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Review of Management and Economical Engineering, Vol. 6, No. 5
SOLVING ECONOMIC MODELS BY MEANS OF LINEAR
PROGRAMMING WITH OBJECTIVE FUNCTION
LINEAR ON INTERVALS
Dorin LIXĂNDROIU
TRANSILVANIA University of Brasov
lixi.d@unitbv.ro
Abstract: The paper analyzes the defining modality in the linear programming
models of objective functions linear on intervals by means of binary variables. This
allows the accomplishment of some optimization models of productive processes,
which bear a closer relation to reality.
Keywords: economic modeling, linear programming, objective function linear on
intervals.
1.
INTRODUCTION
In the economic models that lead to linear programming, the objective function
to be optimized (maximization or minimization) is linear. For instance, in the
models for optimizing production, for a volume of production x and a variable unit
cost c, the objective function has the form: z  c  x .
This approach does not bear any relation to reality when the fixed costs vary on
intervals or the direct variable costs (or the unit receipts) differ on disjoint intervals
of the amount of produced or sold products.
In order to write the objective functions in these complex economic situations,
we will consider the cost functions linear on intervals in what follows [WIL, 1993],
[GIA, 2003].
2.
THE COST FUNCTION LINEAR ON INTERVALS
The form of the objective function presented above cannot be accepted when
the production decision also involves a fixed cost, i.e. the partial production cost of
the good considered is 0 , if the production is zero; if x>0, it equals c  x  K , . For
solving, we introduce the binary variable y , defined as follows:
y = 0, if x =0
y = 1, if x >0.
This allows the introduction of the fixed cost in the objective function, which will
take the following form:
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Min z, with z  c  x  K  y  ....
(1)
The connection between x and y can be achieved by the supplementary restriction:
x  M  y,
where M is a constant greater or equal to the maximum value that x can take.
Generalization.
For any objective function (cost or income) linear on intervals, the fictitious
variables xk can be introduced, which take values on the interval ( M k1 , M k ] , with
M0 =0.
On each interval, the variable cost is constant, and the values xk will all be
zero, excepting the interval which includes the value x. Figure 1 presents an example
of objective function of the total cost. This is neither concave (the average
production cost does not rise when x increases), nor convex (average production cost
does not decrease when x increases).
Figure 1. The total cost function
The generalization allows the approach of the uniform rebates (figure 2) and of the
progressive rebates (figure 3) in supply activities.
Figure 2. The total cost function in case of a uniform rebate
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Review of Management and Economical Engineering, Vol. 6, No. 5
Figure 3. The total cost function in case of a progressive rebate
When building up the model, the variable x will be replaced by x1  x2  x3  x4 .
The objective function for the relative part of this production cost becomes:
Min z,
with
z  c1  x1  K 1  y1   c2  x2  K 2  y 2   c3  x3  K 3  y3   c4  x4  K 4  y4   ...
under restriction: 0  x1  M 1  y1
M 1  y 2  x2  M 2  y 2
M 2  y 3  x3  M 3  y 3
(2)
M 3  y 4  x4  M 4  y 4
y1  y 2  y3  y4  1
In figure 4, we present the graph of a concave cost function (the function is nondecreasing).
Figure 4. The concave total cost function
Remark.
If x varies discontinuously, for a variation ε of the cost function, we have to decide
whether x = Mi belongs to the interval i or i+1. Supposing the upper bounds of the
intervals are excluded, there will be:
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0  x1  M 1     y1
M 1  y 2  x2  M 2     y 2
M 2  y3  x3  M 3     y3
M 3  y4  x4  M 4     y4
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(3)
y1  y 2  y3  y4  1
The last restriction does not allow that x belong to the interval (Mi_- ε, Mi).
Model (2) can be generalized for a number of N intervals:
z   ci  xi  K i  yi 
Min z, with
i
under restrictions:
M i 1  yi  xi  M i  yi , for each i=1,2,...,N
 yi  1
(4)
i
Model (4) contains 2  N  1 restrictions. If the number of intervals is greater than
3, we can have a more efficient alternative statement, which requires N+1
restrictions [GIA, 2003], starting from the idea that each xk included between Mk-1
and Mk can be represented as a linear combination having the form:
(5)
xk   k 1  M k 1   k  M k , with  k 1   k  1
3.
AN EXAMPLE OF APPLICATION
We now consider the issue of the distribution of “clients” (warehouses) to the
production or distribution centers. We suppose there are m production/ distribution
centers and n warehouses. We consider that all the production/ distribution centers
are open and the capacity of each center does not vary in the lapse of time
considered. If we re-arrange the distribution, new warehousing locations could be
considered, as well as closing some of the existing warehouses. Thus, it is necessary
to introduce a cost function linear on intervals, like the one in figure 1. We assume a
warehouse could be supplied by several production/ distribution centers.
We note:
Qi , i = 1,2,...,m – the present production/ distribution capacity of the center i
Dj , j = 1,2,...,n – the demand of the warehouse j
Cij, i = 1,2,...,m, j = 1,2,...,n, - the cost of the full satisfaction of the demand of
warehouse j by the center i
qij, i = 1,2,...,m, j = 1,2,...,n, - the amount of the demand of warehouse j met by
center i
The objective minimization function will be the sum of the production and
distribution costs of all the centers. In order to have a clear presentation, reference
will be made in what follows only to a single production/ distribution center out of
the m centers, which will be represented as α. It is assumed that it has a capacity of
Qα = 7000 units at present.
n Cj
 qj
The delivery cost for center α = 
j 1 D j
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The production cost for center α, (in case of a production xα ) will be a linear
function on intervals, for instance:
0 for xα = 0
100 + 0.25 xα for   x  5000
150 + 0.15 xα for 5000  x  9000
Remark. The value of 9000 units can be interpreted as a maximum value of the
production of center α, which cannot be exceeded.
According to the before-mentioned statements, two new variables will be
introduced, associated to the production of the center α, which will be represented as
xα1 , xα2 x  x 1  x 2  , and two binary variables represented as yα1 , yα2.
The production cost for the center a can be found in the objective function under the
following form:
0.25  x 1  100  y 1   0.15  x 2  150  y 2 
Correspondingly, the following restrictions are added:
0  x 1  5000  y 1
5000  y 2  x 2  9000  y 2
Conclusion. Defining some economic models with a closer relation to reality can
lead to complex statements, which actually comprise basic structures that can be
applied without any difficulty.
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