Production Planning Through Multi-Objective De Novo Programming
with Variable Prices
ZORAN BABIĆ1, TUNJO PERIĆ2, BRANKA MARASOVIĆ1
1Faculty of Economics Business and Tourism,
University of Split,
Cvite Fiskovića 5, 21000 Split,
CROATIA
2Faculty of Economics and Business,
University of Zagreb,
Trg J. F. Kennedy 6, 10000 Zagreb,
CROATIA
Abstract: - This paper presents a novel multi-objective De Novo programming approach developed to address
production planning problems. The proposed model incorporates increasing costs or quantity discounts for
certain raw materials which were not considered in previous approaches presented in scientific literature. The
efficiency of the proposed methodology was tested using a bakery production planning example. The multi-
objective De Novo programming model was solved using various multi-objective programming approaches: the
original De Novo programming approach, several goal programming approaches, and the global criterion
method. The results indicate the successful application of the proposed methodology in solving production
planning problems, with no significant difference in the efficiency of the applied multi-objective programming
methods.
Key-Words: - De Novo programming, multi-objective, variable prices, bakery
Received: March 11, 2023. Revised: June 29, 2023. Accepted: July 10, 2023. Published: July 21, 2023.
1 Introduction
About thirty-seven years ago, [1], introduced a new
concept into mathematical programming, a special
approach to optimization, and called it De Novo
programming. In standard mathematical
programming problems resources are set in advance
and the work to be done is to "optimize a given
system". However, the De Novo programming
approach suggests a way of "designing an optimal
system". In De Novo, resource quantities are not
given, since they are available if we have enough
money. The maximum quantities of resources are
limited by the available budget, which is an
important new element of De Novo.
De Novo is generally more effective in solving
problems than the standard programming model.
For example, multi-objective problems, [2], and
price changing, i.e. increasing costs of raw
materials, or quantity discounts, [3], [4], are the
production situations that can be processed very
successfully with the De Novo methodology
providing satisfactory solutions.
Since this new approach was initiated, [1], De
Novo programming has been developing rather
slowly. Many articles and promising ideas related to
De Novo come from authors from the Far East: [5],
[6], [7], [8], [9], [10], [11]. However, the author of
this approach has not abandoned the De Novo idea
and has continued to engage with it in many of his
later works, [12], [13], [14]. He also included it (as a
single or multi-objective approach) among the eight
concepts of optimization, [15], where classic
optimization is only a special case. [16], introduced
a meta-goal programming approach for solving the
multi-objective De Novo programming problem. In
[17], [18], authors tried to introduce some new
constraints in De Novo multi-objective problems
and presented a new way of solving the problem
using an extension of the STEM method. In two
recent papers, the author considers multi-objective
De Novo linear programming problems, [19], [20],
while in a third paper, [21], a project portfolio using
the hybrid approach of data envelopment analysis
and De Novo optimization was considered. Multi-
objective De Novo programming problems were
considered in recent times by [22], [23], while, [24],
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in 2021 presented the exhaustive literature review of
De Novo programming.
This paper aims to present a multi-objective De
Novo programming model that takes into account
variable prices, which were not considered in
previous approaches presented in scientific
literature. While the transformation to the
continuous knapsack problem is not possible in such
cases, this paper proposes a solution with
appropriate simplifications. Previous research, such
as that by [3], [4], has focused on the single-
objective De Novo programming model with
variable prices, but the current study expands on this
by considering a multi-objective approach.
After the introduction part of the paper, Section
2 gives a brief overview of the multi-objective De
Novo programming problem, and in Section 3
situations with variable prices are presented. In
Section 4 a multi-objective model in a bakery will
be formulated while in Section 5 this model is
solved using the founder Milan Zeleny’s, original
approach. Then, some different approaches
involving goal programming and the global criterion
method for solving multi-objective problems are
presented. Section 6 presents the discussion of the
results, and finally, in Section 7, there is the
conclusion.
2 Multiple objective De Novo
programming
The multiple-objective De Novo programming
model, [14], has the following form:
Max Z=CX
s.t. AX – b = 0 (1)
T
p b B
,0Xb
C is a
( , )qn
matrix comprising the coefficients of q
objective functions, and A is a
( , )mn
matrix of
technological coefficients, defining the usages of
resource i upon producing the product type j. Vector
b is the m-dimensional vector of unknown resource
variables, X is the n-dimensional vector of decision
variables, p is the m-dimensional vector of the unit
prices of m resources, and B is the given total
available budget. The solution to the problem (1) is
to find the optimal allocation of budget B and the
distribution of raw materials (resources) with which
we can maximize the values Z = CX of the product
mix.
The main difference between the usual linear
programming model and the De Novo formulation
lies in the treatment of the resources. In the De
Novo programming model resources are not given
in advance but they become decision variables bi.
2.1 Zeleny’s Approach
From (1) follows
TT
p AX p b B
and, defining the n row vector of unit costs
,
T
V p A
the problem (1) can be transformed into:
Max Z = CX
s.t. VX ≤ B, X ≥ 0 (2)
where
1,, Tq
q
Z z z R
and
1,, Tn
n
V V V p A R
.
Most authors solve multi-objective De Novo
models according to suggestions from [1]. Namely,
they construct an auxiliary model (the meta-
optimum problem) involving the minimum budget
quantity to achieve the ideal values of all of the
objective functions. After that, the optimum-path
ratio, which is the ratio of the given budget B and
the minimum budget obtained by this auxiliary
model (3), is calculated. This ratio is then used to
obtain the final solution for the model with the
previously given budget B.
Let
* max , 1, ,
kk
z z k q
, be the optimal
value for the k-th objective of the problem (1)
subject to VX ≤ B, X ≥ 0, and let
1
* *, , * T
q
Z z z
be the q-objective value for
the ideal system with respect to B. Then, the meta-
optimum problem can be constructed as follows:
Min
VX
,
s.t.
*, 0CX Z X
(3)
After solving this problem optimal solution X* is
obtained and consequently
**B VX
,
* *.b AX
The value B* represents the minimum budget to
achieve Z* through x* and b*. Since
*,BB
the
optimum-path ratio for achieving ideal performance
Z* for a given budget level B can be defined as
*
B
rB
(4) According to this ratio, the actual final solution
(or optimal design) can be obtained by the following
calculation:
*, *X r X b r b
and
*Z r Z
. (5)
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3 The Varying Cost of Raw Materials
In a situation of varying the prices of the same
resource, model (1) can be transformed in such a
way that it includes resources with variable prices.
