# LINEAR PROGRAMMING OF BASIC ECONOMIC PARAMETERS USED AT REENGINEERING IN SMALL

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```					 INTERNATIONAL JOURNAL OF MANAGEMENT (IJM)
International Journal of Management (IJM), ISSN 0976 – 6502(Print), ISSN 0976 –
6510(Online), Volume 4, Issue 2, March- April (2013)
ISSN 0976-6502 (Print)
ISSN 0976-6510 (Online)
Volume 4, Issue 2, March- April (2013), pp. 31-43
IJM
Journal Impact Factor (2013): 6.9071 (Calculated by GISI)
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LINEAR PROGRAMMING OF BASIC ECONOMIC PARAMETERS
USED AT REENGINEERING IN SMALL AND MEDIUM
ENTERPRISES

Prof. Dr Slobodan Stefanović
High School of Applied Professional Studies,
Vranje, Serbia

Faculty for strategic and operational management

ABSTRACT

In economic terms, linear programming is a mathematical technique used for
selecting one among more possible economic decisions that will have the greatest
efficiency. Most production issues have been solved by a linear programming method,
also performed here, and a model of linear programming of economic parameters in re-
engineering of small and medium enterprises, for their greater efficiency, is presented.

Key words: linear programming, re-engineering, economic parameters, model.

1.0. INTRODUCTION

Linear programming is a mathematical method for selecting an optimal solution
among larger number of possible solutions. In mathematical terms, linear programming
is a mathematical analysis of optimum problem. These are mathematical methods used
for seeking the maximum (or minimum) value of a linear function, with previously given
limits expressed by a system of linear equations and inequalities.

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International Journal of Management (IJM), ISSN 0976 – 6502(Print), ISSN 0976 –
6510(Online), Volume 4, Issue 2, March- April (2013)

2.0. A MODEL OF LINEAR PROGRAMMING OF ECONOMIC PARAMETERS OF
RE-ENGINEERING

Before we consider theoretical basis of linear programming through an example of
forming a mathematical model for problems of linear programming of economic parameters
of re-engineering. For using linear programming, we shall present the basic groups of
economic parameters of re-engineering comprising ratios of production, labor in its
organization and capital.

GROUP 1 PARAMETERS – GENERAL HOLDERS OF ECONOMIC EFFECTS OF RE-
ENGINEERING AND ANALYSIS OF NECESSARY ORGANIZATIONAL CHANGES IN
SMALL AND MEDIUM ENTERPRISES AND HYPOTHESES (parameter X1)

GENERAL HOLDERS OF ECONOMIC EFFECTS OF RE-ENGINEERING:

∑ UKN
5

pred .troskovi    = (NEE NO = NEE NP ) + NEE NP + NEE SP + NEE PR + NEE PP + NEE NSR . (1)
i =1

Cartesian product of integration rules of computer integrated manufacture CIM:

∑P + ∑T + ∑I + ∑C
n              n          n          n

R PP   →   INTCIM                       i          j          k          l   ,                    (2)
i =1           i =1       i =1       i =1

HYPOTHESES OF A RE-ENGINEERING MODEL AUXILIARY HYPOTHESES:
PARAMETERS - NEE FS , NEE KP , NEESK , NEE CP .

MAIN HYPOTHESIS:

An initial frame for implementation of procedures of logic system reengineering: PHASE I:
Establishing initial values of logic performances NEE FS IFAZA , PHASE II: Measuring logic
system performances NEE FS IIFAZA .
Economic parameters for providing functioning of entrepreneurship for implementation of re-
engineering NEE KP .
Economic parameters for providing functioning of all structures and quality systems in small
and medium enterprises in re-engineering management:

NEESK = NEEIM + NEE IS + NEE TO + NEE D + NEE DUR + NEE PPP + NEE DI .

