92
1.07
0.91
0.00
0.01
5.99
1.46
0.00
0.00
0.00
0.00

Number of overall best/optimum
0
0
0
0
0
0
0
7
7
13
13

(b)

MRLSb
Cplex
ALGO
HLB
AHLB
DPH

Number of problems
20
20
20
20
20
20

Number of best
18
6
20
19
20
20

Average deviation
0.002
0.04
0.00
0.0003
0.00
0.00

Number of overall best
0
0
3
3
18
18

The MOSER et al. [3] approach is heuristic and is based on Lagrange’s multiplier method. Khan et al. [4] use an iterative improvement procedure, namely, KLMA based on the concept of aggregate resources, which is presented in [5]. Both of these methods have failed to find the optimum/best solution for group-1 instances, but KLMA identifies the optimum solution for the problem I06. The average deviations of KLMA and MOSER [3] approaches from optimum/best solutions are 1.46% and 5.99%, respectively.

Hifi et al. [7] has developed two greedy approaches for MMKP, namely, constructive procedure (CP) and complementary constructive procedure (CCP). The detailed description of CP and CCP can be found in [9]. To compare the performance of DPH, we consider the best solution of Hifi et al.’s [7] algorithm referred to herein as HMS. The HMS determines the optimum solutions for I01 and I06 only, and the average deviation of HMS from the optimum/best is known to be 1.92%. Hifi et al. [8] presented two more algorithms for MMKP, namely, reactive local search (RLS) and modified reactive local search (MRLS). RLS approach improves the solution obtained by CCP. The core of the algorithm is mainly based on two strategies, namely, degrading and deblocking. The detailed procedures are available in [8]. RLS presents the optimum solutions for 4 instances, namely, I01, I02, I05, and I06. The average deviation is 1.07%. The modified versions of RLS are known as MRLSa and MRLSb. MRLSa has obtained 4 optimum solutions (I01, I02, I03, and I06), and average deviation is 0.91%. For MRLSb, the average deviation is zero and finds optimum/best solutions for all the test instances except I12; the deviation from the best solution is 0.002%. The best results of Cplex Solver are also compared with DPH; Cplex obtained 8 optimum/best solutions out of 13 instances (I01 to I06, I10, and I12), and the average deviation is 0.01%.

Cherfi and Hifi [9, 10] have developed three approaches for solving MMKP, namely, ALGO, HLB, and AHLB. The column generation method (ALGO) identified the optimum/best solution for all the instances of group-1 problems. It has reached overall best solution for 7 instances (I01 to I07). Cherfi and Hifi [9, 10] have used three time limits for HLB. We consider the best solution among the three time bounds. HLB has found optimum/best solution for all test instances in group  1 and obtained 7 overall best solutions (I01 to I07). AHLB has also been executed with two time limits, and the best results are considered for this comparative study. AHLB provides optimum/best solution for the entire test instances in group  1 and has got overall best for all the 13 instances. DPH was set to perform maximum 10 iterations and got the optimum/best solution for all the test instances. Since the working environment is different, we have not compared the running time of all the algorithms. DPH reaches high-quality solutions within 10 seconds, whereas HLB and AHLB needs more computational time to achieve this kind of high-quality solutions. ALGO, HLB, and AHLB were coded in C++ and run on an Ultra-Sparc 10. The maximum time requirement for this high-quality solution, ALGO, HLB, and AHLB is found to be 1200 sec.

For group-2 instances, the optimum solutions are not available in the literature because of hardness of the problem. The computational results of MRLSb, Cplex, ALGO, HLB, AHLB, and DPH are reported in Table 3(b). From the best solutions of Hifi et al. [8], we observe that ALGO, AHLB, and DPH reached the best for all the 20 instances, and the other algorithms, HLB, MRLSb, and Cplex, have reached 19, 18, and 6 instances of group  2, respectively. In the same time ALGO, HLB, AHLB, and DPH have given the improved solutions for some of the test instances which are superior to Hifi et al. [8]. In Table 3(b), we have also presented the number of overall best solutions for these test instances.

From Tables 3(a) and 3(b), we conclude that DPH is an efficient technique than the other algorithms. The reason is that DPH reached these solutions in 10 seconds (maximum) for all the test instances, where ALGO, HLB, and AHLB reached with more computational time [9, 10]. The time complexity of our algorithm is , where is the number of iterations (we fixed as 10).

4.1. Salient Features of DPH

This heuristic is used to reduce the search space to find the near-optimal solutions of the MMKP. The computational complexity is , and the space complexity is . It reaches the optimum or near-optimum point in iterations, where is the number of groups. Due to dominance principles, this heuristic identifies the zero-value variables instantaneously. DPH takes maximum CPU time of 10 seconds for large-size problem. From the comparison table we observe that our algorithm is the effective one.

5. Conclusion

In this paper, we presented the dominant principle-based heuristic approach for tackling the NP-hard 0/1 multichoice multidimensional knapsack problem. This approach has been tested on 33 state-of-art benchmark instances and has led to give optimal/best solutions for all the test instances given in the literature. The maximum computational time is 10 seconds, where the other recent algorithm requires 1200 seconds. This heuristic is with complexity, and it requires iterations to solve the MMKP. The experimental data show that the optimality/best achieved by this heuristic is always close to 100 percentages. The basic idea behind the proposed scheme may be explored to tackle other NP-hard problems.

Acknowledgment

The authors wish to express their gratitude to the referees for their careful reading of the manuscript and helpful suggestions.

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