MCDM 2006, Chania, Greece, June 19-23, 2006
THE LINEAR MULTIPLE CHOICE KNAPSACK PROBLEM WITH TWO
CRITERIA: PROFIT AND EQUITY
George Kozanidis
Systems Optimization Laboratory
Dept. of Mechanical & Industrial Engineering
University of Thessaly
Pedion Areos, 38334 Volos, Greece
E-mail: [email protected]
Emanuel Melachrinoudis
Dept. of Mechanical & Industrial Engineering
Northeastern University
Boston, MA 02115, USA
E-mail: [email protected]
Keywords: linear multiple choice knapsack, balanced resource allocation, equity, multiobjective linear
programming, nondominated frontier
Summary: We study an extension of the Linear Multiple Choice Knapsack (LMCK) Problem
that considers two criteria. The problem can be used to find the optimal allocation of an
available resource to a group of disjoint sets of activities, while also ensuring that a certain
balance on the resource amounts allocated to the activity sets is attained. The first criterion
maximizes the profit incurred by the implementation of the considered activities. The second
criterion minimizes the maximum difference between the resource amounts allocated to any two
sets of activities. We present the mathematical formulation and explore the fundamental
properties of the problem. Based on these properties, we develop an efficient algorithm that
obtains the entire frontier of nondominated solutions. The algorithm is very efficient compared
to generic multiple objective linear programming (MOLP) algorithms. We present theoretical
findings which provide insight into the behavior of the algorithm, and report computational
results which demonstrate its efficiency.
1. Introduction
The multiple choice knapsack problem is a very commonly used variation of the classical knapsack
problem with numerous applications. Many solution techniques that have been developed for the integer
version of the problem rely on the existence of a fast solution algorithm to its linear relaxation. Besides
being used as a relaxation to its integer counterpart, the linear version of the problem models itself
practical applications too. Typical examples include location, production scheduling, and transportation
management problems (Lin, 1998).
In this work we present a solution algorithm for the linear multiple choice knapsack problem with two
criteria, profit and equity. The main advantage of the algorithm is that it is able to obtain the entire
frontier of nondominated solutions. The results of the computational experiments that were conducted
demonstrate that the algorithm exhibits high efficiency, and additionally, that the computational time
savings increase fast as the problem size increases. For the needs of this work, we term BLMCK
1
(Biobjective Linear Multiple Choice Knapsack) the problem under consideration, and we formulate it as
follows:
Max P = ∑ ∑ pki xki
(1)
Min f
(2)
k∈S i∈Rk
s.t.
∑∑c
k∈S i∈Rk
∑x
ki
i∈Rk
x ≤b
(3)
ki ki
≤ lk , k ∈ S
(4)
L ≤ ∑ cki xki ≤ U , k ∈ S
(5)
U −L≤ f
(6)
U , L, f ≥ 0; xki ≥ 0, i ∈ Rk , k ∈ S
(7)
i∈Rk
The decision variables xki represent continuous activities that belong to |S| disjoint sets. The specific set a
variable belongs to is denoted by its first index, k. Set S contains the indices of all these disjoint sets. For
a specific value of k in S, set Rk contains the indices of the decision variables that belong to set k. The
positive parameters pki and cki represent the profit and cost, respectively, incurred per unit of application
of the activity represented by xki. Parameter b is also a positive quantity, denoting the available amount of
resource that can be used towards the implementation of the considered activities.
The objective function (1) maximizes total profit, while constraint (3) limits the total resource amount
used to a maximum value b. The multiple choice constraints (4) restrict the sum of all variables within
each set k to at most lk. Constraints (5) define the auxiliary decision variables U and L in such a way that
cki xki , belongs to
the total resource amount allocated to any single set k (also called cost of the set),
∑
i∈Rk
the interval [L,U]. The width of this interval is restricted to at most f by constraint (6). The second
objective function (2) minimizes f. Constraints (7) ensure that all variables are nonnegative.
The remainder of this work is organized as follows: In Section 2 we explore the fundamental properties
of the problem, and based on them we introduce an efficient algorithm that obtains its entire set of
nondominated solutions. We also present theoretical results that give insight into the algorithm. In
Section 3 we examine the computational complexity of the algorithm and report computational results
which demonstrate its performance in the average case. Finally, in Section 4 we summarize the
contribution of the present work and point to directions for future research.
