ecomod.net - Equilibrium price model for... · Web viewglobal financial crisis, mutual debts, price level, matrix games. Introduction The proposed algorizms [1,2,3] present only algorizms

Equilibrium price model for mutual debts1

Yadulla HASANLI,

Head of the Laboratory “Modeling of Social-Economic Processes”

of the Institute of Control Systems of the Azerbaijan National Academy of Sciences, dr.professor

E-mail: [email protected], [email protected], [email protected]

Mobile tel:((+994 50) 3263995

Abstract

Starting from the middle of 2008 when the global financial crisis started, we can observe the formation of mutual debts, problems in financing of the real sector and as a result devaluation of the debts and especially securities. As a result the following question is rising: how the mutual debts should be valued in order to hold the market equilibrium and these debts could be normaly repaid? We tried to find the answer to this question in this work.

The mutual debts (receivable and payable balances) between the countires were presented in matrix forms and the indexes of their existing debts were identified through the application of the theory of matrix games.

Key word: global financial crisis, mutual debts, price level, matrix games.

Introduction

1 This study was carried out by the support of Science Foundation of State Oil Company of Azerbaijan Republic iin 2013.

mailto:[email protected]



The proposed algorizms [1,2,3] present only algorizms of problem’s mathematical solution, in other words the effective sequence (route) of payments

of debts without considering the economic meaning of the problem. If to look through the problem’s economic meaning we can say that the prices play

the significant role in the event of debts. The economic system forms the prices and keeps the balance through mutual influence of demand and

supply. The prices posses the decisive factor as a main element of “Invisible hand”. In the long term period the balance in the market is achieved

through the change of level of prices. In the short term period and the nonperfect competition the monopoly estranges the prices from the balance level.

Estranging of the prices from the balance level as a result leads to the event of debts. Emergence of debts would inflate the size of non payments with

forming the multiplicative effect. If we look through the global finance crisis beginning from the middle of 2008 we can primarily observe the

increasing of mutual debts, the problems arising in the financing of real sector and as a result we can see the searching by the system its balance with

reduction the value of debts, especially securities. If we would look through the natural form of exchanging of goods and services with some

abstraction of the problem, it would not be so difficult to see the lack of opportunity of the permanent balance of the market and not arising the debts.

So, we can take the variation from the balance level of the market as a main factor of arising of debts. In the long-term period the market’s tendency to

the balance changes the level of prices, in other words, directs from the violation level of balance of prices to the balance level of prices in economics.

This process causes the event of debts. For example, the existence of the monopoly prices in the market creates the inefficency of the activities of other

sectors and, in turn, this causes the increasing of payments for the purchased goods with increasing prices, not returning the kredits as well. Then the

following question arises: in existing mutual debts in which level should the prices of the goods of subjects be in order to provide the balance in the

market and not arising of debts?

We will try to find the answer to this question below.

Economico-mathematical definition of the task

Let us describe the mathematical form of the balanced prices model of mutual debts with the matrix games and linear programming problem[4]. Let, pi

denote the price level of the debts of i-th subject (country). Then the debt of the j -th subject to be obtained from 1-st subject will be a1j*p1, the debt to

be obtained from the 2-nd subject will be a2j*p2, the debt to be obtained from the i -th subject will be aij*pi and finally the debt to be obtained from the n

-th subject will be anj*pn. For not having the debts (kreditor debts) of an enterprise to any other enterprise the sum of all debts to be obtained by j-th

enterprise should be equal to the sum of all its debts (accounts receivable) to be paid to other enterprises:

x1 j p1+x2 j p2+…+xij pi+…+xnj pn=b j p j , j=1,2 ,... , n (1)

There are n variables(pi) and n equations of the system of equations (1).