Economic parameters for providing competitiveness of a product price on the market
NEE CP created in manufacturing conditions by applying re-engineering:

NEE CP = ∑ NEE CP (i = 1...7) .
7

i
i =1

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GROUP 2 PARAMETERS – FORMING PRICES OF THE PRODUCTS FROM SMALL
AND MEDIUM ENTERPRISES THROUGH THE PRODUCTION FACTOR

1.   Manufacture costs, i.e. value of goods to be expressed in money TR P ,
2.   Value of money material used for calculating the value of goods VNM ,
3.   Size of price measure used as a unit of value measurement CN ,
4.   Supply and demand ratio PT .
GROUP 3 PARAMETERS – NON-ECONOMIC FACTORS VEF
GENERAL FINANCIAL PARAMETERS OF A SMALL OR MEDIUM ENTERPRISE
FUNCTION, CAUSATIVELY INFLUENCING IMPLEMENTATION OF RE-
ENGINEERING

1. Marketing function (R M ) ,
2. Scientific research function (R NR ) ,
3. Management approach in production planning function R Md| ,            (   )
4.   Function of financial planning for developing an enterprise (R P ),
5.   Function of planning and business control (R PKP ) ,
6.   Economic – financial function (R EFF ) ,
7.   Function of connecting incomes with performance (R VZP ) .

3.0. COMPREHENSIVE REVIEW OF BASIC ECONOMIC FACTORS IN
FORMING A RE-ENGINEERING MODEL
For forming a re-engineering model in small and medium enterprises in view of
economic factors, it is required to include all significant analysis comprising:
(1) ANALYSIS OF NECESSARY ORGANIZATIONAL CHANGES IN SMALL AND
MEDIUM ENTERPRISES (ANOP)

This analysis includes Cartesian product of integration rules of computer integrated
manufacture (CIM) – of an enterprise and given set of components:

∑P + ∑T + ∑I + ∑C
n          n          n           n

R PP   →   INTCIM          i          j          k           l   .
i =1       i =1       i =1        i =1

(2) ANALYSIS IN VIEW OF A RESEARCH SUBJECT – HOLDERS OF ECONOMIC
EFFECTS OF RE-ENGINEERING (AAPI)
Analysis in view of research subject refers primarily to a method of implementation
of re-engineering in small and medium enterprises and includes the following costs and
savings comprising the following economic parameters: a parameter NEE NO = NEE NP ,
NEE NO - a parameter of economic effect holder referring to new organizational model and it
exclusively depends on NEE NP - a parameter of economic effect holder referring to nature of
new businesses in new organization of small and medium enterprise by applying re-
engineering; parameters created by implementation of systematic support to new organization
of small or medium enterprise; /parameter NEE SP - a parameter of economic effect holder

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referring to implementation of systematic support; parameter NEE PR - a parameter of
economic effect holder referring to implementation of “pilot solution”; parameter NEE PP - a
parameter of economic effect holder referring to invested time required for implementation of
planned changes and application of plan in implementation of re-engineering in phases in an
enterprise; parameters being general, but referring to training of employees for new process
and new work system with planned changes and application of plan in implementation of re-
engineering in phases; /parameter NEE NSR - a parameter of economic effect holder referring
to costs due to training of employees for new work system. In view of analysis, total costs

∑ UKN
5

Previous .COSTS    resulting from application of a research subject, and defined as economic
i =1
parameter holders in implementation of re-engineering, are expressed by adding all these
listed parameters:

∑ UKN
5

Previous. COSTS   = (NEE NO = NEE NP ) + NEE NP + NEE SP + NEE PR + NEE PP + NEE NSR . (3)
i =1

(3) ANALYSIS OF BASIC HYPOTHESES (AOH)

The analysis of basic hypotheses includes following economic parameters of costs
resulting from implementation of re-engineering in business activities and staff revitalization,
i.e. adjusting organizational structure in terms of ownership restructuring, and defines it as
four economic parameters in defining auxiliary hypotheses, as follows:
1. parameters of financial recovery of small or medium enterprise NEE FS ; this
parameter includes 2 phases: PHASE I: Establishing initial values of logic
performances NEE FS IPHASE , PHASE II: Measuring logic system performances
NEE FS IIPHASE .
2. parameters for providing functioning of entrepreneurship for implementation of re-
engineering NEE KP .
3. parameters for providing functioning of all structures and quality systems in small and
medium enterprises by implementation of re-engineering NEE SK ; NEE SK =
NEE IM + NEE IS + NEE TO + NEE D + NEE DUR + NEE PPP + NEE DI .
4. parameters for providing competitiveness of a product price on the market NEE CP
created     in   manufacturing      conditions      by     applying     re-engineering:

NEE CP = ∑ NEE CPi (i = 1...7) .
7

i =1

Analyzed economic parameters determine justification and prove the main hypothesis
of a re-engineering model, i.e. “By forming a basic model of re-engineering with financial
analysis method – production factors, we can classify all economic parameters actively
influencing in a price of finished product being a result of manufacture in small and
medium enterprises”.

