New thinking of multi-objective programming with changeable space--in search of excellence.
Huang, Jih-Jeng ; Tzeng, Gwo-Hshiung
Introduction
It has been over 60 years since the simplex algorithm was proposed
by Dantzig (1947) as a discipline for handling optimization problems. In
the same year, von Neumann (1947) developed the theory of duality as a
linear optimization solution and applied it in the field of game theory.
The original motivation of mathematical programming was the need to
solve complex planning problems in wartime operations. After World War
II, this technique was extended to solve various industrial optimization
problems with great success. However, traditional mathematical
programming models, whether single-objective programming (SOP) or
multi-objective programming (MOP), consider optimizing a fixed objective
function(s) under given constraints or resources. Therefore, it can only
be viewed as an optimal tool within a system rather than as a tool to
optimize the system itself. More specifically, most progress in
traditional optimization methods has focused on algorithms or
convergence properties to derive the optimal solution within a system,
not on evolving the optimality concept itself, nor on expanding the
notions of true optimization (Zeleny 1998).
The postulate of traditional mathematical programming not only
simplifies problems themselves, but also reflects the situation which
companies faced in the past. As time has gone by, new practical problems
have become more complicated than in the past. An increasing number of
practical phenomena supports the necessity for changeable parameters of
mathematical programming. For example, the emergence of dispatched
workers or leased workers provides the flexibility of a company to
adjust its human-resource allocation. In addition, a system can
dynamically tune up its production resource allocation through satellite
manufacturers or outsourcing. Hence, modern systems and problems should
be considered to re-design and re-shape according to their flexibility,
not their fixation (Zeleny 1998).
Another obvious case to justify the necessity for changeable
parameters in mathematical programming is virtual organizations. Virtual
organizations can be regarded as a network of independent firms that
join together to capture specific market opportunities (Davidow, Malone
1992; Lambrechts et al. 2009). Each firm in a virtual organization
contributes core competences and shares resources. Hence, the resources
and production capabilities of a firm in a virtual organization are
changeable and variable. Hence, mathematical programming models should
also be flexible enough to deal with these problems. That is, we should
relax the parameters of mathematical programming models from fixation to
changeability.
De Novo programming was proposed by Zeleny (1982, 1986) to first
extend traditional mathematical programming to relax the assumption of
fixed resource allocations. The main concept of "De Novo
programming" is to reconstruct the original constraints by
incorporating the information of resources' unit prices. Hence, the
result of De Novo programming can determine the optimal resource
allocation of a system and its objectives so that trade-offs among
objectives can be appropriately eliminated (Zeleny 1990; Huang et al.
2005, 2006; Chen, Tzeng 2009).
Although Shi (1999) proposed the concept of a possible debt to De
Novo programming, there are some significant differences between the two
models. Shi's model (Shi 1999) enables the available resources to
be flexible. Then, a possible debt can borrow additional money from a
bank with a fixed interest rate to achieve the preferences of multiple
decision makers and the corresponding objectives. On the other hand, the
proposed models try to achieve the desired levels which are set by a
decision maker under the objective of the minimum total debt. In
addition, the proposed models consider the situations of the changeable
parameters of the objective and technological coefficients.
In addition, the concept of the "changeable space" of
mathematical programming was also proposed by (Yu, ChiangLin 2006;
ChiangLin et al. 2007; Yu, Chen 2010, 2012) to illustrate that the
parameters of a system may in practice be changed via investment,
outsourcing or production improvement. However, they did not develop a
complete model to resolve the mathematical programming problems of
changeable parameters, also called "changeable space". The
basic concept of changeable decision space and aspiration level can be
shown as in Fig. 1.
[FIGURE 1 OMITTED]
1. De Novo programming
De Novo programming can be considered as an optimal method for
solving resource allocation problems. Traditionally, a multiple
objective resource allocation problem (Hackman, Platzman 1990) could be
formulated to maximize the following multi-objective knapsack problem:
max z = Cx; (1) s.t. Ax [less than or equal to] b, x [greater than
or equal to] 0, (1)
where: C is the objective coefficient matrix; x denotes the
variable vector; A denotes the technological coefficient matrix and the
b vector denotes the maximum limited resource portfolios.
However, traditional resource allocation models may be
inappropriate in the real world. First, since the components of b are
determined in advance, the optimal solution heavily depends on whether
resources are appropriately allocated. In addition, although it is
rational to allocate resources using the above model in a hierarchical
system, when resources can be brought from markets, the factor of a
resource's unit price should be considered, and so the traditional
methods are no longer suitable. In order to deal with this issue, De
Novo programming was proposed.