2. Solution Methodology
2.1 BLMCK Properties
The algorithm we propose for the solution of BLMCK is divided into two phases. In the first phase, the
second objective is ignored and an optimal solution to Problem LMCK is obtained. Starting from this
solution in the second phase, the algorithm incorporates the equity objective and evaluates the
nondominated tradeoff rate (see Steuer, 1986) between the two objectives, as the second objective is
allowed to decrease continuously to zero. Taking advantage of the problem’s special structure, the
algorithm returns its entire nondominated frontier upon termination.
2
For the solution of Problem LMCK in Phase I, we utilize Algorithm LMCK (see Kozanidis and
Melachrinoudis, 2004). Two important properties exist (see Kozanidis and Melachrinoudis, 2004) that
enable us to eliminate in advance several decision variables and reduce this way the size of the problem.
We can visualize graphically the decision variables of each set as points on a two-dimensional graph,
where the x-axis represents the cost of a variable and the y-axis its profit (see Fig. 1 in Kozanidis and
Melachrinoudis, 2004). The variables which are not eliminated in each set define the left upper hull of all
variables that belong to this set.
An important difference between Problems BLMCK and LMCK is that, for Problem BLMCK, variables
which qualify for elimination according to these two properties, but belong to the right upper hull of the
associated multiple choice coefficient space should not be apriori eliminated, because they may take a
positive value in a Pareto optimal solution. Therefore, Algorithm BLMCK, which is introduced in the
next section, does not eliminate in advance such variables. For the remainder of this work, we use the
terminology introduced by Kozanidis et al. (2005).
2.2 Solution Procedure
Let’s assume that the second objective is suppressed from the BLMCK formulation. Note that the
presence of constraints (5) and (6) alone does not affect the feasible region of the problem, since they
artificially enlarge the dimensionality of the original space. In the solution obtained when the resulting
LMCK problem is solved using Algorithm LMCK, f is greater or equal to the maximum difference
realized between the costs of any two sets. We use the terms maxcost and mincost to denote the
maximum and minimum cost, respectively over all sets. Internal is a set whose cost is different from
maxcost and mincost. Upper is a set whose cost is equal to maxcost and lower a set whose cost is equal to
mincost. Based on this terminology, at the end of Phase I, f is greater or equal to (maxcost-mincost).
After the equity objective is incorporated in Phase II, f should be minimized. In order to minimize f, the
algorithm needs to modify the current resource allocation to the multiple choice sets. There are five
candidate options for decreasing f, while also retaining optimality with respect to the first objective. Any
option different than these will lead to a suboptimal solution with respect to the first objective. These
options, displayed in Figure 1, are introduced next. A set is called increasing if the application of an
option increases its cost, and decreasing if the opposite is true. The increasing and decreasing slope of a
multiple choice set are defined in Kozanidis et al. (2005).
Option A
Option B
Option C
Option D
Cost
Cost
Cost
Cost
f
f
f
f
Option E
Cost
f
Figure 1: The five candidate options for decreasing f
Option A: Decrease the resource amount allocated to all upper sets and reallocate the recovered amount
to the internal or lower set with the maximum increasing slope.
Option B: Decrease the resource amount allocated to the internal or upper set with the minimum
decreasing slope and reallocate the recovered amount to the lower sets.
3
Option C: Decrease the resource amount allocated to all upper sets and reallocate the recovered amount
to the lower sets.
Option D: Decrease the resource amount allocated to all upper sets.
Option E: Increase the resource amount allocated to all lower sets (considered only if a positive resource
residual is available).
Let ∆P and ∆f be the marginal differences realized in total profit and in f, respectively, when any one of
these options is applied. Note that, by construction, ∆P is nonpositive and ∆f strictly negative, since the
initial solution is optimal for LMCK (∆P < 0), and we only consider ∆f < 0. Therefore, the ratio ∆P/∆f is
always nonnegative. Out of the five options, the algorithm should apply the one resulting in the minimum
total profit decrease per unit decrease in f. This will ensure that the first objective will have the maximum
possible value for the new improved value of the second objective.
After deciding on the option that should be applied by choosing the one with the minimum ∆P/∆f ratio,
the algorithm must determine when the application of this option should stop. The application of an
option stops when an appropriate stopping condition applies. This happens when at least one of the ∆P/∆f
ratios of the five options changes, or when f becomes equal to 0. In the latter case, this is because at that
point all possible values for f have been considered. In the former case, this is because at that point the
new ∆P/∆f ratios have to be computed and compared again.