Matrix A of technological coefficients is the
(m, n) matrix. Let the matrix AC of type
(m1, n) be the block matrix from matrix A which
contains the technological coefficients of resources
that have constant prices, and let block matrix AV,
type (m2, n), be the matrix with coefficients of
resources with variable prices. Of course,
12
m m m
. Without loss of generality, it can be
supposed that matrix AC is located in the first m1
rows from matrix A, or the matrix form:
C
V
A
AA
.
Besides vector b of resource variables, a new
vector d can be included. It presents the quantity of
resources if they cross the border values when the
prices of resources become higher or lesser than the
original prices. Both of these vectors are divided
into two parts, one for resources with constant
prices and the other for variable prices, i.e. in matrix
form:
C
V
b
bb
,
0
V
dd
.
Resource quantity for the resources with
variable prices is also divided into two parts: first is
the quantity that is purchased with the original price
()
V
b
and second is the additional quantity
()
V
d
that
is purchased with higher or lesser price. Therefore,
the total quantity of resources with variable prices
will be
()
VV
bd
.
The price vector will be divided similarly. Let
pC be the first part of the price vector for resources
that have constant prices, and pV is the second part
for resources that have variable prices. Besides that,
the additional vector pV’ whose coefficients are the
prices for an additional quantity of resources exists,
or in matrix form:
C
V
p
pp
,
0
''V
pp
.
In these vectors, bC and pC are type
1
( ,1)m
and
,,
V V V
b d p
,
'V
p
are type
2
( ,1)m
where, of course,
12
m m m
.
The model (1) with an additional quantity of
resources for the resources with variable prices can
now be transformed in:
Max
Z CX
s.t.
0AX b d
(6)
'
TT
p b p d B
, , 0X b d
or
Max
Z CX
00
0
CC
V V V
Ab
X
A b d
00
'
TT
CC
V V V V
pb B
p b p d
, , , 0
C V V
X b b d
or
Max
Z CX
0
CC
A X b
(7)
0
V V V
A X b d
'
T T T
C C V V V V
p b p b p d B
, , , 0
C V V
X b b d
.
For the resources with constant prices from (7),
there follows:
TT
C C C C
p b p A X
.
In that way, the budget equation in (7) can be
written as:
'
T T T
C C V V V V
p A X p b p d B
, or
'
TT
C V V V V
V X p b p d B
(8)
where
T
C C C
V p A
is the row vector of the unit
costs of resources with constant prices.
In the end, the multi-objective model with
variable prices in the simplified form can be
presented as:
Max
Z CX
0
V V V
A X b d
(9)
'
TT
C V V V V
V X p b p d B
, , 0
VV
X b d
In this paper two cases of variable prices will
be considered: the increasing costs of raw materials
and the quantity discounts offered for volume
purchases.
The increasing costs of raw materials
Let us assume that k raw material can be
purchased at the price pk but only for a quantity
lower (or equal) than Q. The price of k raw material
above that quantity, Q is pk' > pk. The relation for the
k raw material can now be transformed in:
1 1 2 2k k kn n k k
a x a x a x b d
(10)
with the additional constraint bk Qk , where dk is
the additional quantity of the k raw material with the
unit price pk'.
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There is no need to specify that bk should reach
the maximum value of Qk first, before allowing dk
greater than zero. The optimization model ensures bk
reaches the maximum value of Qk because of the
lower penalty, i.e. lower price pk.
Quantity discounts offered for volume
purchases
Quantity discounts offered for volume purchases
may be formulated in a somewhat different way. Let
us assume that for the k resource (bk) the valid price
is pk as long as the purchased quantity is below Qk,
and the discounted price pk' is valid for the entire
quantity if the purchased quantity is higher than (or
equal to) Qk.
Let
bk, pk – the amount and price of k raw material
if it is purchased at less than the quantity discount
volume;
dk, pk' – the amount and price of k raw material
if it is purchased at the quantity discount.
The new model for k raw material, instead of one
equation (10), has relations:
*10
k k k
b Q y
(11)
20
k k k
d Q y
(12)
20
kk
d My
(13)
where M is a very large positive number, or the
upper limit for the procurement of the resource k,
and Qk* is a number that is slightly lower than Qk.
The variables yk1 and yk2 are binary variables (0 or
1), for which the following applies:
12
1
kk
yy
(14)
The problem of the mutual exclusivity of
variables bk and dk can be introduced as follows:
If dk = 0 (there is no quantity discount) then the
relation bk < Qk has to be true (i.e. the necessary
amount of resources is below the one needed for the
discount). Similarly, if bk = 0 then the relation dk ≥
Qk has to be true (we arrived at the quantity of
resources needed for a discount).
Then, if yk1 = 1 from relation (14), it follows
that yk2 = 0. The equations (11), (12), and (13) then
become
*
kk
bQ
,
0, 0.
kk
dd
The last two relations then assure that dk = 0.
Moreover, since Qk* is slightly lower than Qk, bk is
strictly lower than the limit Qk.
In line with the same equation, if yk2 = 1 then
yk1 = 0. Equations (11), (12), and (13) become
bk ≤ 0 , dk ≥ Qk , dk ≤ M.
Constraint bk ≤ 0 and the non-negativity constraint
on variable bk ensure that bk = 0, and dk is greater
than (or equal to) the discount limit Qk and smaller
than the big positive number M (or another upper
limit defined in advance).
This way of introducing quantity discounts is
useful especially if quantity discounts appear in
several stages, i.e. if there are several classes in
which a supplier approves different quantity
discounts, [25].
Taking into consideration costs for additional
quantities of raw material a new budget equation
can be defined:
'
TT
p b p d B
1
'
m
i i k k
i k K
p b p d B
(15)
Set K contains the indices of raw materials with
increasing or discounted prices, i.e.
11, ,K m m
. (16)
In accordance with the relation from the model (7) it
follows:
'
T T T
C C V V V V
p b p b p d B
or
1
1
'
m
i i k k k k
i k K k K
p b p b p d B
(17)
For resources with constant prices follows:
TT
C C C C C
p b p A X V X
, and so the budget
constraint becomes as in relation (8):
'
TT
C V V V V
V X p b p d B
or
1
'
n
Cj j k k k k
j k K k K
v x p b p d B
(18)
where
Cj
v
are the unit costs of the resources with
constant prices used for producing product j.