De Novo programming was proposed by Zeleny (1982, 1986) to
re-design or re-shape a given system to achieve an ideal point based on
given resources (subject to constraints). He suggested that trade-offs
are properties of an inadequately designed system and thus can be
eliminated through designing a better, preferably optimal system.
The usefulness of De Novo programming is that it can consider the
following MOP problem (Zeleny 1982). Assume a manufacturer produces two
different products, suits and dresses, in quantities x and y. Each of
them requires five different resources, nylon through golden thread,
according to technologically determined requirements. Unit prices of
resources are also given, as shown in Table 1.
If two objectives, namely profit ([f.sub.1]) and quality index
([f.sub.2]), are considered by the company, we can formulate the
following two-objective mathematical programming problem as:
max [f.sub.1] = 400[x.sub.1] + 300[x.sub.2];
max [f.sub.2] = 6[x.sub.1] + 8[x.sub.2];
s.t. 4[x.sub.1] [less than or equal to] 20,
2[x.sub.1] + 6[x.sub.2] [less than or equal to] 24,
12[x.sub.1] + 4[x.sub.2] [less than or equal to] 60,
3[x.sub.2] [less than or equal to] 10.5,
4[x.sub.1] + 4[x.sub.2] [less than or equal to] 26,
[x.sub.1], [x.sub.2] [greater than or equal to] 0,
where [f.sub.1] and [f.sub.2] denote the objectives: profit and
quality index, respectively. Let the two objectives be equally
important. Then, if we employ the compromise solutions and set p = 2, we
can obtain the optimal solution: [x.sub.1] = 3.9837, [x.sub.2] = 2.5163,
[f.sub.1] = 2348.37 and [f.sub.2] = 44.03. However, the above solution
may be unsatisfactory to the decision makers, due to the inappropriate
resource portfolio.
Therefore, Zeleny (1998) proposed the concept of an optimal
portfolio of resources, which is the design of system resources with a
sense of integration, so that there are no trade-offs in a newly
designed system. The original idea of De Novo programming was that
productive resources should not be engaged individually and separately
because resources are not independent. Later, various issues, such as
considering fuzzy coefficients (Li, Lee 1990a, b, 1993; Lee 1994), other
optimum-path ratios (Shi 1995) and the 0-1 programming problem (Kim et
al. 1993), have been proposed to deal with more complicated situations.
Based on the above concept, via breaking the constraints of fixed
resources, De Novo programming can eliminate trade-offs between
objectives to achieve the aspiration solution.
The procedures of De Novo programming can be described as follows
(Zeleny 1982):
1. Find the aspiration level vector ([z.sup.u]) by solving each
objective function of a system separately as
max [z.sup.u.sub.k] = [c.sub.k] x, k = 1,...,m; s.t. Vx [less than
or equal to] B, x [greater than or equal to] 0, (2)
where: [z.sup.u] = [z.sup.u.sub.1] [z.sup.u.sub.2] ...
[z.sup.u.sub.m]] denotes the aspiration level vector; V = pA denotes the
unit cost vector; p is the resource's unit price vector and B
denotes the total budget.
2. Identify the minimum budget B* and its corresponding resource
allocation ([x.sup.*] and [b.sup.*] = A[x.sup.*]) with the aspiration
level, such as:
min [B.sup.*] = V[x.sup.*];
s.t. Cx = [z.sup.*],
x [greater than or equal to]0. (3)
3. Use the optimum-path ratio (r) to obtain the final solution (z,
x and b).
z = r x [z.sup.*]; (4)
x = r x [x.sup.*] (5)
and
b = r x [b.sup.*], (6)
where r = B / [B.sup.*]. (7)
If we reconsider the previous example and deal with that problem
using De Novo programming, we can obtain the optimal resource allocation
as: x = 4.03 and y = 2.54, [b.sub.1] = 16.12, [b.sub.2] = 23.3,
[b.sub.3] = 58.52, [b.sub.4] = 7.62 and [b.sub.5] = 26.28 under the
total budget B = 2600 to generate [f.sub.1] = 2375 and [f.sub.2] = 44.5
(ideal point). Comparing this result with the previous example, we can
find that two objective values deriving from De Novo programming are
better than that of traditional MOP, due to the appropriate resource
allocation.