At each Phase II iteration, the algorithm developed for Problem BLMCK repeats the following
procedure. First, it selects the option that will be applied. This is done by selecting out of the five
candidate options the one whose ratio ∆P/∆f is minimum (in what follows, we refer with ∆P/∆f to the
minimum such ratio). Next, the algorithm decides the extent of the current iteration by finding the
stopping condition that applies first. This determines the end of the current iteration and the beginning of
the next one. The algorithm terminates with the entire nondominated frontier when f drops to 0. The
following result is crucial for the development of the algorithm for Problem BLMCK.
Proposition 1: The initial solution, obtained at the end of Phase I, is Pareto optimal for Problem BLMCK
if and only if ∆P/∆f > 0.
Proof: Since this solution is optimal for the single objective problem, the result follows from the
definition of Pareto optimality and the fact that no other option exists, resulting in a smaller ∆P/∆f ratio.
The situation is illustrated in Figure 2, which displays the optimal solution obtained at the end of Phase I
in criterion space. In the first case, ∆P/∆f = 0 for this initial solution. This means that the single objective
problem has multiple optimal solutions and at least one of them is satisfied with f ’< f. Since we can move
to a solution with a better value for the second objective without worsening the value of the first
objective, the initial solution is not Pareto optimal. Therefore, the initial, together with all intermediate
solutions, are dominated by the solution obtained at the end of the first iteration (P = P1 for all of them).
In the second case, ∆P/∆f > 0. This means that in order to move to a solution with a better value for the
second objective, we must worsen the value of the first objective. Therefore, the initial solution is Pareto
optimal. Additionally, all intermediate solutions are Pareto optimal (nondominated line segment), since P
decreases strictly as f decreases. Of course, the ratio ∆P/∆f can never be negative, as explained above.
Next, the algorithm developed for Problem BLMCK is formally introduced.
Algorithm BLMCK
Phase I (Optimal solution of LMCK)
Without eliminating variables which belong to the right upper hull of the associated multiple choice
coefficient space, use Algorithm LMCK to find the optimal solution to the LMCK problem that results
when the second criterion is suppressed.
Phase ΙΙ (Solution of ΒLMCK)
while (f > 0) do {
select option that minimizes ΔP/Δf
find stopping condition and compute |Δf|
4
iterate, update solution and find new value of f
if ΔP/Δf > 0, add the initial and all intermediate solutions of the iteration that was
just completed to the set of nondominated solutions of the problem
}end while
P
P
Case 2: ΔP/Δf > 0
Initial solution Pareto optimal
Case 1: ΔP/Δf = 0
Initial solution not Pareto optimal
P1
P2
P1
f2
f1
f
f2
f1
f
Figure 2: Determining whether the solution obtained at the end of Phase I is Pareto optimal or not
2.3 Insight into Algorithm BLMCK
To get further insight into Algorithm BLMCK, assume that the second objective becomes Min (U − L).
Then, we can replace this objective by constraint (6), and parametrically change the value of f, in order to
get the entire nondominated frontier of the problem. Essentially, this is the parametric e-constraint
reduced feasible region method for obtaining the entire nondominated frontier of a MOLP problem (see
Steuer, 1986). Based on this observation, the validity of the algorithm is summarized in the following
result:
Proposition 2: Algorithm BLMCK correctly obtains the entire nondominated frontier of Problem
BLMCK.
Proof: Phase I of Algorithm BLMCK terminates with an optimal solution to Problem LMCK. If LMCK
has an alternative optimal solution with a better f-value, a nondominated criterion vector (P,f) will be
determined in Phase II. Otherwise, the starting criterion vector is nondominated. At each Phase II
iteration, the algorithm applies the option for which the marginal decrease in total profit is the minimum
possible per unit decrease in f. Therefore, at the end of each iteration, the first objective has the maximum
possible value for the current value of the second objective. This means that optimality is maintained
throughout for the first objective with respect to the current value of the second objective. Since f is
changed continuously, as soon as its value drops to 0, the algorithm has identified the optimal values of
the first objective for all possible values of the second objective. Therefore, the entire nondominated
frontier of Problem BLMCK has been obtained.