Since the same raw material has a different price
variable, the income from a product unit is no longer
constant. Therefore, if among the objective
functions, there exists a net income equation,
maximizing the sum of cj xj, (where cj is the unit
profit for articles) would not be an accurate measure
of net income. For that reason, the net income
equation should be recalculated as the difference
between sales and the total cost of materials, where
the objective function will include materials at both
prices. If S is the vector of the selling prices and sj is
the selling price of j product, the net income
objective function is defined as follows:
( ' )
T T T
Max z S X p b p d
, or
11 '
nm
j j i i k k
j i k K
Maxz s x p b p d
(19) In that equation, dk (k K) remains for those
materials which in additional quantities can be
bought only at a higher price, or the quantities of
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raw materials if we bought them with quantity
discounts.
Similarly, as in the budget equation, the second
part of these equations follows:
''
T T T T T
C C V V V V
p b p d p b p b p d
,
and, because
TT
C C C C C
p b p A X V X
, from
relation (8) follows:
''
T T T T
C V V V V
p b p d V X p b p d
.
In that way, the net income objective function
becomes:
'
T T T
C V V V V
Max z S X V X p b p d
or
1
' (20)
n
j Cj j k k k k
j k K k K
Max z s v x p b p d
4 Case Study
A new multi-objective De Novo programming
problem with the variable prices of raw materials
will be explained using the example of production
planning in a bakery, which produces twenty
different products, [3]. Table 1 and Table 2 present
the list of articles (Table 1) and the list of raw
materials (Table 2) with some new data for
formulating a new objective function. The tables are
provided in the Appendix of the paper.
Table 1 presents the selling prices for each
product, the weights of the articles, the amount of
flour in each item, and the lower and upper bounds
for monthly production. In Table 2, 27 different raw
materials are used in the production of these articles.
Table 2 also presents the purchasing prices for every
one of them and for the last four raw materials for
which we have variable prices. The technological
coefficients, i.e. the amounts of raw materials in one
unit of articles (𝑎𝑖𝑗) are presented in Table 3.
The raw materials R26 and R27 (Wheat flour T-
850 and Wheat flour T-550) can be purchased at a
discounted price if the purchased quantity is greater
than 14200 kg (Q26 = 14200 kg or 14.2 t) for the
first and greater than 60000 kg (Q27 = 60000 kg or
60.0 t) for the second raw material. This reduced
price is valid for the entire quantity supplied, i.e.
26 ' 2.3004p
and
27 ' 2.244p
.
In addition to this, let us consider the increasing
costs for yeast (R24) and corn concentrate Aurelia
(R25) as follows: The limit of yeast purchased at a
lower price is Q24 = 2000 kg (2.0 t), while this limit
for corn concentrate is Q25 = 1600 kg (1.6 t). The
purchase price of the additional quantity of yeast is
p24' = 7.7616, and of corn concentrate p25' = 14.6364
(12% higher) monetary units.
Suppose that the budget which is available for
purchasing these raw materials is 300 000 m.u.
In accordance with Table 1, Table 2, and Table
3 a multiple objective linear programming problem
with two objective functions (net income and total
production measured by flour consumption) can be
formulated. This problem has twenty-seven raw
material constraints and one budget constraint. In
addition to this, with the intention to introduce
variable prices into the model, it appears some
additional constraints for the raw materials that have
quantity discounts - relations (11), (12), and (13).
This model has 20 integer decision variables for the
quantities of types of bread products xj and 27
continuous resources variables bi (among them b13 is
also an integer – the number of eggs). Besides that,
introducing variable prices requires some additional
variables for resources that have variable prices, i.e.
di is the additional quantity of the i-th raw material.
In the end, there are some binary variables yki, which
provide the mutual exclusivity of the amount of
resources that would be purchased at lower or upper
prices. In addition to that, there exist 40 lower and
upper bounds constraints for the 20 types of bread
products and two constraints for resources with
increasing costs (b24 and b25).
If xj is the production quantity of j bakery
product, then the first objective function, which
takes increasing costs and quantity discounts into
consideration, has the following form, as in equation
(19):
20 27
111 '
j j i i k k
j i k K
Maxz s x p b p d
,
24,25,26,27K
The second objective function is the total bakery
production measured by flour consumption. The
coefficients for this objective function are given in
Table 1 (
2j
c
), and this function is:
20
22
1jj
j
Max z c x
The budget and raw material constraints are as
follows:
27
1
' , 24, ,27
i i k k
i k K
p b p d B K
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20
10, 1, ,27
ij j i i
ja x b d i
The constraints for the discounted prices for the
last two raw materials (k = 26, 27) in accordance
with the relations (11) – (14) (bk and dk in kg) are:
26 26,1
14199 0by
,
26 26,2
14200 0dy
,
26 26,2 0d M y
, and
27 27,1
59999 0by
,
27 27,2
60000 0dy
,
27 27,2 0d M y
.
For binary variables yk1, yk2 holds
26,1 26,2 1yy
and
27,1 27,2 1yy
.
In addition, there are two constraints (in kg) for
resources that have increasing costs:
b24 2000 and, b25 1600.
Of course, due to the nature of the data from Table
1, all articles have lower and upper bounds (40
additional constraints). If the lower and upper
bounds are Lj and Uj, the complete De Novo
programming multi-objective model for bakery
production has the following form:
Model M1
20 27
111 '
j j i i k k
j i k K
Maxz s x p b p d
,
24,25,26,27K
(21)
20
22
1jj
j
Max z c x
(22)
subject to
27
1
'
i i k k
i k K
p b p d B
(23)
20
10, 1, ,27
ij j i i
ja x b d i
(24)
*10, 26,27
k k k
b Q y k
(25)
20, 26,27
k k k
d Q y k
(26)
20, 26,27
kk
d My k
(27)
12
1, 26,27
kk
y y k
(28)
, 24,25
kk
b Q k
(29)
, 1, ,20
j j j
L x U j
(30)
where variables xj, j = 1,…, 20, and b13, are
integers, while variables
12
, ; 26,27
kk
y y k
are
binary variables, and all decision variables are
nonnegative.
Now some simplifications can be introduced.
Namely, 27 constraints for raw materials can be
reduced by the inclusion of the majority of these
constraints in the budget constraint. Model (7) is as
follows:
0
CC
A X b
, or
CC
b A X
for resources with constant prices. This relation can
be included in the budget constraint and thus the
budget constraint from model (7) becomes
'
TT
C V V V V
V X p b p d B
,
as in model (9), where
T
C C C
V p A
.
Vector VC can be calculated from initial data.