The procedures of De Novo programming proposed by Zeleny (1982)
only derive one possible Pareto solution. Formally, we can formulate De
Novo programming as:
max Cx; (8)
s.t. p'Ax [less than or equal to] B,
Ax [less than or equal to] b,
x [greater than or equal to] 0. (8)
Next, we can solve the above problem using traditional MOP models,
such as goal programming or compromise solutions, to derive the optimal
solution of De Novo programming.
Another kind of extension for MOP is to consider changeable
parameters of MOP models. De Novo programming, like traditional
mathematical programming, also postulates that the parameters of models,
such as technological coefficients and objective coefficients, are
fixed. However, these parameters may be changeable if we consider
exogenous factors, such as increasing investment, improving efficiency
and time advancement (Chiang Lin et al. 2007).
Chiang Lin et al. (2007) proposed a single-objective programming
model with changeable parameters as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where: [C.sub.j1] denotes the original [j.sup.th] objective
coefficient; [C.sub.j2] is the [j.sup.th] new objective coefficient
caused by y, which could be investment in advertisement, service and
distribution channels, etc.; [d.sub.k1] denotes the original [k.sup.th]
available resources and [d.sub.k2] is the [k.sup.th] new extra resource
caused by z, which could be making alliances, outsourcing, etc. Usually,
y and z have upper limit constraints, namely y' and z',
respectively, or Eq. (9) will become an unbounded problem.
Let us modify the previous example (as shown in Table 2) to
demonstrate Yu's model. Assume the firm can increase its objective
coefficients and resources through investment (y) and purchase (z). The
profits of products [x.sub.1] and [x.sub.2] can be increased by 3 and 4
units separately and the quality index of products [x.sub.1] and
[x.sub.2] can be increased 0.3 and 0.4 separately through the
investment. In addition, the number of units of each resource can be
increased through the unit purchase. Finally, the investment and
purchase levels should be less than 7 units, separately, and the total
of the investment and purchase is limited to 10.
Although Yu's model only considers the single-objective
situation, we can simply transform it to a multi-objective situation by
using the compromise solutions. Here, we use the compromise solution and
set p = 2 to calculate the problem as follows:
max [f.sub.1] = 400[x.sub.1] + 300[x.sub.2] + y(3[x.sub.1] +
4[x.sub.2]);
max [f.sub.2] = 6[x.sub.1] + 8[x.sub.2] + y(0.3[x.sub.1] +
0.2[x.sub.2]);
s.t. 4[x.sub.1] [less than or equal to] 20 +0.3z,
2[x.sub.1] + 6[x.sub.2] [less than or equal to] 24 + 0.3z,
12[x.sub.1] + 4[x.sub.2] [less than or equal to] 60 + 0.3z,
3[x.sub.2] [less than or equal to] 10.5 + 0.3z,
4[x.sub.1] + 4[x.sub.2] [less than or equal to] 26 + 0.3z,
0 [less than or equal to] y, z [less than or equal to] 7,
y + z [less than or equal to] 10,
[x.sub.1], [x.sub.2], y, z [greater than or equal to] 0.
The optimal solution can be calculated as: y = 7, z = 3, [x.sub.1]
= 4.1655, [x.sub.2] = 2.5595, [f.sub.1] = 2593.19 and [f.sub.2] = 57.80.
From the above result, besides resource reallocation, we know that the
firm can also increase its profit and quality index through outside
investment or resource purchase.
Although the previous models extended traditional mathematical
programming to deal with more practical problems, they cannot satisfy
the purpose of this paper. Let us depict Figure 2 to highlight the
purpose of this paper as follows.
[FIGURE 2 OMITTED]
The original idea of De Novo programming is to re-allocate
production resources so that a system's trade-offs can be
eliminated, and the ideal point can be achieved. However, the question
arises, what if the ideal point still cannot satisfy the decision
makers? Although ChiangLin et al. (2007) proposed another model to
resolve the above problem, their model still cannot ensure that the
desired level, which is wanted by the decision maker, can be achieved.
This might be true if adding the effects of y and z still cannot achieve
the desired point. In addition, what if a company cannot consider the
effects of y or z? Maybe the above questions should be transformed into
the problem that if the desired point is wanted by the decision maker,
how can the system be adjusted or re-designed to achieve that point?
Decision makers may prefer to know what to do to achieve the desired
point (aspiration level) via a re-designed system than what to do to
optimize a fixed system.