Opened form of (1) will be as follows:

{ x11 p1+ x21 p2+…+x i1 p i+…+xn 1 pn=b1

x12 p1+ x22 p2+…+x i2 pi+…+xn 2 pn=b2

…………………………………………………….x1 n p1+x2 n p2+…+ x¿ pi+…+ xnn pn=bn

In the matrix form the equation will be as follows:

Ap=b (2)

Here,

AT=( x11 x21 … xn1

⋮ … ⋮x1 n ⋯ xnn

), pT=( p1 , p2 , …, pn), bT=(b1 , b2 , …,bn)

If the determinant of ‖X ij‖ matrix is not zero then the system has a uniqe solution.

p=A−1 b (3)

The found solutions of the system (1) - p1, p2, .., pi ,..., pn correspondingly give the level of balance values (index) of the enterprise’s debts.

Let us now consider the balance values of the kreditor debts of an enterprise:

{ x11 t1+x12 t 1+…+x1i t1+…+ x1 n t1=a1

x21 t2+x22 t 2+…+x2 i t 2+…+x2nt 2=a2

……………………… …………………………….xn 1t n+ xn2 tn+…+x¿ tn+…+xnn tn=an

(4)

If write in the form of vector matrix,

The solution of the system (4) will be as follows:

t i=ai

(x11+x12+…+x1 i+…+x1 n), i=1,2 ,... , n

Finding an optimal solution of the task and the economic mining of the solution.

Let pi represents the value level of the debts of i-th enterprise. Due to x ij represents the debt of j-th enterprise to i-th enterprise the product x ij∗pi will

characterize real market price of the debt of j-th enterprise to the i-th enterprise. Thereby the balanced value (market value) of debts (accounts

receivable) to be obtained by i-th enterprise from n enterprises will not be less than the debts (kreditor debts) of the i-th enterprise to be paid to other

enterprises. It means that,

x1 j p1+x2 j p2+…+xij pi+…+xnj pn≥ b j , j=1,2 , …,n (5)

Certainly, enterprises are interesting in the minimal value of their debts (accounts payable).

f =∑i=1

n

ai∗pi →min (6)

pi can not receive a negative value because of the representing a value index.

pi ≥0 , i=1,2 , …, n (7)

The economical meaning of the task (5)-(7) is that the value of debts of the enterprises (debts to be received) should be so minimal that

enterprises are able to settle its kreditors debts at least.

Let represent the task (5)-(7) in opened form. Considering that x ii=0 , i=1,2,…, n . Then,

{ 0∗p1+x21 p2+…+x i 1 pi+…+xn1 pn ≥ b1

x12 p1+0∗p2+…+x i 2 pi+…+xn2 pn ≥ b2

…………………………………………………….x1 n p1+x2 n p2+…+x¿ p i+…+0∗pn≥ bn

(8)

f =a1 p1+a2 p2+…+a i pi+…+an pn→ min (9)

p1≥ 0 , p2≥ 0 , …, pn≥ 0 (10)

If represent in matrix-vector form,

AT p≥ b (11)

p≥0 (12)

f =ap → min (13)

Here,

AT=‖X ij‖=( x11 x21… xn 1

⋮ … ⋮x1n ⋯ xnn

), pT=( p1 , p2 , …, pn), bT=(b1 , b2 , …,bn)

If in an optimal solution of the task (11)-(13) p¿=( p1¿ , p2

¿ ,…, pn¿ )=(1,1 ,…,1) , that is pi

¿=1 ,i=1,2 ,…,n, this means that the shadow and

objectively stipulated values (actual value) of debts equal to its nominal value an economics is in a balanced state.

f min=a1 p1¿+a2 p2

¿+…+ai p i¿+…+an pn

¿=∑i=1

n

a i=∑i=1

n

bi (14)

Obviously, in this case,

f min

∑ bi

=1

This case takes place when a i=bi.

Example: If we receive pi¿=0 in an optimal solution, this means that the value of debts of i-th enterprise iz zero. In other words, the i-th enterprise

completely has a position as a monopolist and due to the enterprise sales its products in high level than the balanced level of the market there is not a

market value with the complete competition and should not be paid any debt to it. 0< pi¿<1 , this means that the i-th enterprise has sold its products at

partially higer prices than the balanced value of the market and the enterprises- purchasers enterprices have loaded into debts, so the market prices of

debts (accounts receivable) to be settled to i-th enterprise is less than their nominal prices.