Namely vector
1 23
,, T
C
p p p
, i.e. the vector for
the resources with constant prices, is presented in
the first 23 rows the of price column in Table 2.
Matrix AC is formed from the first 23 rows from
Table 3 and components of vector VC are calculated
and presented in Table 4. All input data are
presented in Appendix 1. Finally, budget constraint
is:
1
'
n
Cj j k k k k
j k K k K
v x p b p d B
as in relation (18).
Only four raw material constraints that have
increasing costs (b24 and b25) or discounted prices
(b26 and b27) will remain. In such a way the model
will have 23 constraints and 23 decision variables
less than the initial model, but the solution obtained
will remain the same. Of course, the values of the bi
variable that are substituted have to be obtained by
the inclusion of xj variables (j = 1,…,20) in the
initial relations (24) from the previous complete
model M1.
Because the first objective function, in addition
to twenty xj variables, also contains raw material
variables, the same relation from model (7) in the
first objective function can be included. Then the
new form of the first objective function as presented
in relation (20) will be obtained:
1
'
n
j Cj j k k k k
j k K k K
Max z s v x p b p d
In the first objective function of the reduced model
there now remains only eight raw material variables
bk and dk, where
24,25,26,27kK
, as well as
twenty xj variables.
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Following the substitution of these 23 raw
material constraints in the budget constraint and in
the first objective function a reduced and much
simpler model is obtained.
The reduced model is as follows (the binary
variables yki are now marked as y1, y2, y3, and y4):
Model M2
Max z1 = Max (9.452482x1 + 10.629x2 +
+ 9.524034x3 + 7.412533x4 + 7.807031x5 +
+ 9.989509x6 + 9.476482x7 + 10.33086x8 +
+ 8.817266x9 + 8.819602x10 + 4.528776x11 +
+ 5.541641x12 + 5.501423x13 + 4.451883x14 +
+ 4.004302x15 + 4.800361x16 + 5.484859x17 +
+ 4.62133x18 +4.824314x19 + 5.540952x20 –
– 6.93b24 –7.7616d24 – 13.068b25 –14.6364d25 –
– 2.706b26 – 2.3004d26 – 2.64b27 – 2.244 d27)
Max z2 = Max (0.41248x1 + 0.29988x2 +
+ 0.31001x3 + 0.4857x4 + 0.486x5 + 0.48592x6+
+ 0.41248x7 + 0.35115x8 + 0.2145x9 + 0.2413+
+ x10 + 0.086x11 + 0.0625x12 + 0.09354x13 +
+ 0.06561x14 + 0.04335x15 + 0.06573x16 +
+ 0.06327x17 + 0.03667x18 + 0.04592x19 +
+ 0.053 x20)
s.t.
Budget constraint:
(B) 0.723518x1 + 0.231x2 + 1.791966x3 +
+ 0.471467x4 + 0.076969x5 + 0.066491x6 +
+ 0.723518x7 + 0.52914x8 + 2.498734x9 +
+ 1.356398x10 + 0.103224x11 + 0.110359x12 +
+ 0.978577x13 + 0.060117x14 + 0.051698x15 +
+ 0.059639x16 + 0.419141x17 + 0.0.41867x18 +
+ 0.371686x19 + 0.939048x20 + 6.93b24 +
+ 7.7616d24 + 13.068b25 + 14.6364d25 +
+ 2.706b26 + 2.3004d26 + 2.64b27 + 2.244d27 ≤
≤ 300000
Constraints for resources with variable prices:
(R24) 0.008x1 + 0.008x2 + 0.0076x3 + 0.0119x4+
+ 0.01047x5 + 0.01191x6 + 0.008x7 +
+ 0.00896x8 + 0.009x9 + 0.0086x10 +
+ 0.00252x11 + 0.00187x12 + 0.002 x14 +
+ 0.00133x15 + 0.00199x16 + 0.00186x17 +
+0.0022x18 +0.0027x19 + 0.003x20 b24 d24 = 0
(R25) 0.12852x2 + 0.013x20 - b25 - d25 = 0
(R26) 0.184x1 + 0.1551x3 + 0.3399x4 + 0.316x5 +
+ 0.184x7 + 0.30165x8 + 0.125x9 b26 d26 = 0
(R27) 0.102x1 + 0.29988x2 + 0.155x3 + 0.170x5+
+ 0.48592x6 + 0.102x7 + 0.241.3x10 + 0.086x11 +
+ 0.0625x12 + 0.093.54x13 + 0.06561x14 +
+ 0.04335x15 + 0.06573x16 + 0.06327x17 +
+ 0.036.67x18 +0.04592 x19 + 0.053x20 b27 – d27 = 0
Constraints for the resources with quantity
discounts:
b26 14199 y1 ≤ 0
d26 14200 y2 ≥ 0
d26 M y2 ≤ 0
y1 + y2 = 1
b27 59999 y3 ≤ 0
d27 60000 y4 ≥ 0
d27 M y4 ≤ 0
y3 + y4 = 1
Lower and upper bounds:
, 1, ,20
j j j
L x U j
b24 2000, b25 1600
where variables
, 1, ,20,
j
xj
and b13, are
integers, while variables y1, y2, y3, y4 are binary,
and M is a great positive number, and all decision
variables are nonnegative.
5 Solving the Model
5.1 Zeleny’s Approach
The first step in solving this model is to obtain the
optimal solutions for each objective function
separately. These optimal solutions are obtained
with MATLAB and are presented in Table 5 and
Table 6 (Appendix 2). Of course, the available
budget is 300000 m.u. for both solutions.
The optimal values of the objective functions are:
z1* = 2143888.1 m.u.; z2* = 98457.5 (kg),
and in both solutions the available budget is
completely spent.
In these tables, the required quantities of
resources (in kg) are presented. In both solutions,
we have quantity discounts for raw materials 26 and
27 since we purchase these two types of flour over
limited quantities (Q26 = 14200 kg and Q27 = 60000
kg). The twenty-fourth raw material has to be
purchased over the limited quantity and so the
quantity over the limit (d24) is purchased at a higher
price. For R25 in both solutions, the limit is not
exceeded (Q25 = 1600 kg) and the whole quantity
(b25) will be purchased at a lower price.
Of course the final values of the resource
variables that are not presented in Model 2
(b1 to b23) have to be obtained by the inclusion of xj
variables (j = 1,…,20) in the initial relations (24)
from the previous complete model M1.
After that, the meta-optimal problem (3) is
formulated. It is shown in the relations below:
Min
B VX
s.t.
