This document summarizes a study that used a genetic algorithm to solve the multidimensional multiple choice knapsack problem (MMKP) and measured its performance against traditional approaches. The genetic algorithm was able to obtain near-optimal revenue solutions for large-scale MMKP problems in less time than traditional methods like Branch and Bound with Linear Programming (BBLP), Modified Heuristic (M-HEU), and Multiple Upgrade of Heuristic (MU-HEU). While the revenue obtained was nearly the same across all methods, the genetic algorithm had significantly better timing complexity and its effectiveness increased as the problem constraints grew larger.

1. International Conference on Electronics, Computer and Communication (ICECC 2008)
University of Rajshahi, Bangladesh
ISBN 984-300-002131-3
Solving Multidimensional Multiple Choice Knapsack Problem By Genetic Algorithm &
Measuring Its Performance
Shubhashis K. Shil1
, A. B. M. Sarowar Sattar2
, Md. Waselul Haque Sadid3
,A. B. M. Nasiruzzaman4
,Md.
ShamimAnower5
1-3
Department of CSE & 4-5
EEE, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh
E-mail: sks_engr_ruet@yahoo.com1
, nasir_zaman_eee@yahoo.com4
Abstract: The objective of this paper is to evolve
simple and effective method for the multidimensional
multiple choice knapsack problem (MMKP), which is
capable of obtaining near optimal revenue and of
course, better timing complexity than that of
traditional approaches for a large-scale system. In
optimization, genetic algorithm (GA) has some well-
known advantages. Some traditional methods are
well known to solve MMKP such as Branch and
Bound with Linear Programming (BBLP), Modified
Heuristic (M-HEU), Multiple Upgrade of Heuristic
(MU-HEU). In this study, a similar investigation for
solving MMKP is considered using simple Genetic
Algorithm. The performance of results of revenue
obtained with all of the methods are nearly same but
from the view point of calculation time GA is much
superior and its effectiveness increases as the
constraints of the problem are increased.
Key Words: Genetic Algorithm, MMKP, MU-HEU,
M-HEU, BBLP.
1. INTRODUCTION
The multidimensional multiple-choice knapsack
problem (MMKP) is a more complex variant of the
0–1 knapsack problem (KP). Since 0-1 KP is an NP-
Hard problem algorithms for finding exact solution
of the MMKP are not suitable for most real-time
decision-making applications, such as quality
adaptation and admission control for interactive
multimedia systems, or service level agreement
management in telecommunication networks.
The MMKP is a variant of knapsack problem that is
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.
Let there be n groups of items. Group i has li items.
Each item of the group has a particular value and it
requires m resources. In mathematical notation, let vij
be the value of the jth item of the ith group, r= (rij1 ,
rij2, …,rijm) be the required resource for the jth item of
the ith group and R=(R1, R2,...,Rn) be the resource
bound of the knapsack.
Mathematically the problem is stated as follows:
V=maximize ∑∑= =
n
i
li
j
ijijvx
1 1
(Objective function)
So that,
k
n
i
li
j
ijkij Rrx∑∑= =
≤
1 1
(Resource constraints)
where, V is the value of the solution, k=1,2,…, m,
and the picking variables are ,
∑=
=
li
j
ijx
1
1
and to reduce calculation time.
Fig. 1 illustrates an example of MMKP. We have to
pick exactly one item from each group. Each item
has two resources, r1 and r2. The objective of
picking items is to maximize the total value of the
picked items subject to the resource constraints of the
knapsack in minimum time, that is
∑ 1(r of picked items) 15≤ and,
∑ 2(r of picked items) 15≤
Fig. 1 An example of MMKP.
The MMKP has been studied by many researchers
and methods have been presented in several classical
papers [1], [2], [3], [4], and [5], and in well
established textbooks [6], and [7]. These classical
methods require specific assumptions on the
objective function such as continuity,
differentiability. Moreover, these methods are slowly
convergent.
484