It is obvious that if the parameters of a system remain constant,
the best solution is the ideal point if there is no trade-off among
objective functions. However, the desired point (aspiration level),
which is better than the ideal point, can never be achieved. Hence, if
we want to achieve the desired point (aspiration level), we have to
"upgrade" the parameters of the system. Usually, a company may
improve its objective/technological coefficients by importing new
equipment and methods or by enlarging its fixed resources or promoting
its human resources by innovating technology or increasing the total
budget, or both, or other ideas. Next, we develop possible models to
re-design or re-shape a system to achieve the desired point (aspiration
level) according to the concept of changeable space, including decision
space and objective (outcome) space.
2. MOP with changeable parameters
In practice, there are three basic ways for a system to achieve its
desired point (aspiration level): 1) increasing budgets; 2) improving
objective coefficients; and 3) upgrading production efficiency. Next, we
discuss each situation and develop the corresponding optimal model as
follows. In the first situation, a company can make some financing
decisions, e.g. to borrow money from the bank or to issue company
bonds/stocks to the capital market in order to enlarge/enrich available
budgets so as to increase R&D in innovation and creativity for the
value-created knowledge economy. Then, the problem of achieving the
desired point (aspiration level) is equivalent to minimizing the extra
budget under the given objectives space and decision space
(constraints).
Assume a company has n objectives to be achieved and m products are
produced. We can incorporate the concepts of financing decisions into
MOP and formulate the following model:
<Model 1: MOP with changeable budgets>
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)
where: [c.sub.ij] denotes the [j.sup.th] coefficient of the
[i.sup.th] objective function; [[f.sup.**.sub.1](x) denotes the desired
value of the [i.sup.th] objective; p denotes the unit price vector of
resources; B is the original budget and [??] denotes the extra budget
obtained from financing decisions.
Example 1. Let us reconsider the initial example of producing suits
and dresses. If the objective functions and constraints are constant, we
can obtain the optimal solution from De Novo programming as [f.sub.1] =
2375 and [f.sub.2] = 44.5. However, the decision maker feels unsatisfied
with the results and hopes to increase [f.sub.1] (profit) from 2,375 to
2,600 and [f.sub.2] (quality) from 44.5 to 60, respectively. Hence, one
way to achieve the desired solution is to borrow money from the capital
markets via financing decisions. Table 3 presents the information
related to Example 1.
The problem of Example 1 is to derive the minimum extra budget
which can achieve the desired point and determine the corresponding
resource allocation. Next, we can formulate the following linear
programming to solve the problem of Example 1 as:
min [??]
s.t. 400[x.sub.1] + 300[x.sub.2] = 2600,
6[x.sub.1] + 8[x.sub.2] = 60,
30 x 4[x.sub.1] + 40 x (2[x.sub.1] + 6[x.sub.2]) + 9.5 x
(12[x.sub.1] + 4[x.sub.2]) + 20 x 3[x.sub.2] +
10 x (4[x.sub.1] + 4 [x.sub.2]) [less than or equal to] 2600 +
[??],
[x.sub.1], [x.sub.2] 0.
By solving the above problem, we obtained that the extra budget
needed is [??] = 376, producing [x.sub.1] = 2 and [x.sub.2] = 6. The
corresponding resource allocation can be calculated as [b.sub.1] = 8,
[b.sub.2] = 40, [b.sub.3] = 48, [b.sub.4] = 18 and [b.sub.5] = 32. The
corresponding profit and quality index are exactly equal to 2,600 and
60, respectively.
Comparing the result with De Novo programming and the proposed
models, it can be seen that the proposed method can find a way to
achieve the desired point which cannot be achieved by using De Novo
programming. It is clear that the only way to go beyond the ideal point
is to ask for outside help. Therefore, if the system hopes to achieve
the desired point, it needs an extra $376. However, if the extra budget
needed is equal to zero, it means that the original system is good
enough to achieve the desired point, and the proposed model reduces to
De Novo programming.
Besides borrowing money from capital markets, a company can also
achieve its desired goal through improving the objective coefficients of
a system, e.g. economics of scale, electronic commerce, total quality
management (TQM) and eliminating middle agencies. In this situation, a
company should consider the unit improving cost of each objective
coefficient and determine the optimal budget allocation between the
improving costs and production resources. Then, we can develop a MOP
model with changeable objective coefficients as follows:
<Model 2: MOP with changeable objective coefficients>
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [p.sup.c.sub.ij] denotes the unit upgrading cost with respect
to the [j.sup.th] product coefficient of the [i.sup.th] objective
function and [[??].sub.ij] is the [j.sup.th] upgrading product
coefficient of the [i.sup.th] objective function.