*CX Z
0X
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where budget B is now the function of twenty xj
variables and eight raw material variables, as can be
seen from relation (18) or the budget constraint in
the simplified model M2.
Metaoptimal model – M3
Min B = 0.723518x1 + 0.231x2 + 1.791966x3 +
+ 0.471467x4 + 0.076969x5 + 0.066491x6 +
+ 0.723518x7 + 0.52914x8 + 2.498734x9 +
+ 1.356398x10 + 0.103224x11 + 0.110359x12 +
+ 0.978577x13 + 0.060117x14 + 0.051698x15 +
+ 0.059639x16 + 0.419141x17 + 0.0.41867x18 +
+ 0.371686x19 + 0.939048x20 + 6.93b24 +
+ 7.7616d24 + 13.068b25 + 14.6364d25 +
+ 2.706b26 + 2.3004d26 + 2.64b27 + 2.244d27
s. t.
11
* 2143888.1zz
22
* 98457.5zz
,
and in addition to all the remaining constraints that
are presented in Model M2 (constraints for raw
materials with increasing costs and quantity
discounts, and all lower and upper bounds
constraints).
The optimal solution to this meta-optimal
problem is presented in Table 7. The final values of
the resource variables that are not presented in this
model (resource variables with constant prices b1 –
b23) have to be obtained by the inclusion of xj
variables (j = 1,…,20) in the relations (24) from
model M1.
The optimum-path ratio is
* 300000 308 0.97378257 307r B B
and the optimal design has to be obtained with the
calculation
*X r X
. Since the solutions for
product quantity (xj) must be integers, the obtained
values for these variables are first rounded to
integers. The quantities of raw materials (bi) are
obtained so that integers xj are included in the initial
raw material relations (24) from model M1.
According to that, the optimum-path ratio
transformation
( *)b r b
was not used for
obtaining the raw materials quantities. The problem
is that the solution obtained in that way is not a
feasible solution for the original problem. Namely,
some product quantities have values lower than the
lower bounds from Table 1 (x1, x2, x3, x9, x10, x13, x18,
and x20). This final design is presented in Table 8.
Therefore, that way of solving this problem is
not appropriate for problems that have integer
variables and lower or upper bounds for the decision
variables. For that reason, this multi-objective
problem will be solved with some other approaches
below.
Multi-objective De Novo programming with the
meta-optimal solution is not the only available
approach. Several multi-objective decision-making
techniques can be used for obtaining the best
compromise solution. Below some goal
programming approaches will be presented and, in
the end, an approach that uses the global criterion
method has been shown.
5.2 Goal Programming Approaches
Goal programming is an extension of linear
programming models and includes the achievement
of target values (goals) for each objective, instead of
the maximization or minimization of the objective
functions.
The structure of the i-th goal is as follows, [26]:
()
i i i i
f X n p t
(31)
where:
()
i
fX
- mathematical expression for the i-th
attribute (X is the vector of decision variables).
i
t
- target value for the i-th attribute, i.e. the
achievement value that the DM considers as
satisfying for the i-th attribute.
i
n
- negative deviation variable, i.e. quantification
of the under-achievement of the i-th goal.
i
p
- positive deviation variable, i.e. quantification of
the over-achievement of the i-th goal.
The first goal programming variant is known as
weighted goal programming (WGP). In the WGP
model deviations from the target values are assigned
weights (ui and vi) according to their relative
importance to the decision maker and minimized as
an Archimedean sum. The formulation of this
variant is as follows:
1
1
min ( )
q
i i i i
ii
z u n v p
k
(32)
s.t.
()
i i i i
f X n p t
,
1, ,iq
,
XS
where S is the set of model constraints.
The function z of weighted deviational
variables is known as the achievement function.
Each deviational variable in the achievement
function is divided by a normalization constant ki.
This allows us to overcome the problem of
incommensurability. Namely, deviations from
objectives measured in different units must not be
summarized together. In this paper, a normalization
technique based on the difference between the
positive and negative ideal solution is presented
[27]. For this task a payoff table is presented and
from which it is easy to identify ideal and anti-ideal
solutions or optimistic and pessimistic objective
values. In the bakery problem, the payoff table is
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presented in Table 9. From that table, the ideal and
anti-ideal solutions are as follows:
2143888.10,98457*( .49)Z
- Ideal
.(1895180.6 10,92119.51)Z
- Anti-ideal
The normalized constant is now calculated as
follows:
*
i i i
k z z
, and in the bakery problem it
is:
12143888.1 1895180.6 248707.5k
292119.9845 51 6337.99 87.4k
Finally, the weighted goal-programming model
for solving multiple objective De Novo
programming problems can be formulated. In our
problem, there are two maximization objectives and
for their target levels the ideal levels zi* has been
taken. If positive ideal values are used for
maximization-type objectives, then the positive
deviation (pi) should be zero. This is because the
objective values cannot be greater than the ideal
values. In addition to that, let the weights ui be equal
for both objectives and
UI = 10000. Consequently, the achievement
function in the WGP model has the following form:
12
10000 10000
min 248707.5 6337.98
z n n
12
min 0.040208 1.57779z n n
s. t.
1 1 1
( ) * 2143888.1z X n z
2 2 2
( ) * 98457.49z X n z
and with all other constraints from model M2
(budget constraints, constraints for raw materials
with increasing costs and quantity discounts, and all
lower and upper bounds constraints).
The optimal solution of that model is presented
in Table 10 and the values of the objective function
are presented in the last row of the table. They are
obtained with the following relations:
1 1 1
( ) * 2066814.* 9z WGP z n
m.u.
2 2 2
( ) * 97426.54*z WGP z n
kg.
Another goal programming variant, which was
applied in the De Novo multi-objective model, is the
so-called Min-max goal programming, [28], [29]. In
Min-max goal programming, the maximum
deviation from amongst the weighted set of
deviations is minimised rather than the sum of the
deviations themselves. The mathematical expression
of that variant is as follows:
min zD
s.t.
1( ) , 1, ,
i i i i
iu n v p D i q
k
(33)
()
i i i i
f X n p t
,
1, ,iq
,
XS
.
For the normalization constant ki the same relation
as in WGP has been taken. For the same reason, as
in WGP (if we take
*
ii
tz
) pi = 0. In addition to
that, let the weights ui be equal for both objectives
and
ui = 10000 like before in the WGP model.
Consequently, our Min-max GP model is:
min zD
s.t.