2. International Conference on Electronics, Computer and Communication (ICECC 2008)
University of Rajshahi, Bangladesh
ISBN 984-300-002131-3
Genetic Algorithms (GA) are general search
technique based on the mechanism of natural
selection and natural genetics. In GA, search points
are represented by genetic strings. The search
process uses probabilistic transition rules instead of
deterministic ones as in traditional optimization
method. The search is conducted over a population of
solutions. Moreover, no specific assumptions on the
objective function, such as continuity,
differentiability, etc. are required in GA.
Genetic Algorithm is different from traditional or
conventional methods in four ways:
1. GA work with a coding of the parameter set, not
the parameters themselves.
2. GA search from a population of points, not a
single point.
3. GA use payoff (objective function) information,
not derivatives or other auxiliary knowledge.
4. GA use probabilistic transition rules, not
deterministic rules.
In several papers [1] the 0-1 classical knapsack
problem is solved using GA, which is one-
dimensional.
In this paper, analysis is done for multi-dimension
and multiple choice knapsack problem. In view of
the above, the main thrust of the research work
presented in this paper is to propose a new but
effective method for solving MMKP using GA.
2. STRUCTURE OF SOLUTION TECHNIQUE
2.1 Pseudo Code of General Genetic Algorithm
Begin
//Initialize the generation counter
t→0;
//Create a random population within the
bound and initialize it
Initialize_Population (p(t));
//Evaluate the fitness of the generated
population with respect to targeted function
Evaluate_Population (p(t));
//Order population to find the best solution
Order_Population (p(t));
While (not termination-condition satisfied)
Begin
//Select the relatively fir population
(roulette wheel)
Select_Population (p(t) from p(t-
1));
//Crossover operation
Apply_Crossover (p(t));
//Mutation operation
Apply_Mutation (p(t));
//Evaluate the fitness of the new
generated population
Evaluate_Population (p(t));
//Combine the result of the new
generation with that of the previous one
Alternate_Generation (offspring of
p(t) + parent of p(t));
//Order the generation with the
previous one to find the best solution
Order_Generation (offspring of p(t)
+ parent of p(t));
//Increase the generation counter
t→t+1;
End
End
2.2 Working Steps of Genetic Algorithm
The design problem can be formulated as the
following optimization problem:
1. Minimize the fitness function
1/(maximize ∑∑= =
n
i
li
j
ijijvx
1 1
)
subjected to constraint k
n
i
li
j
ijkij Rrx∑∑= =
≤
1 1
.
2. In every generation 30 individuals are taken.
3. Roulette Wheel Method of reproduction is used.
4. Single cross-site of crossover is used with
crossover rate 0.5 and mutation rate 0.05.
5. As a terminating condition number of generation
is chosen to be 100, 200, 300, and so on.
3. RESULT AND DISCUSSION
In Table 1 and Fig. 2 comparisons between BBLP,
M-HEU, MU-HEU and GA from the view points of
revenue and required time keeping number of items
per group 10 and number of resources 10, are given.
485

3. International Conference on Electronics, Computer and Communication (ICECC 2008)
University of Rajshahi, Bangladesh
ISBN 984-300-002131-3
Table 1 Comparison among various methods for
solving MMKP varying no. of generation
100 150 200 250 300 350 400 450 500
0
1
2
3
4
5
6
7
8
9
10
No. of Generation
R
e
qu
ire
d
T
im
e
(s
e
c
)
BBLP
M-HEU
MU-HEU
GA
Fig. 2 Graphical comparison of various methods for
MMKP
From the table it is evident that BBLP is superior
from the point of view of revenue but it is poor from
required time of calculation. M-HEU & MU-HEU
produce comparatively better solutions based on
required time. But GA is superior from the view
point of required time and it also produces near
optimal revenue probably not better all time.
4. CONCLUSION
This paper presents the comparisons of GA, BBLP,
M-HEU and MU-HEU algorithms for solving
MMKP. A new approach for solving the challenging
optimal MMKP is presented. The results are
compared with the results obtained using traditional
methods. The results are quite similar except that GA
is much faster than the traditional ones. It is now
quite clear from the above discussion that the GA
approach shows potential and stern concentrations
are the need of the day because of its probabilistic
nature.
REFERENCES
[1] S. Khuri, T. Back, and J. Heitkotter, “The Zero-one
Multiple Knapsack Problem and Genetic
Algorithms”, Proceedings of The ACM Symposium
of Applied Computing (SAC94), 1994.
[2] M Hifi, M Michrafy, and A Sbihi, “Heuristic
algorithms for the multiple-choice
multidimensional knapsack problem” Journal of
the Operational Research Society, pp. 1323–1332,
2004.
[3] Shahadat Khan, Kin F. Li, Eric G. Manning and
Md Mostofa Akbar, “Solving the knapsack
problem for adaptive multimedia systems” Studia
Informatica Universalis, SIU-2002, pp. 161–182,
Oct. 2002.
[4] M. M. Akbar, Eric G. Manning, Gholamali C.
Shoja, S.
Khan, “Heuristic Solutions for the Multiple-Choice
Multi- Dimension Knapsack Problem”
International Conference on Computational
Science, San Fracsisco, California, pp 659-668,
May, 28-30, 2001.
[5] R. Parra-Hernandez and N. Dimopoulos,, “A new
Heuristic for Solving the Multichoice
Multidimensional Knapsack Problem,” IEEE
Transaction on Systems, Man and Cybernetics.
Part A: Systems and Humans, 2002.
[6] David E. Goldberg, Genetic Algorithms in Search,
Optimization and Machine Learning, Pearson
Education, 2002.
[7] S.Rajasekharan, S. A. Vijayalekshmi Pai, Neural
Networks, Fuzzy Logic & Genetic Algorithms,
Prentice Hall of India, 2003.
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