Example 2. Following the previous example of producing suits and
dresses, if the company cannot borrow money from capital markets, but
also hopes to increase [f.sub.1] (profit) from 2,375 to 2,600 and
[f.sub.2] (quality) from 44.5 to 60, another way is to improve its
objective coefficients through possible strategies or technologies.
Therefore, we assume that the unit improving costs of the objective
coefficients are $0.200, $0.289, $2.225 and $2.487, respectively, as
shown in Table 4.
Next, we can formulate the following mathematical programming to
consider achieving the desired points via improving the objective
coefficients as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Solving the above problem, we can obtain the extra budget [??] = 0.
The result means that no extra budget is needed for achieving the
desired point. Then, we can also derive [x.sub.1] = 4.43, [x.sub.2] =
2.70, [[??].sub.11] = 3.51, [[??].sub.12] = 0.18, [[??].sub.21] = 1.44,
and [[??].sub.22] = 2.00. In addition, the corresponding resource
allocation can be assigned as [b.sub.1] = 17.72, [b.sub.2] = 25.06,
[b.sub.3] = 17.72, [b.sub.4] = 8.10, [b.sub.5] = 28.52, profit = 2600
and the quality index = 60. We should highlight that if [??][not equal
to] 0, it means that we still cannot achieve the desired point via
improving objective coefficients. Therefore, it indicates that extra
budget is also needed for the system to achieve the desired point.
Otherwise, we should consider another possibility.
The last situation discussed here is that a company may expand its
outcome space through upgrading the technology coefficients of a system.
For example, a company can adopt business process reengineering (BPR),
new information technologies or enterprise resource management (ERP) to
increase its production efficiency, i.e. upgrading its technology
coefficients. Hence, we can conceptualize the above description to
formulate the following model:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
upgrading technological coefficient matrix and [p.sup.a.sub.kj] is the
unit upgrading cost with respect to the [j.sup.th] technology
coefficient of the [k.sup.th] constraint.
Example 3. We can follow the previous example of producing suits
and dresses to address the following problem. If the company still
cannot achieve the desired point through adding extra budget or
improving objective coefficients, the last possible way is to update the
technological coefficients of the system. Therefore, we assume each unit
updating cost of the technological coefficients can be defined and
presented as shown in Table 5.
Incorporating the information of the unit updating cost of the
technological coefficients, we can formulate the following mathematical
programming model:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Solving the above problem, we can obtain the extra budget [??] = 0.
The result means that no extra budget is needed for achieving the
desired point. Then, we can also derive [x.sub.1] = 2.42, [x.sub.2] =
5.69, [[??].sub.11] = 2.03, [[??].sub.21] = 1.27, [[??].sub.22] = 0.30,
[[??].sub.31] = 0.27, [[??].sub.32] = 0.25, [[??].sub.42] = 0.26,
[[??].sub.51] = 0.25 and [[??].sub.52] = 0.26. In addition, the
corresponding resource allocation can be assigned as [b.sub.1] = 4.76,
[b.sub.2] = 34.20, [b.sub.3] = 49.72, [b.sub.4] = 15.59, [b.sub.5] =
30.36, profit = 2600 and quality index = 60.
In the previous models, we can use one of the three ways to try to
achieve the desired point. In practice, three situations could exist
simultaneously. Therefore, a more general model of changeable parameters
can be considered to incorporate the previous three situations as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
If [??] = 0, it means that the desired point can be achieved
without increasing the budget. The decision maker can improve objective
co efficients (when [[??].sub.ij] [not equal to] 0), update the
technological coefficients (when [[??].sub.ij] [not equal to] 0) or both
(when [[??].sub.ij] [not equal to] 0 and [[??].sub.ij] [not equal to] 0)
to achieve the desired point. On the other hand, if B [not equal to] 0,
it means that the original system cannot be achieved through updating
the objective or the technological coefficients. Hence, the only way is
to increase the budget through borrowing money from capital markets.
Note that the proposed models enable the objective and
technological coefficients to be changeable. Although this
characteristic makes the model more flexible, it may suffer from
problems of computation and solubility. Hence, we propose the concept of
convex programming, and the corresponding proof to justify that our
models is rational and solvable as follows.
Definition 1. A mathematical programming is called convex
programming if and only if its objective and constraint functions are
convex. Note that it is clear that least-squares and linear programming
are special kinds of convex programming.