Algorithm BLMCK is related to Algorithm LMCKE (Kozanidis et al. 2005). The differences between the
two algorithms are the following. Algorithm LMCKE solves a single objective linear program and, as a
result, returns a unique optimal solution. On the other hand, Algorithm BLMCK solves a MOLP
problem, by computing the entire set of nondominated solutions, which in the general case is an infinite
set. In Algorithm LMCKE, f is a parameter with specific fixed value, while in Algorithm BLMCK it is a
decision variable. In Algorithm BLMCK, an iteration is carried out as long as f > 0. On the other hand,
Algorithm LMCKE terminates when the desired value of f is obtained. Additionally, the stopping
conditions that apply in each of the five options are not the same for the two algorithms. This is mainly
due to the fact that some of the stopping conditions of Algorithm LMCKE do not act as stopping
conditions in the case of Algorithm BLMCK. Finally, Algorithm LMCKE incorporates upper bounds on
the optimal cost of each set, while this is not true for Algorithm BLMCK.
5
At each Phase II iteration, Algorithm BLMCK moves to a new solution with a strictly lower f−value.
Implicitly, it visits an infinite number of intermediate solutions corresponding to the infinite number of
intermediate values that f takes between its initial and its final value. Proposition 1 clearly extends to each
of these iterations. Hence, there are two distinct cases. If ΔP/Δf = 0 at the beginning of an iteration, then
the initial and all intermediate solutions will be dominated by the solution obtained at the end of this
iteration. Whether this last solution will be Pareto optimal or not depends on the value of ΔP/Δf that will
be computed at the next iteration. If ΔP/Δf > 0, then the initial and all solutions encountered from that
point on, until termination of the algorithm, will be Pareto optimal. This is due to the following
interesting result:
Proposition 3: Let q and t be two Phase II iterations of Algorithm BLMCK, such that q is carried out
before t. Let also rq be the value of the ratio ΔP/Δf for iteration q and rt the value of the ratio ΔP/Δf for
iteration t. Then, rq ≤ rt.
Proof: Consider the feasible region of Problem BLMCK in criterion space. This region is an open
polyhedron in the nonnegative orthant of the two dimensional plane, since the two objectives always take
nonnegative values, and there always exists an infinite number of feasible values for variable f. The profit
objective takes its maximum value at the end of Phase I of Algorithm BLMCK. As the value of the equity
objective decreases in Phase II, the value of the profit objective is always nonincreasing. The validity of
the proposition results from the fact that the feasible region of Problem BLMCK in criterion space is
convex and the set of its nondominated solutions is connected, since this is a MOLP problem.
Proposition 3 implies that the ratio ΔP/Δf can only be equal to 0 for a sequence of iterations at the
beginning of Phase II. Once this ratio takes a positive value, it will remain positive until termination of
the algorithm. Additionally, this result states that, for any two Phase II iterations, the ratio of the one
which was performed earlier cannot be larger than the ratio of the one which was performed later. This is
an interesting result which agrees with practical experience. The intuition is that the more we need to
"squeeze" f in order to attain higher level of equity, the higher price (in terms of profit) we have to pay.
To put it in a different way, a unit of resource balance costs less when the solution at hand is unbalanced
than when it is somewhat balanced. As the balance level of the solution at hand increases, this unit cost
increases too.
When f drops to 0, the entire nondominated frontier of the problem has been obtained and the algorithm
terminates. Clearly, for this final solution, the following result holds, since we cannot improve any of the
two objectives without worsening the value of the other.
Corollary 1: The solution obtained from Algorithm BLMCK when f drops to 0 is Pareto optimal.
Two interesting observations with respect to Algorithm BLMCK should be made at this point. The first is
that it is possible for a solution in which money is simply “wasted” without reaching a higher profit in
one of the multiple choice sets to be Pareto optimal. It can be assumed that each set will prefer a solution
with a higher profit for this particular set as compared to a solution with a smaller individual profit, but a
higher individual cost. The decision maker should always keep this in mind when reviewing the set of
nondominated solutions in order to decide on the one of maximum utility. Even so, under certain
conditions this situation will never come up. For example, this situation will never come up when the
upper and the left upper hull in each multiple choice coefficient space coincide. This is because, in that
case, only nonnegative values will be possible for the increasing slope of each set, therefore the profit of
a set will never decrease when its cost increases.