1
10000
248707.5nD
1
0.040208 nD
2
10000
5337.98nD
2
1.57779 nD
1 1 1
( ) * 2143888.1z X n z
2 2 2
( ) * 98457.49z X n z
with all other constraints from model M2.
The optimal result of that model is presented in
Table 11. The objective functions values are
presented in the last row of the table obtained with
the following relations:
1 1 1
( max) * 2081877.21*z Min z n
m.u.
2 2 2
( max) * 96877.21*z Min z n
kg.
5.3 The Global Criterion Method
Lastly, a multi-objective De Novo model can be
solved using the Global Criterion method, [30],
[31].
This method develops a global objective
function made up of the sum of the deviations of the
values of the individual objective functions from
their respective ideal values as a ratio to that of the
ideal value, [32]. It can be said that the Global
criterion model minimizes the distance to the ideal
solution by using Minkowski’s Lp metric. The
mathematical formulation is as follows (the
assumption is that all the objective functions have to
be maximized).
Min
1
( *) ( )
( *)
p
qii
ii
z X z X
FzX
(34)
where
( *)
i
zX
is the value of the objective function
i at its individual optimum
*X
,
()
i
zX
is the
function itself, and
(1 )pp
is the integer
value exponent that serves to reflect the importance
of objectives. Of course, all constraints must be
included in the model. Below the case p = 1 is
presented, which implies that equal importance is
given to all deviations.
The objective function in the bakery model is:
Min
2
1
()
2( *)
i
ii
zX
FzX
=
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= 2 Max
2
1
()
( *)
i
ii
zX
zX
where
11
( *) * 2143888.1z X z
and
22
( *) * 98457.49z X z
.
Because the objective function coefficients in
that function are too small, they are multiplied by
106 and solved only the maximum problem. The
optimal solution remains the same but, in the end,
the optimal value of the objective function has to be
divided by 106 and subtracted from 2. In the end
objective function for the global criterion method
(maximisation) is as follows:
Max (8.598459x1 + 8.003596x2 + 7.59108x3 +
+ 8.390612x4 + 8.577669x5 + 9.594857x6 +
+ 8.609654x7 + 8.385363x8 + 6.291351x9 +
+ 6.564639x10 + 2.985886x11 + 3.219647x12 +
+ 3.516151x13 + 2.742925x14 + 2.308067x15 +
+ 2.906689x16 + 3.200982x17 + 2.528028x18 +
+ 2.716658x19 + 3.122838x20 – 3.23244b24 –
– 3.62034d24 – 6.09547b25 – 6.82704d25 –
+1.26219b26 – 1.073d26 – 1.23141b27 –1.0467d27)
Of course, all other constraints from model M2 must
be taken into consideration. The optimal solution of
that model is presented in Table 12. Table 13
presents the optimal values of objective functions
obtained by the three presented approaches.
6 Discussion of Results
From the results obtained, we can see that the multi-
objective De Novo programming model has shown
high application efficiency in solving production
plan optimization problems. The efficiency and
flexibility provided by the proposed model cannot
be achieved by modeling the problem using
standard mathematical programming models. In
standard mathematical programming problems,
resources are predetermined and the work to be
done is to "optimize a given system." In contrast, the
De Novo approach suggests a way of "designing an
optimal system." In De Novo, resource quantities
are not predetermined, as they are available if we
have enough money. The maximum quantity of
resources is limited by the budget, which is an
important new element of De Novo. The presented
model and obtained results show that variable
resource prices can be successfully incorporated into
the multi-criteria De Novo model.
From the results presented in Table 13, it can
be seen that for all three approaches, the values of
both objective functions are very close to the ideal
value, especially for the second objective function.
The Weighted Goal Programming approach
provides a solution in which the first objective
function achieves 96.4% of the ideal value, and the
second objective function achieves 98.95% of the
ideal value. The Min-max approach provides a
solution in which the first objective function
achieves 97.11% of the ideal value, and the second
objective function achieves 98.39% of the ideal
value. Finally, the third approach, Global Criterion
Method, provides a solution in which the first
objective function achieves 97.06% of the ideal
value, and the second objective function achieves
98.45% of the ideal value.
This indicates that, according to the criterion of
deviation of the obtained solution from the ideal
values of the objective functions, it cannot be said
that any one of these methods is more or less
efficient than the others.
7 Conclusion
Compared to the standard programming model, De
Novo is generally more effective in solving
problems. For example, multi-objective problems,
and price changing, i.e. the increasing costs of raw
materials, or quantity discounts, are production
situations that can be processed very successfully
with De Novo methodology and provide satisfactory
solutions.
The model presented in this paper (M1)
indicates a high application efficiency when using
De Novo multi-objective programming in solving
production plan optimization problems in various
production companies. Here is presented how this
model can be applied to one such company, i.e. a
bakery that produces twenty different articles and
uses twenty-seven different raw materials. Since the
prices of raw materials in the production program
vary, the multi-objective model could not be easily
transformed into a simple knapsack problem, as is
usual with multi-objective De Novo problems.
Namely, some of the raw materials have different
price variables and their equations cannot be
substituted in the budget equation. For that reason, a
new simplification which reduces our set of
constraints is introduced, so solving the model
becomes much easier (in the case of the bakery the
set of constraints and the set of decision variables by
23 constraints and variables altogether has been
reduced).
The inclusion of variable prices in the multi-
objective De Novo programming model is a further
innovation. None of the papers that deal with multi-
objective De Novo programming present a De Novo
model which involves variable resource prices.
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Future work on this topic will investigate other
production situations in which the De Novo multi-
objective programming model with increasing
resource prices and quantity discounts can be
applied.