Definition 2. Suppose a function f: [R.sup.n] [right arrow] R is
twice differentiable. Then f is convex if and only if its domain, D(f),
is a convex set and its Hessian matrix is positive semidefinite.
As we know, convex programming is a special kind of non-linear
programming. Like linear programming, it generally has no analytic
solution. However, for a convex optimization problem, all locally
optimal solutions are globally optimal. Hence, there are many
algorithms, such as interior-point method, which can be used for solving
these problems reliably and efficiently. In addition, the time
complexity of the convex programming is approximately max{[n.sup.3],
[n.sup.2]m, F}, where n is the number of variables, m is the number of
constraints, and F is the cost of the evaluating objectives, constraints
and their first and second derivatives. In a general situation, the
complexity of a convex programming model belongs to the polynomial
problems. It can be easily solved by today's computers.
The proposed models belong to convex programming.
Proof:
Equations (10)-(12) are special cases of Equation (13). Hence, if
Equation (13) is a convex programming problem, Models 1-3 are convex
programming problems.
First, we want to examine the Hessian matrices of the objective and
constraints of Equation (13). For convenience of the operations, we
assume
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then, the Hessian matrix of each equation is calculated,
separately, as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Clearly, all of the matrices above are positive semidefinite.
Hence, the problem of Equation (13) is a convex programming problem.
Similarly, we can prove that Models 1-3 are examples of convex
programming. This proof also justifies the rationale and solubility of
the proposed models.
Next, we propose the related findings according to the results of
the illustrated examples.
3. Discussion
Traditional MOP is used to solve optimal problems with a given and
fixed system. However, an increasing number of flexible
systems/organizations, such as team-based and virtual organizations, are
being proposed to replace traditional ones. Therefore, the problem of
system optimization should be extended from inside to outside the
system. That is, we should design a new system rather than optimizing a
given system through changing the system parameters. In practice, these
methods include BPR, outsourcing, dispatching workers, etc.
In this paper, we develop three MOP models for decision-makers to
design or plan a system. The first model retains the original parameters
of objective and technological coefficients but only considers borrowing
money from capital markets. The second model enables objective functions
to be changeable so that a system can improve its objective coefficients
to achieve the desired point. The last model considers that the
technological coefficients of a system are changeable. Therefore, the
desired point can be achieved through updating its technological
coefficients.
Next, we can illustrate how to expand the changeable spaces for
achieving the desired point (aspiration level), as shown in Fig. 3. Note
that in Fig. 2, the left and right quadrant charts denote the decision
and objective spaces, respectively. The curves pointed from the decision
space to objective space indicate the corresponding corner solutions of
the programming problem. In addition, Fig. 4 displays the sets of
different objective spaces. It can be seen that the objective set is
gradually enlarged if we release the parameters of the traditional
mathematical programming problems. That is, the optimal solution can be
upgraded by considering changeable parameters of mathematical
programming.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
It can be seen that the decision spaces of the illustrated example
are expanded gradually through resource re-allocation or changeable
parameters. Therefore, the corresponding objective spaces are changeable
and create the possibility of achieving better solutions. As we know,
the original decision space will cause the Pareto solution, due to the
trade-off between objectives. Although the De Novo programming enables
the possibility of resource re-allocation to obtain a better result, it
still causes a De Novo trade-off because of the fixation of the system
parameters. Therefore, the only way to achieve the desired point is to
expand the decision space via changeable parameters. Then, we can obtain
the desired point without trade-off between objectives.
Here, we compare the three models, De Novo, Yu's model and the
proposed model, as follows. Although the three models share the same
purpose of system design, there are several significant differences
between them. First, De Novo incorporates the variables, resource unit
prices and total budget, to re-allocate the available resources. On the
other hand, Yu's model enables the objective coefficients and
resources to be changeable via outside investment or time. However, the
proposed model considers all possible situations of parameter
changeability. In addition, the purpose of the proposed model is to find
a way with the minimum budget to achieve the desired objectives set with
minimum budget. In contrast, De Novo and Yu's model derives the
optimal solution under re-given objectives and/or constraints. The
comparisons of these models are presented in Table 6.
In addition, the relationship between these models can be
represented as shown in Fig. 5.
Recently, the concept of habitual domains (HD) has been proposed
and supports the flexible or changeable parameters of decision spaces.
Habitual domains (Yu 1980, 1984, 1985, 1990, 1991) refer to the set of
human's thinking, judging, responding, experience, and knowledge.