The second interesting observation that should be pointed out is the following. Among the set of Pareto
optimal solutions of the problem, there may exist some with a positive resource residual, although
additional activities exist which have not been selected for implementation. When this situation is
encountered, a positive resource amount has been preserved, because there is no way this can be used to
increase total profit, while also maintaining the current level of equity. As the number of considered
activities in each set increases, the probability that this situation will come up decreases. This is because,
6
at the presence of more candidate activities in each set, it is more likely that it will be possible to divide
the unused resource amount among the sets in a way that total profit increases, while also maintaining the
current level of equity.
The above two observations imply that it is wiser to give to the profit objective higher priority than to the
equity one. Usually, very high level of equity should be sought when this is absolutely desirable, and the
resulting loss in profit can be tolerated.
3. Computational Implementation
3.1 Complexity Analysis
Let r be the number of multiple choice sets, Nk the number of variables in multiple choice set k, N
=
N k and Nmax = max N k . As shown in the analysis of Kozanidis et al. (2005), the worst case
∑
k∈S
k ∈S
complexity of Phase I is O(N logNmax) + O(N logr) = O(N logm), where m = max (Nmax, r). The work
needed at each Phase II iteration to find the ratios ΔP/Δf of the five options, find the optimal value of ∆f,
and update to get the new solution that results is O(r). The number of Phase II iterations depends on the
number of times each of the different stopping conditions applies. In what follows, the average-case
performance of the algorithm is examined by analyzing the computational experiments that were
conducted.
3.2 Computational Experiments
Algorithm BLMCK was coded in C/C++ and tested on a Pentium IV/2.5 GHz processor. The results
obtained are presented in Tables 1 and 2. In these tables, Nk was assumed to be the same for all k. Both r
and Nk varied between 150 and 600 in steps of 150. This means that the biggest problems solved
contained 360,000 decision variables. Two types of problems were tested. In type A problems, variables
that should be eliminated were also considered among the initial variables of the problem. Problems of
this type resemble the problems that arise in real world applications. In type B problems, the variable
coefficients were randomly generated in such a way that no variable qualified for elimination. Problems
of this type provide good insight into the performance of the algorithm, yet would rarely arise in practice.
The algorithm exhibited high variability for type A problems. The computational time for different
instances of the same size varied significantly for both Phases I and II. As a result, 50 random instances
were solved for each problem size. The results reported in columns 3-6 of Table 1 are the average and
maximum time in seconds needed for completion of Phase I and termination of the algorithm,
respectively. The last column of this table presents the average percentage of total variables that were
eliminated. It is clear from this column that the vast majority of the initial variables were eliminated.
Table 2 presents similar results as those of Table 1, for type B problems. The variance exhibited among
problems of the same size was much smaller in this case and as a result, 30 instances were solved for
each problem size. The random variate generation ensured that all variables belonged to the upper hull of
the associated coefficient space.
7
Table 1: Computational times (in seconds) for type A problems
Phase I
r
150
150
150
150
300
300
300
300
450
450
450
450
600
600
600
600
Nk
150
300
450
600
150
300
450
600
150
300
450
600
150
300
450
600
Avg
0.02002
0.03502
0.05390
0.06974
0.04064
0.07486
0.10914
0.14280
0.06108
0.11324
0.16804
0.21972
0.08288
0.15386
0.22058
0.29382
Total Time
Μax
0.040
0.041
0.100
0.100
0.060
0.110
0.150
0.200
0.101
0.171
0.221
0.291
0.120
0.211
0.291
0.380
Avg
0.04750
0.06528
0.08270
0.10114
0.28426
0.28992
0.35974
0.39756
0.98578
1.38052
1.22900
1.49870
2.43124
2.88496
2.99664
3.33722
Percentage
eliminated
Μax
0.090
0.101
0.130
0.141
0.561
0.631
0.681
0.942
2.573
3.204
3.415
6.059
6.409
10.786
8.833
9.654
95.2%
97.3%
98.1%
98.4%
95.3%
97.3%
98.1%
98.5%
95.1%
97.3%
98.0%
98.5%
95.1%
97.3%
98.1%
98.5%
Table 2: Computational times (in seconds) for type B problems
Phase I
r
150
150
150
150
300
300
300
300
450
450
450
450
600
600
600
600
Νk
150
300
450
600
150
300
450
600
150
300
450
600
150
300
450
600
Avg
0.11847
0.32083
0.64423
1.03803
0.26263
0.73273
1.39070
2.26900
0.40063
1.16713
2.27997
3.80067
0.58820
1.72463
3.47400
6.96337
Total Time
Max
0.160
0.331
0.751
1.162
0.350
0.872
1.503
2.384
0.471
1.282
2.364
3.925
0.701
1.813
4.287
9.754
8
Avg
0.13480
0.33817
0.65893
1.05607
0.39693
0.87617
1.52767
2.42580
0.88970
1.68743
2.79707
4.39387
1.82340
3.00337
4.79493
8.69147
Max
0.180
0.341
0.761
1.182
0.511
1.002
1.662
2.544
1.011
1.852
2.874
4.537
1.963
3.144
6.520
11.106
3.3 Discussion of Results
The variability exhibited for type A problems becomes clear from the results of Table 1. We observe that,
for some problem sizes, the average time needed for termination of the algorithm is almost 25% of the
corresponding maximum value. On the other hand, as it is clear from Table 2, type B problems exhibit a
much lower variability. This is because, in type B problems, a much larger number of activities are
considered in each set, since no variable is eliminated. Consequently, the sets look similar with each other
and the distribution of the available resource is much more balanced in the solution obtained after
termination of Phase I, which in turn simplifies the task of Phase II. Therefore, the total computational
effort does not differ very much for different instances of the same size.