If pi¿>1 then this means that i-th enterprise has sold its products at the price less than the balanced value of the market and has loaded into the

higher kreditor debts. For this reason the shadow and market prices of debts (accounts receivable) to be settled to i-th enterprise are higher than

nominal prices.

If, f min=a1 p1

¿+a2 p2¿+…+ai p i

¿+…+an pn¿<∑

i=1

n

bi (15)

Then it means that the shadow and market prices of debits dont’t allow accounts payable have been paid off. Considering that the economics always

strives to the balance, then we can conclude that level of prices (inflation) have to be increased in order to accounts payable have been fallen and paid

off.

Obviously, in this case

f min

∑ bi

<1

If, f min=a1 p1

¿+a2 p2¿+…+ai p i

¿+…+an pn¿>∑

i=1

n

bi (16)

Then this means that the shadow and market prices of debts allow accounts payable not only have fully been paid off, even some money excess have

been created.

In this case,

f min

∑ bi

>1

Already money is coming in the banking and financial sectors. Considering that banks create money with multiplicative effect, the risks of the real

sektor are increasing due to not meeting them the banks credits supply. Not being returned after a while risky kredits to the banking and finance sectors

results in the impeded real sector financing and crisis in financial sector, in other words, the prices fall significantly. It should be noted that the

sfinancial crisis starting with fall of oil prices in the world market at the beginning of 2008 took place in connection with surpluses of money in USA

banks. The surpluses of money in banks was the reason for granting of risky mortgage credits which later resulted in non repayment of these credits.

This, in turn, caused the problem in the financing of the real sector and resulted in finance crisis. The prices of oil and other goods was considerably

fallen. The market price of accounts payable are increasing and the currency reservs for their repayments start to decrease.

If f min

∑ bi

=1 , then this means that the existence of debts doesn’t cause any crisis threat. The economics is in balanced state. If f min

∑ bi

<1 , then this

means that the balance has been broken and its restoring can be caused with increasing the prices. Considering that the economic strive for balance, we

can conclude that the level of prices have to be increased (inflation) in order for accounts payable have fallen and repayed. That is the real sector crisis

is anticipated. If f min

∑ bi

>1 , this means that balance has been broken. Counting that the system always trends toward the balance then the falling of

prices can cause balance. That is the crisis of the finance sector is necessary to cause the balance.

Let us denote by BI the ratio f min

∑ bi . BI can be characterised as crisis index

BI=f min

∑ b i (17)

Conjugate task for the task (15)-(17) shows the objective conditional value (market price) of the accounts payable of the enterprise.

{ x11 t 1+ x12 t2+…+x1 i t i+…+x1nt n ≤a1

x21 t 1+ x22 t2+…+x2 i ti+…+x2n tn ≤ a2

……………………………… ……… …………….xn 1 t1+xn 2t 2+…+xn3t i+…+xnn t n≤ an

(19)

z=b1t 1+b2 t2+…+bi t i+…+bnt n →max (20)

t 1≥ 0 , t2 ≥0 ,…, t n≥ 0 (21)

According to the mutual inter-enterprises debts (debitor-accounts payable or non payments matrix) let us consider the linear programming task

(15)-(17) and (19)-(21) being mainly dual to each-other. Firstly, note that according to the non payment matrix total amount of all country non

payments in 1998 is 2789661.2 thousand manat (AZN).

The results obtained from the solution of the task (15)-(17) on the base of year 1998 information with applying Simplex method are given in

Appendix ....

The values of variables and target function obtained from the solution of the task are given in the Table 1.

Table 1.