Therefore, it is clear that HDs play a key role in affecting human
behaviour. To improve the quality of decision-making, people should
think about two major problems: 1) how to polish the existing HDs; 2)
how to expand the existing HDs. Recently, many papers have been proposed
to efficiently expand HDs, known as competence set analysis (Yu, Zhang
1989, 1990, 1992; Li, Yu 1994), and practical applications (Huang et al.
2012; Larbani, Yu 2009a, b, 2012a, b). These methods can be used to
significantly improve the quality of decision-making.
There are four elements within a habitual domain:
1. The potential domain (PD)--the collection of ideas and actions
that can potentially be activated;
2. The actual domain (AD)--the set of ideas and actions that are
actually activated;
3. The activation probabilities (AP)--the probabilities that ideas
and actions in PD also belong to AD;
4. The reachable domain (RD)--the set of ideas and actions that can
be attained from a given set in an HD.
Since decision processes depend on the evolution of HDs, an
expanding HD can result in effective decisions and preferred solutions.
Hence, the expanding HD can be considered as the changeable parameters
of the system.
Next, we compare the differences between goal programming and the
proposed method as follows. Goal programming was proposed by Charnes and
Cooper (1961) to deal with linear MOP problems. The idea of goal
programming is to seek a solution which is nearest to the ideal point.
Thus, a decision maker should first assign the targets or goal to each
objective and then minimize the "distance" from the targets to
the objectives (usually we can use [L.sub.p]-norm to define the distance
between the targets and the objectives) to find the solution.
The differences between goal programming and the proposed methods
can be described as follows. First, the purpose of goal programming is
to find a solution, which is located in Pareto solutions, and is closest
to the ideal point. However, the purpose of the proposed models is to
find out a way that can achieve the desired point via possible changes.
Hence, the optimal solution of goal programming is usually worse than
the ideal point because of tradeoffs among objective functions. On the
other hand, the proposed models can ensure that the desired point, which
is better than the ideal point, can be achieved. In sum, goal
programming optimizes the objective functions within a given system,
i.e. inside optimization. However, the proposed models minimize extra
budget with a flexible system, i.e. outside optimization.
Secondly, goal programming optimizes the objective functions of a
system with fixed parameters. That is, goal programming solves the
problem of what to do to obtain the optimal solution of a system. On the
other hand, the proposed models tolerate the parameters of a system
being changeable so that the desired point can be achieved through
possible adjustments. Hence, goal programming is more suitable for
dealing with "what to do" problems, while the proposed models
try to answer "how to do" problems.
Finally, the essence of goal programming is to determine an optimal
solution in an MOP problem. Therefore, like other traditional MOP
methods, such as compromise solutions, it is an optimization method.
However, the essence of the proposed models seeks to change or modify
the original system so that the desired point can be achieved. Hence,
the proposed method can be regarded as an optimal system-design method
rather than an optimization method.
On the other hand, we can also highlight the differences between De
Novo programming and the proposed models. De Novo programming
incorporates the information of resources' unit prices into
traditional mathematical programming to design a better system. On the
other hand, the proposed method not only considers resources' unit
prices but also relaxes other parameters, such as objective and
technological coefficients, to be changeable. In addition, the optimal
solution of De Novo programming is the ideal point. However, the
proposed method seeks ways to achieve the desired point.
Finally, we can highlight the philosophical difference between the
proposed method and other MOP methods, including De Novo programming,
from the perspective of formal analysis (Tzeng, Huang 2011). All
traditional MOP models can be considered as normative models which focus
on the problems that decision makers should ideally resolve. However,
the proposed method should be regarded as a prescriptive model which
considers the methods that decision makers ought to use to improve their
decisions.
Conclusions
Traditional MOP problems focus on the optimization within a system.
However, more and more technologies and strategy developments enable
systems to be re-designed or re-shaped to perform better. Hence, we
should extend traditional MOP from a normative model to a prescriptive
model. In this paper, we propose three kinds of MOP with changeable
parameters, i.e. budget, objective coefficients and technological
coefficients, to help decision-makers to achieve their desired points.
doi: 10.3846/20294913.2013.860931
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Jih-Jeng Huang (a), Gwo-Hshiung Tzeng (b)
(a) Department of Computer Science & Information Management,
Soochow University, Taipei, Taiwan
(b) Institute of Management of Technology, National Chiao Tung
University, Hsinchu, Taiwan
(c) Graduate Institute of Urban Planning, National Taipei
University, San Shia, Taiwan
Received 02 August 2012; accepted 30 June 2013
Corresponding author Jih-Jeng Huang
E-mail: jjhuang@scu.edu.tw
Jih-Jeng HUANG is an Associate Professor of Computer Science and
Information Management at Soochow University, Taiwan, and teaches
research method, multivariate analysis, and capital asset and pricing
models, among others. He received his PhD in Information Management from
the National Taiwan University. He has published on these interests
widely in journals and conference proceedings. His current research
interests include multiple criteria decision making, knowledge
management, behavioural economics and finance, and data analysis.