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6510(Online), Volume 4, Issue 2, March- April (2013)

(4) ANALYSIS OF FORMING PRICES OF THE PRODUCTS FROM SMALL AND
MEDIUM ENTERPRISES THROUGH THE PRODUCTION FACTOR (AFCP)

I) Price rate of products from small or medium enterprises
It depends on the following most significant economic parameters causatively influencing
results of re-engineering, as follows:
• manufacture costs, i.e. value of goods to be expressed in money TR P ,
• value of money material used for calculating the value of goods VNM ,
• size of price measure used as a unit of value measurement CN ,
• supply and demand ratio PT
• non-economic factors VEF .

Depending on a method and conditions for forming product prices, we distinguish the
following types of economic factors affecting value of the product price and contributing to
monitoring re-engineering in the field of product price:

1. free market prices – formed by action of supply and demand law onto free competitive
market CN ST ,
2. monopoly prices – formed and determined by monopoly, i.e. one or few manufacturers
that dictate the offer, and accordingly a price itself CN M ,
3. administrative prices – prices determined by the state CN A ,
4. mixed prices – prices formed under influence of market and administrative means CN MŠ .

II) Production function (P) ,

Production function is different from technology, being a technological efficiency, and
defines the maximum of production range, resulting from every possible combination of production
factors.
P = f (x, y, z.....) ,

The simplest is possible, to express production function by Domar’s method:

K
P=     ,                                   (4)
k

Walrase – Leontiev production function includes several factors. According to this function,
production sector product is:
X ij
Pj =       ,                                 (5)
a ij

Douglas production function, expressed in general form as:

P = C Ra          Kb,                             (6)

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Therefore, in general case:
P1 = C     (p    X )x   Yy   Zz ,                       (7)

P1 = p x    C Xx        Yy   Zz ,                       (8)

P1 = p x    P.                              (9)

II.1. Production possibility frontier ( G Pr M
II.2. Law of diminishing returns ZO Pr
II.3. Marginal productivity of production factors
II. 3.1. Total, mean and marginal product (marginal analysis)

Mean product is obtained by dividing total product of a factor and the amount of consumption of
that factor, as in formula:
U prxi
Pprxi =        .                             (10)
xi

Marginal product is increase in total product at unit change in the amount of consumption of a
factor:
U Prxi
Uprxi 1    ∆U Prx
G Prx =            =         .                         (11)
xi xi 1       ∆x

II. 3.2. Total, mean and marginal costs

Total costs UTr are neither homogeneous nor increase at all production levels proportionally
with the increase of the output. Total production costs are divided into fixed UFTr ; variable
UVTr costs, and therefore total production costs may be expressed by the formula:

UTr = UFTr + UVTr .                              (12)

Marginal costs are expressed with the following pattern: GTr = UVTr .

UVTr
If there are no unit changes in the production range: GTr =        , Q – production range.
Q

The enterprise will engage all production factors until there is equalization between marginal
physical product at last money unit spent for each production factor:

CFPx CFPy          CPFz
=     = .... =      .                          (13)
Cx   Cy            Cz

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III) Demand for production factors

Marginal revenue is a change in total enterprise revenues, when production range changes by unit. On
this basis, we have:
UPd
-   VGP (marginal revenue of factors) =         ,
R
UTr
-   GPd (marginal revenue) =     ,
R
UTr
-   GTr (marginal cost) =     .
Q
IV) Production factor offer
V) Production factor market
VI) Production factor procurement and product sale on the perfectly competitive market
VII) Perfect competitiveness in product factor procurement, but monopoly in product
sales
VIII) Bilateral monopoly
IX) Price – production factor incomes (gain, rent, interest and profit)