For type A problems and fixed Nk, the percentage of total execution time devoted to Phase II seems to
increase as the total number of sets increases. On the other hand, for fixed r, the percentage of total
execution time devoted to Phase I seems to increase as the number of variables in each set increases. The
same behavior is observed for type B problems. Therefore, the number of variables in each set has a
greater impact on the computational effort of Phase I, while the number of sets on the computational
effort of Phase II.
Additionally, the percentage of total time devoted to Phase I is much smaller in type A problems than in
type B problems of the same size. This is not surprising, since the ordering of variables is the dominating
operation in Phase I, while the arrangement of sets is the dominating operation in Phase II. Hence, the
algorithm spends a larger percentage of the total time on ordering the variables in type B problems than
in type A problems. On the other hand, the algorithm spends a larger percentage of the total time on
arranging the sets in type A problems than in type B problems.
For the same total number of variables, the total computational effort increases as the number of sets
increases. This is more evident in type A problems, where for fixed total number of variables, an increase
in r results in an increase of the expected number of variables that are not eliminated (see Sinha and
Zoltners, 1979). On the other hand, for fixed r, the number of variables that remain after elimination does
not increase very fast for type A problems, even when the value of Nk increases from 150 to 600.
Therefore, the number of sets seems to be more crucial in the total computational effort of the algorithm
than the number of variables in each set.
Comparing the computational effort needed for the solution of a type A instance with that of a type B
instance of the same size, we observe that with very few exceptions, this is always smaller in type A
problems. This is mainly due to the fact that in type B problems no variable is eliminated and therefore a
larger computational effort is needed in Phase I for the ordering of the variables and the construction of
the multiple choice lists. On the other hand, when we compare the net time needed for execution of Phase
II this is smaller in type B than in type A problems of the same size.
The entire nondominated frontier of a 150X150 type A instance is shown in Figure 3. The feasible region
of the problem in objective space consists of the polyhedron formed by the nonnegative portions of the
two axes and this frontier, after it is extended to the right with a straight horizontal line. This is always an
open polyhedron, since there exists an infinite number of feasible values for f. The nondominated frontier
is always concave, ensuring that the feasible region of the problem in objective space is convex.
The result of Proposition 2 can be clearly visualized in these two diagrams. As we move towards the left
on the nondominated frontier, the marginal profit sacrificed in order to attain the same decrease on f
increases. Thus, the higher level of equity we want to attain, the higher price we have to pay. Let P(f ) be
the y-coordinate of the nondominated solution whose x-coordinate is equal to f. The above discussion
implies that the following result holds:
Corollary 2: P is an increasing piecewise linear concave function of f.
9
Figure 3: Nondominated frontier of a 150X150 type A problem
The graph of the nondominated frontier can be very useful in the decision making procedure. The
decision maker is presented with the whole nondominated set and the nondominated tradeoff rate (∆P/∆f
as a function of f) and can select the nondominated vector with his/her best tradeoff between the two
objectives. The nondominated vector could even be on a nondominated edge, not necessarily an extreme
point. Hence, the decision maker can decide directly on this diagram which solution gives the best
compromise between the two objectives. Of course, once this decision has been made, a specific Pareto
optimal solution can be found by modifying appropriately the terminating condition of Algorithm
BLMCK, if the history of iterations has not been stored.