Gwo-Hshiung TZENG is a Distinguished Chair Professor of National
Taipei University, Taiwan. He received a Bachelor's degree in
Business Management from the Tatung Institute of Technology (1967); a
Master's degree in Urban Planning from Chung Hsing University
(1971); and a PhD in Management Science from Osaka University, Japan
(1977). He is Editor-In-Chief of the International Journal of Operations
Research and the International Journal of Information Systems for
Logistics and Management, among others. His current research interests
include statistics, multivariate analysis, networks, routing and
scheduling, multiple criteria decision making, fuzzy theory,
hierarchical structure analysis for application to technology
management, energy, environment, transportation systems, transportation
investment, logistics, location, urban planning, tourism, technology
management, electronic commerce, and global supply chains.
Table 1. Resource allocation of Zeleny's example
Technological coefficients
Unit Resource [x.sub.1] = 1 [x.sub.2] = 1 No. of
price units
30 Nylon 4 0 20
40 Velvet 2 6 24
9.5 Silver thread 12 4 60
20 Silk 0 3 10.5
10 Golden thread 4 4 26
Table 2. Modified example to demonstrate Yu's model
Technological coefficients
Resource [x.sub.1] = 1 [x.sub.2] = 1 No. of Unit Purchase
units Benefit
Nylon 4 0 20 0.3
Velvet 2 6 24 0.3
Silver 12 4 60 0.3
thread
Silk 0 3 10.5 0.3
Golden 4 4 26 0.3
thread
Table 3. Information table for Example 1
Technological coefficients
Unit Resource [x.sub.1] = 1 [x.sub.2] = 1 No. of units
price
30 Nylon 4 0 [b.sub.1]
40 Velvet 2 6 [b.sub.2]
9.5 Silver thread 12 4 [b.sub.3]
20 Silk 0 3 [b.sub.4]
10 Golden thread 4 4 [b.sub.5]
Table 4. Information table for Example 2
Technological
Objective coefficients coefficients
x = 1 y = 1 Unit Resource [x.sub.1] [x.sub.2]
price = 1 = 1
400 ($0.200) 300 30 Nylon 4 0
($0.29)
6 ($2.225) 8 ($2.487) 40 Velvet 2 6
9.5 Silver 12 4
thread
20 Silk 0 3
10 Golden 4 4
thread
Objective coefficients
x = 1 y = 1 Unit No. of
price units
400 ($0.200) 300 30 [b.sub.1]
($0.29)
6 ($2.225) 8 ($2.487) 40 [b.sub.2]
9.5 [b.sub.3]
20 [b.sub.4]
10 [b.sub.5]
Table 5. Information table for Example 3
Objective Technological
coefficient coefficients
x = 1 y = 1 Unit Resource [x.sub.1] [x.sub.2] No. of
price = 1 = 1 units
400 300 30 Nylon 4 ($0.5) 0 [b.sub.1]
6 8 40 Velvet 2 ($0.5) 6 ($0.27) [b.sub.2]
9.5 Silver 12 ($0.27) 4 ($0.26) [b.sub.3]
thread
20 Silk 0 3 ($0.25) [b.sub.4]
10 Golden 4 ($0.25) 4 ($0.25) [b.sub.5]
thread
Table 6. The differences between the system design of the models
De Novo Yu's model
Changeable Budget Resource and
parameters Objective
Optimality via Resource Outside investment
Reallocation or time
Optimal solution Model determined Model determined
Purpose Optimality under Optimality under
given constraints given constraints
Objectives Original objectives Original objectives
Philosophy What What
The proposed
Changeable Budget, Resource,
parameters Objective and Technology
Optimality via Both
Optimal solution Determined by decision-makers
Purpose Optimality by any possibility
Objectives Minimum debt
Philosophy How
Fig. 5. The relationship between the models
The Proposed Model
--Technological coefficients changeable
--Achieving the desired level via the minimum (extra) budget
De Novo Programming Changeable Model
--Resource reallocation --Objective changeable
--Resource unit price --Resource changeable
--Via budget --Via investment or time