4.0. AN EXAMPLE OF THE LINEAR PROGRAMMING ECONOMIC EFFECTS OF RE-
ENGINEERING BY USING THE PROPOSED MODEL

Let us suppose that a small or medium enterprise manufactures two products, P1 and P2 . For
manufacturing these products, it is required to understand general economic parameters of re-
engineering with using group parameters M 1 , M 2 and M 3 for their production. For manufacturing
one unit of the product P1 , it takes 2 hours for machine work with using general economic
parameters of re-engineering M 1 , e.g. 1 hour of machine work with using general economic
parameters of re-engineering M 2 and e.g. 2 hours of machine work M 3 with using general
economic parameters of re-engineering. Machine capacities are expressed in available hours of each
machine in observed time period and they amount: 100 hours for machine 1, 120 hours for machine 2
and 120 hours for machine 3. A small or medium enterprise realizes income of 6 dinars per product
unit P1 , and 4 dinars per product unit P2 . All these data are shown in Table 1.
The problem is the following: which quantities of P1 and P2 products should be
manufactured in small and medium enterprises by using economic parameters of re-engineering to
realize maximum income by using economic parameters of an enterprise in re-engineering function at
given limiting conditions (capacities). From the problem set like this, we can see that quantities of P1
and P2 products that should be manufactured according to optimal production program by using
economic parameters of re-engineering are unknown. We shall mark general economic parameters of
re-engineering and parameters of analysis of necessary organizational changes affecting manufacture
of a product P1 , produced in a small or medium enterprise with X 1 , and the amount of P2 product
produced by using the same economic parameters and analysis as in previous case with X 2 , and
form a mathematical model for this problem.
The income that will be realized on P1 product will be 6X 1 , and on P2 product 4 X 2 , for a total
Z0 = 6X1 + 4X 2,
where z 0 is total income, and entire formula will be called a criteria function, or optimality
criterion of economic parameters of reengineering.

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According to the defined problem, we seek for the maximum value of total revenue, so
we have the following criterion function:
(max); z0 = 6 x1 + 4 x2 .
In a similar way, we shall formulate limiting factors, for each group of parameters individually.
It takes 2 hours for the machine 1 to produce one unit of a product P1 with using above
mentioned economic parameters of re-engineering, i.e. 2 x1 hours to produce total amount of this
product. This machine also produces the product P2 , and it takes 1 ⋅ x 2 hours for a machine 1 to
produce x 2 units of this product. Total time for manufacturing whole quantity of products P1 and
P2 amounts 2 x1 + x 2 . Certainly, total used hours for the machine 1 for manufacturing products
P1 and P2 cannot be more than total available amount being 100 hours for the observed time
period. In that way we get an inequality:
2 x1 + x2 ≤ 100.
In the similar way, we obtain the corresponding inequalities through which capacities of
machines 2 and 3 are expressed:
x1 + 3x 2 ≤ 120,
2 x1 +2 x 2 ≤ 120.
In its nature, production range cannot be negative, and the previous conditions should be
complemented by a condition that x1 and x 2 variables cannot be negative, i.e. x1 ≥ 0, x 2 ≥ 0.
In this way, we obtained a mathematical model from certain expressions for previously
formulated problem of determining the amount of products produced in small or medium
enterprises by using economic effects of re-engineering. It consists of a function for optimality
criterion of economic parameters of re-engineering.
z 0 = 6 x1 + 4 x2 ,
whose maximum value should be find with the following limitation system:

2 x1 + x 2 ≤ 100, 
x1 + 3x 2 ≤ 120,  
(1,4)
2 x1 + 2 x 2 ≤ 120,
x1 ≥ 0, x 2 ≥ 0.  

NOTE 1: Lines with arrows indicate connection between economic parameters of re-
engineering, i.e. that they affect all products and their production values and revenues equally.
NOTE 2: For smooth operation of technical systems (machines) M 1 , M 2 and M 3 , all
production conditions with limiting factor systems have been met: a) production issues, b)
transport issues, c) location issues, d) distribution issues, e) establishing plans for foreign trade, f)
various structural problems solved.
Our task is to define such values for variables x1 and x 2 that will satisfy the system of
inequalities and ensure that criterion function reaches its maximum value.
We simplified the problem because there are only two variables, and therefore it can be solved
graphically. This method of solving enables presentation of the problem essence, and before we
start its analytical solving. The graphical method for solving linear problems may be applied
only if a problem has no more than two variables.