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Appendix 1. Tables with input data
Table 1. List of articles
Mark
Article name
Selling
prices sj
Flour per
unit (in kg) c2j
Weight
(in kg)
Monthly Amount
lower
upper
A1
Rye mixed round
10.176
0.41248
0.60
1580
4890
A2
Corn mixed
10.86
0.29988
0.60
10520
17100
A3
Bread with sunflower seeds
11.316
0.31001
0.50
1580
4210
A4
Wheat mixed semi-white
7.884
0.4857
0.65
13150
40390
A5
Wheat half-white bread - folk
7.884
0.486
0.65
9210
23670
A6
Wheat white sandwich
10.056
0.48592
0.65
65750
123080
A7
Rye mixed long
10.2
0.41248
0.60
2630
9460
A8
Wheat mixed Sun
10.86
0.35115
0.60
1710
6000
A9
Swedish bread
11.316
0.2145
0.50
1320
2760
A10
Wheat mixed bread - Zagora
10.176
0.2413
0.50
1710
4260
A11
White rolls of mini salty
4.632
0.086
0.10
9210
23670
A12
White rolls - milk roll
5.652
0.0625
0.08
1970
5670
A13
Stuffed pastry layered cheese
6.48
0.09354
0.10
2370
5260
A14
White rolls with salt
4.512
0.06561
0.07
2240
6310
A15
White rolls round kaiser
4.056
0.04335
0.06
7890
22090
A16
White mini rolls
4.86
0.06573
0.09
7890
24450
A17
White pastry croissant
5.904
0.06327
0.07
1970
4730
A18
Donut
5.04
0.03667
0.07
15780
42080
A19
White pastry - trace
5.196
0.04592
0.05
1580
4340
A20
Stuffed rolls
6.48
0.053
0.08
1970
7890
Table 2. List of raw materials
Mark
Raw material
Prices
(per kg)
Mark
Raw material
Prices
(per kg)
Variable
prices
R1
Rye flour
3.96
R15
Rum aroma
49.8828
R2
Wheat flour T-110
2.706
R16
Goldperle TBM
39.6
R3
Kitchen salt
1.848
R17
Vanilla sugar
16.8828
R4
Additive panifarin
36.3
R18
Butter aroma
151.3776
R5
Wheat germs
19.6152
R19
Rye sourdough
15.6684
R6
Grandma mix
26.3868
R20
Cheese for bakery
19.14
R7
Suvita
20.7108
R21
Enhancer
24.948
R8
Sugar
6.996
R22
Wiener note
30.228
R9
Pure corn grits
9.24
R23
Grainpan Max
14.9952
R10
Edible oil
9.24
R24
Yeast
6.93
7.7616
R11
Margarine BV
11.88
R25
Corn concentrate
13.068
14.6364
R12
Margarine Tropic
12.54
R26
Wheat flour T-850
2.706
2.3004
R13
Eggs (pieces)
0.924
R27
Wheat flour T-550
2.64
2.244
R14
Marmalade
10.824
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Table 3. Norms - The amount of raw material in grams in one unit of articles
Mark
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
R1
126.48
126.48
49.5
89.5
R2
145.8
R3
8
8
6.84
9.5
9.52
9.52
8
7.85
6.8
R4
4
4
7.14
R5
34.4
R6
19
R7
57
R8
R9
68.9
R10
3.57
R11
R12
R13
R14
R15
R16
4.2
R17
R18
R19
4
3.8
4
R20
R21
2
1.52
2.38
2.38
1.96
1.06
1.3
R22
R23
143
R24
8
8
7.6
11.9
10.47
11.91
8
8.96
9
8.6
R25
128.52
R26
184
155.1
339.9
316
184
301.65
125
R27
102
299.88
155
170
485.92
102
241.3
Table 3. Norms - continuation
Mark
P11
P12
P13
P14
P15
P16
P17
P18
P19
P20
R1
R2
R3
2
0.94
1.73
1.29
0.89
1.33
1.24
0.55
1
1
R4
R5
R6
R7
R8
2
0.94
1.73
1.29
1.33
1.33
3.1
1.47
5
R9
R10
3.67
R11
3
2
1.33
1.93
5
5
3
R12
46.29
12.5
19
R13
0.15
0.2
0.1
R14
10
R15
0.31
0.73
R16
0.8
0.8
0.8
R17
0.73
R18
1
R19
R20
20
33
R21
2
0.31
1
1
1
1
R22
3.12
R23
R24
2.52
1.87
2
1.33
1.99
1.86
2.2
2.7
3
R25
13
R26
R27
86
62.5
93.54
65.61
43.35
65.73
63.27
36.67
45.92
53
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Table 4. Selling prices and unit costs for the articles
Mark
Article name
Selling
prices sj
Unit costs for the first
23 raw materials – VC
T
C C C
V p A
j Cj
sv
A1
Rye mixed round
10.176
0.723518
9.452482
A2
Corn mixed
10.86
0.231
10.629
A3
Bread with sunflower seeds
11.316
1.791966
9.524034
A4
Wheat mixed semi-white
7.884
0.471467
7.412533
A5
Wheat half-white bread - folk
7.884
0.076969
7.807031
A6
Wheat white sandwich
10.056
0.066491
9.989509
A7
Rye mixed long
10.2
0.723518
9.476482
A8
Wheat mixed Sun
10.86
0.52914
10.33086
A9
Swedish bread
11.316
2.498734
8.817266
A10
Wheat mixed bread - Zagora
10.176
1.356398
8.819602
A11
White rolls mini salty
4.632
0.103224
4.528776
A12
White rolls - milk roll
5.652
0.110359
5.541641
A13
Stuffed pastry layered cheese
6.48
0.978577
5.501423
A14
White rolls with salt
4.512
0.060117
4.451883
A15
White rolls round kaiser
4.056
0.051698
4.004302
A16
White mini rolls
4.86
0.059639
4.800361
A17
White pastry croissant
5.904
0.419141
5.484859
A18
Donut
5.04
0.41867
4.62133
A19
White pastry - trace
5.196
0.371686
4.824314
A20
Stuffed rolls
6.48
0.939048
5.540952
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Appendix 2. Tables with optimal solutions
Table 5. Optimal solution for z1 (max profit)
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1580
x15
22090
b9
117.82
b23
188.76
x2
10520
x16
24450
b10
175.85
b24
2000
x3
1580
x17
4730
b11
211.46
d24
328.31
x4
13150
x18
42080
b12
205.26
b25
1377.64
x5
21316
x19
4340
b13
9559
d25
0
x6
122351
x20
1970
b14
420.80
b26
0
x7
2630
b1
947.62
b15
32.18
d26
14200.14
x8
6000
b2
1917.27
b16
82.90
b27
0
x9
1320
b3
1832.42
b17
30.72
d27
75054.63
x10
1710
b4
59.68
b18
4.34
y1
0
x11
23670
b5
58.82
b19
22.84
y2
1
x12