In order to test the performance of Algorithm BLMCK further, we also developed an AMPL model (see
Fourer et al., 2002) for Problem BLMCK. To overcome the difficulty introduced by the second objective,
we used the e-constraint reduced feasible region method for multicriteria optimization (see Steuer, 1986)
and transformed it to a constraint by giving a specific value to f. This way, the second objective was
restricted to a maximum value, instead of being minimized. To ensure that the solution obtained this way
is Pareto optimal, we utilized Corollary 1 and set f equal to 0.
Of course, the main drawback of this approach is that, in contrast to Algorithm BLMCK, it is only able to
find a single nondominated solution and not the entire nondominated frontier of the problem. Even so,
the time Algorithm BLMCK needs to solve the problem is much smaller than the time required by
AMPL. In type A problems with r = 300, Nk = 300, AMPL (version 9.1, using CPLEX solver with
default values) needs approximately 1.5 seconds to find the optimal solution on the same machine. When
r = 450, Nk = 450, this time increases to approximately 5 seconds. For type B problems, the differences
between the algorithm and AMPL are drastically larger. When r = 300, Nk = 300, AMPL needs
approximately 10 seconds to find the optimal solution. When r = 450, Nk = 450, this time increases to 105
seconds. These time savings increase fast as the problem size increases. Therefore, the present algorithm
can be very useful to practitioners for solving real life applications with a large number of decision
variables. The reason that there is such a big difference in computational effort between the two problem
types is probably because in type A problems, the solver manages to detect the fact that the problem is
reducible in size which results in significant time savings.
Vector maximization algorithms (Steuer, 1986), that can alternatively be used to find the efficient set of
the problem, are reasonably expected to consume considerably more time than what AMPL does to find a
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single nondominated solution. This observation makes the efficiency of Algorithm BMCK even clearer,
since it manages to obtain the entire nondominated frontier in a timely and straightforward fashion.
4. Conclusions
In this work, we introduced an extension of the Linear Multiple Choice Knapsack Problem, which
incorporates a second objective. This objective accounts for an equitable allocation of the available
resource. We presented the mathematical formulation of the problem and showed that this problem
structure exhibits several fundamental properties. These were used to develop an optimal two-phased
algorithm for the problem. In Phase I, the algorithm ignores the second objective and enhances an
existing method for the Linear Multiple Choice Knapsack Problem to obtain an initial solution. Phase II
incorporates the second objective and successively obtains the entire nondominated frontier of the
problem.
Our computational results illustrate the efficiency of the algorithm. It outperforms a commercial linear
programming package and its superiority increases with problem size. The algorithm performs well
because it exploits the special structure of the problem. It utilizes the special relationship between the two
objectives. In addition, typical software packages can only solve single objective problems, which means
that they can only find one out of the many Pareto optimal solutions of the problem. This makes the
multiple advantages of the algorithm even more evident.
The present work points to a number of directions for future research. We have illustrated a situation in
which direct tradeoffs between the two objectives can be developed in a biobjective linear program. The
concept of the algorithm can be extended to other multiobjective problems, in which direct tradeoffs
between the objectives can be derived. We believe that the insights gained into this problem structure and
the advantages of the algorithm will prove useful in real world applications that involve a large number
of decision variables.
References
Fourer, R., Gay, D.M. and Kernighan, B.W. (2002) AMPL: A Modeling Language for Mathematical
Programming, Duxbury Press.
Kozanidis, G. and Melachrinoudis, E. (2004) “A branch & bound algorithm for the 0-1 mixed integer
knapsack problem with linear multiple choice constraints” Computers & Operations Research, 31, 695711.
Kozanidis, G., Melachrinoudis, E. and Solomon, M. (2005) “The linear multiple choice knapsack
problem with equity constraints” International Journal of Operational Research, 1, 52-73.
Lin, E. Y-H. (1998) “A bibliographical survey on some well known non-standard knapsack problems”
INFOR, 36, 274-317.
Sinha, P. and Zoltners, A.A. (1979) “The multiple choice knapsack problem” Operations Research, 27,
503-515.
Steuer, R. E. (1986) Multiple Criteria Optimization: theory, computation and application, New York:
Wiley.
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