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Table 1. Parameters influencing on optimality criterion of economic parameters of
reengineering

Time required for producing a                   Capacity of
unit of a product by using                      amounts in
PARAMETER                                                             economic parameters of re-                       hours by
engineering                                        using
P1 - product   P2 - product                    capacities
(P1 =                  (
f ( M1, M 2 , M P2 = f ( M1 , M 2 , M

M 1 = X 1 = X I 1 + X I 2. -                    GROUPE M1 PARAMETERS:                            GENERAL
HOLDERS OF ECONOMIC EFFECTS OF RE-ENGINEERING AND
ANALYSIS OF NECESSARY ORGANIZATIONAL CHANGES IN SMALL AND
MEDIUM ENTERPRISES AND HYPOTHES

1.         GENERAL HOLDERS                    OF     ECONOMIC            EFFECTS              OF         RE-
ENGINEERING:
5                                            X 11                      X 12                X 13
X I 1 = ∑ UKN pred .troskovi = ( NEE NO = NEE NP )+ NEE NP + NEE SP
i =1
2                      1                  100
X 14            X 15              X 16
+    NEE PR + NEE PP + NEE NSR                        and Cartesian product of integration rules
of                 computer                 integrated                 manufacture                         CIM:
 nX 18             X 1.10   X 1.11

X 17
          n X 19   n        n

RPP    → INTCIM ⋅  ∑ Pi + ∑ T j + ∑ I k + ∑ Cl .
 i =1   i =1     i =1     i =1     
                                   
X 1.12        X 1.13          X 1.14
HYPOTHESES PARAMETERS -                          X I 2 , NEE FS , NEE KP , NEE SK k.

M 2 = X 2 = X II 1 + X II 2.                     GROUP M2 PARAMETERS – FORMING
PRICES OF PRODUCTS PRODUCED IN SMALL AND MEDIUM
ENTERPRISES THROUGH PRODUCTION FACTOR and Production factor
offer, Production factor market, Production factor procurement and product sales
on the perfectly competitive market, Perfectly competitiveness in procurement of
production factors, Monopoly in product sales, Bilateral monopoly, Price –
production factors incomes (gain, rent, interest and profit)
PARAMETRI                     SU:               TR P ,            VNM   ,            CN      ,             PT     ,
U prxi
P = f ( x, y, z.....) , Ppr =                         ,,
xi
xi
U Pr xi
Uprxi −1     ∆U Prx UTr = UFTr + UVTr , VGP (marginal revenue of
G Prx =             =       ,                                                                                                  1                     3                   120
xi − xi −1    ∆x
factors) = Uod/R GPd (marginal revenue) = Utr/R, Marginal cost = Utr/Q.

M 3 = X 3 = X III 1 + X III 2.                       GROUP        M3   PARAMETERS                    –    NON-
ECONOMIC FACTORS VEF

GENERAL FINANCIAL PARAMETERS OF A SMALL OR MEDIUM                                                                              2                      2                   120
ENTERPRISE    FUNCTION,      CAUSATIVELY INFLUENCING
IMPLEMENTATION OF RE-ENGINEERING:
(RM ) , (R NR ) , (RMd | ) , (RP ) , (RPKP ) , (REFF ) .
TOTAL REVENUE PER PRODUCT UNIT                                                                                                 6                      4                    -

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International Journal of Management (IJM), ISSN 0976 – 6502(Print), ISSN 0976 –
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The problem is solved by presenting graphically all inequalities from the system of
limitation, in Descartes coordinate Oxy system.
Since variables x1 and x 2 cannot be negative, we use only the first quadrant for
graphical presentation of the limitation system and present each inequality separately.
Let us present the first inequality related to the machine 1. By using common graphic
presentation, we first draw equation 2 x1 + x 2 = 100. This is the line which intersects with
axis x1 in a point M 1 (50.0), and axis x 2 in M ,1 (0.100). The line that goes through these
two points, forms a triangle with tips O, M 1 , M ,1 with axes x1 and x 2 (in Figure 1). All
triangle points meets the first limitation and meet the requirement of non-negativity of
variables.
In the same way, we can set the other limitation through the appropriate equations.
The second limitation intersects with axis x1 in point M 2 (120.0) and with axis x 2 in
point M , 2 (0.40) thus forming a triangle OM 2 M , 2 . Finally, the third limitation intersects
the first axis in point M 3 (60.0), and the second axis in point M , 3 (0.60) and forms a
triangle OM 3 M , 3 .
The solution to the problem has to meet every limitation individually, but also all
limitations taken together. Therefore, all limitations of the system (1,4) form common
area OM 1CM , 2 , being patterned in Figure 1. Each point of common area meets set
limitations, and therefore are the set of the possible solutions. It is required to determine
the solution from the set of possible solutions, that will ensure that the criterion function
(1,1) will reach its maximum value. It can be easily calculated that the extreme points
M 1 , C and M , 2 of the common area correspond with the following criterion functions:

M 1 : x1 =50,        x 2 = 0,          z 0 = 300,
B : x1 =40,          x 2 = 20,         z 0 = 320,
C : x1 =30,          x 2 = 30,         z 0 = 300,
,
M 2 : x1 =0,         x 2 = 40,         z 0 = 160.

Point B, with coordinates x1 = 40 and x 2 = 20, meets the limitation system (1,4)
and ensures maximum value of criterion function (1.4.), and therefore it is the optimal
solution to the problem.
It should be said that it is not necessary to determine the values of the criterion
function for all extreme points to select the optimal solution. This can be achieved by
graphical presentation of the criterion function z 0 and its parallel movements. Therefore,
the maximum value of the criterion function will be reached when its graph is at the
farthest point from the coordinate start while going through at least one point of the
common area.

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X2

M 1 '(0,100)

M 3 '(0,60)

M 2 '(0,30)                      B (40,20)

X 2 =20

0             40     50    60                          100   120
X1
X 1 =40                                                        X 1 +3X 2 =120

2X 1 +X 2 =100         2X 1 +2X 2 =120

Figure 1. The area of optimal economic parameters of reengineering - the area of
O50BM , 2 points

In the end, we can conclude that optimal solution, x1 = 40 and x 2 = 20 shows that a small or
medium enterprise should produce 40 units of the product P 1 and 20 units of the product P2
to realize the maximum possible revenue under given conditions of 320 dinars with using all
economic parameters of re-engineering. Every other production program will yield less
income. This method can be successfully used as a basic method in programming the
production in small and medium enterprises by using all economic parameters of re-
engineering.

5.0. CONCLUSION

Optimality criterion function of economic parameters of re-engineering my be
analytically determined by using common area with common expression:
A: x1 = 5 ⋅ n, x 2 = 0 ⋅ n, z 0 = 6 ⋅ n ;
B: x1 = 4 ⋅ n, x 2 = 2 ⋅ n, z 0 = 6,4 ⋅ n ;
C: x1 = 3 ⋅ n, x 2 = 3 ⋅ n, z 0 = 6 ⋅ n ;
D: x1 = 0 ⋅ n, x 2 = 4 ⋅ n, z 0 = 3,2 ⋅ n ;
This implies that by approximation, the mean value of the optimality criterion function for
the presented case is:
6 + 6,4 + 6 + 3,2
x1 = (2 z + 1) ⋅ n, x 2 = (2 z − 1) ⋅ n, z 0 =                    ⋅ n = 5,4 ⋅ n ; which applies for
4
z = 0,1,2,3.
In Descartes coordinate system, this curve may be presented based on values entered in Table
2 (Figure 2):

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Table 2. Shadow Box field representing optimal area of economic parameters of
reengineering

Point                       M1              B               C           M ,2

x1                       50              40              30              0

x2                       20              20              30              40

z0                       300             320             300         160

Based on this, we can conclude that general expressions of optimality criterion are:

x1 = (2 z + 1) ⋅ n, z = 0,1,2,3.                                    (14)

x 2 = (2 z − 1) ⋅ n, z = 0,1,2,3.                                   (15)

z0 = i ⋅ n ,                                                 (16)

where i depends on optimality criterion of economic parameters of re-engineering.

X2
Tacka M1   B    C         M2'
X1   50   40 30          0                               40

X2   20   20   30        40
30
Z0   300 320 300 160
M2'           20

30   40       50
C                                        X1

160               M1
B
300
320

Z0

Figure 2. Optimality criterion function of linear programming of economic parameters
of programming – spatial curve

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ISSN 1820-7294 UDC 005th
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International Journal of Management Research and Development (IJMRD), Volume
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