5670
b6
30.02
b20
112.41
y3
0
x13
2370
b7
90.06
b21
450.54
y4
1
x14
6310
b8
225.03
b22
17.69
z1* = 2143888.1
Table 6. Optimal solution for z2 (max flour consumption)
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1580
x15
7891
b9
117.82
b23
188.76
x2
10520
x16
7892
b10
64.02
b24
2000
x3
1580
x17
1970
b11
81.50
d24
420.35
x4
35916
x18
15780
b12
171.76
b25
1377.64
x5
23670
x19
1580
b13
3609
d25
0
x6
123080
x20
1970
b14
157.80
b26
0
x7
2630
b1
735.27
b15
12.13
d26
21388.09
x8
1710
b2
5236.55
b16
59.65
b27
0
x9
1320
b3
1951.42
b17
11.52
d27
71097.73
x10
1710
b4
29.05
b18
1.58
y1
0
x11
9210
b5
58.82
b19
22.84
y2
1
x12
1970
b6
30.02
b20
112.41
y3
0
x13
2370
b7
90.06
b21
449.56
y4
1
x14
2243
b8
85.46
b22
6.15
z2* = 98457.5
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Table 7. Metaoptimal solution (Min B, Z ≥ Z*)
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1592
x15
22090
b9
117.82
b23
188.76
x2
10520
x16
24450
b10
79.33
b24
2000
x3
1580
x17
4730
b11
211.46
d24
433.66
x4
20126
x18
15780
b12
206.26
b25
1377.64
x5
23670
x19
4340
b13
4300
d25
0
x6
123080
x20
1970
b14
157.80
b26
0
x7
8475
b1
1688.41
b15
12.99
d26
18392.83
x8
6000
b2
2934.37
b16
61.86
b27
0
x9
1320
b3
1960.43
b17
11.52
d27
75442.03
x10
1710
b4
83.11
b18
4.34
y1
0
x11
23670
b5
58.82
b19
46.27
y2
1
x12
5670
b6
30.02
b20
112.41
y3
0
x13
2370
b7
90.06
b21
484.17
y4
1
x14
6310
b8
186.37
b22
17.69
B* = 308077
Table 8. Optimal solution with optimal path ratio r = B/B* = 0.97378253
(Xj rounded to integer, bi obtained from the original constraints)
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1550
x15
21511
b9
114.72
b23
183.76
x2
10244
x16
23809
b10
77.25
b24
2000
x3
1539
x17
4606
b11
205.91
d24
369.84
x4
19598
x18
15366
b12
200.85
b25
1341.49
x5
23049
x19
4226
b13
4187
d25
0
x6
119853
x20
1918
b14
153.66
b26
0
x7
8253
b1
1644.12
b15
12.65
d26
17910.46
x8
5843
b2
2857.39
b16
60.23
b27
0
x9
1285
b3
1909.02
b17
11.22
d27
73463.90
x10
1665
b4
80.93
b18
4.23
x11
23049
b5
57.28
b19
45.06
x12
5521
b6
29.24
b20
109.45
x13
2308
b7
87.72
b21
471.48
x14
6145
b8
181.48
b22
17.23
B* = 300000
Table 9. Pay-of-table for bakery problem
z1(x1*)
z1(x2*)
2143888.10
92119.51
z2(x1*)
z2(x2*)
1895180.6
98457.49
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Table 10. Optimal solution with weighted goal programming
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
4080
x16
20380
b11
183.85
b25
1377.64
x2
10520
x17
4730
b12
171.76
d25
0
x3
1580
x18
27566
b13
3610
b26
0
x4
33660
x19
4337
b14
157.80
d26
18255.57
x5
19730
x20
1970
b15
12.13
b27
0
x6
102570
b1
735.27
b16
59.65
d27
74543.00
x7
7890
b2
3892.86
b17
11.52
y1
0
x8
5000
b3
1936.16
b18
1.58
y2
1
x9
1320
b4
29.05
b19
22.84
y3
0
x10
1710
b5
58.82
b20
112.41
y4
1
x11
19730
b6
30.02
b21
492.51
n1
77073.20
x12
4730
b7
90.06
b22
17.69
n2
1030.95
x13
2370
b8
164.01
b23
188.76
x14
5260
b9
117.82
b24
2000
x15
18410
b10
64.02
d24
414.01
z1 (WGP) = 2066814.9
z2 (WGP) = 97426.54
Table 11. Optimal solution with goal programming – the Min-max approach (Min D)
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1580
x16
24450
b11
185.74
b25
1377.64
x2
10520
x17
2344
b12
176.44
d25
0
x3
1580
x18
15780
b13
3666
b26
0
x4
22418
x19
1582
b14
167.80
d26
18094.20
x5
23670
x20
1970
b15
12.25
b27
0
x6
123080
b1
947.62
b16
59.65
d27
74567.01
x7
2630
b2
3268.54
b17
11.52
y1
0
x8
6000
b3
1929.63
b18
1.58
y2
1
x9
1320
b4
59.68
b19
22.84
y3
0
x10
1710
b5
58.82
b20
112.41
y4
1
x11
23670
b6
30.02
b21
457.24
n1
62010.89
x12
5670
b7
80.06
b22
17.69
n2
1580.27
x13
2370
b8
165.18
b23
188.76
x14
6310
b9
117.82
b24
2000
x15
22090
b10
79.33
d24
402.19
D*
368.69
z1 (WGP) = 2081877.21
z2 (WGP) = 96877.21
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Table 12. Optimal solution with the Global criterion method
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
Variables
Optimal
values
x1
1580
x15
22090
b9
117.82
b23
188.76
x2
10520
x16
24450
b10
79.33
b24
2000
x3
1580
x17
1871
b11
183.86
d24
403.39
x4
22575
x18
15780
b12
171.77
b25
1377.64
x5
23670
x19
1580
b13
3610
d25
0
x6
123080
x20
1970
b14
157.80
b26
0
x7
2633
b1
948.00
b15
12.13
d26
18148.11
x8
6000
b2
3291.44
b16
59.65
b27
0
x9
1320
b3
1930.68
b17
11.52
d27
74543.63
x10
1710
b4
59.69
b18
1.58
y1
0
x11
23670
b5
58.82
b19
22.86
y2
1
x12
5670
b6
30.02
b20
112.41
y3
0
x13
2370
b7
90.06
b21
487.24
y4
1
x14
6310
b8
164.01
b22
17.69
z* = 0.045929
z1 (Global) = 2080933.08
z2 (Global) = 96931.03
Table 13. Comparisons of the results
z1
z2
Ideal values
2143888.1
98457.49
WGP
2066814.9
97426.54
% from ideal
values
0.964049803
0.989529
Minimax
(Chebishev)
2081877.21
96877.21
% from ideal
values
0.971075501
0.9839496
Global criterion
method
2080933.08
96931.03
% from ideal
values
0.97063512
0.984496
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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