The balanced value of accounts receivable of enterprises of the Republic of Azerbaijann in 1998

Seq. noName of an enterpriseVariables (value index of accounts receivable of enterprises)Value of variablesThe difference of creditor and accounts receivable

1«Azerkimya»

p1

047941.6

2«Azerigaz» Joint Stock Company

p2

0-1656.6

3State Oil Company of Azerbaijan Republic

p3

0-238938.6

4«Azerenerji» Joint Stock Company

p4

0151364.0

5«Azneftkimyamash» State Enterprise

p5

05716.4

6Azerbaijan Railways

p6

86.775487.8

7Ministry of Agriculture

p7

0343741.8

8Absheron Regional Water Join Stock Company

p8

0764.8

9Baku Electric Network

p9

0.9572763.0

10Ministry of Communication and High Technologies

p10

0-37181.0

11«Azerbaijan Air Ways » State Consern

p11

1 766.713395.2

12«Azeravtoyol» State Enterprise

p12

0.015481.2

13«Azerkontrakt» State Enterprise

p13

0.0-740.4

14«Azerbalıq» State Consern

p14

0.0-742.8

15«Azgur» Joint Stock Company

p15

0.0

-1535.0

16Ministry of Media and Information

p16

0.03133.0

17Other ministries, enterprises and organizations

p17

16.326553.6

18Budget organizations

p18

0197452.6

19Taxes to budget

p19

0-349029.2

20To Social Protection, Engagement and Social Protection of Diabled funds

p20

0

-129518.0

21Loan debts to banks

p21

0-194453.4

Function

(Min.) =51122090.0

As we see the value of the target function (f=51122090.0), i.e. the market price of accounts receivable of enterprises is substantially greater than

existing nominal price. The crisis index is greater than one.

BI =2789661.2 /51122090.0=18.3

This means that the prices have to be increased for the returning the system to its balanced state. This is, in turn, is evidenced the necessity of the crisis

of the real sector.

The solution of the optimization problem with application of Simplex method is as follows:

dk fmax(db bi min(kbq) bi

q)

19982789661

.22322993

.0 0.816727610

.0 7.2

The solution of dual problem (the value of accounts payable) is:

2 950 532.0000

Referencies

Hasanli Y.H. Algorithms for resolution of mutual arrears of enterprises // Az.AS News, A series on physics-mathematics, XIX edition, №3-4, Baku, “Elm”, 1999, pp.119-124,

Hasanli Y.H. Mathematical methods of efficiently solving mutual payments problem // Second international symposium on mathematical & computational applications, Baku, 1999, p.15.

Самарский А.А., Михайлов А.П. Математическое моделирование. Идеи. Методы. Примеры. 2-е изд., испр. М.: Физматлит, 2001, 320 с.

David Gele, The Thery of Liner economic Models, MeORAW-Holl, Book Company, INC, NEW-YORK TORONTO LONDON, 1960

AppendixesAppendix 1. The inter-bodies debitor-accounts payable matrix in the Republic of azerbaijan (2008, in thousand AZN)

N1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

10.0 913.6 2540.0 54231.6 5.0 240.2 0.0 1658.6 0.0 36.6 0.0 8.0 0.0 0.0 290.0 0.4 5421.4 68.4 5489 1759.8 11235.4

20.0 0.0 154687.4 0.0 0.0 0.0 0.0 0.0 46.4 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13078.6 2179.8 23722

31638.6 5665.2 0.0 9970.4 10879.6 2957.4 81.2 2510.6 1331.0 414.2 2134.2 1639.2 960.2 0.0 565.2 19.2 156796.0 3332.4 289615.4 75400.4 132647.8

40.0 142510.6 417422.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14068.2 12582 0

50.0 184.2 90.0 1016.8 0.0 0.2 0.0 177.0 0.0 91.0 0.0 0.0 0.0 0.0 0.0 0.0 5528.0 0.0 6340 4072.6 739.2

6596.8 57.8 4098.2 2118.2 150.0 0.0 4022.6 85.6 0.0 14.0 36.2 876.8 26.8 9.2 0.0 3.2 21545.6 2630.0 37134.4 11585 1246.8

71428.6 1191.6 176329.4 98944.4 6.2 116.2 0.0 1601.2 2262.6 77.8 1031.4 120.0 0.0 0.0 0.0 0.0 0.0 0.0 24840.2 15455.6 25198.8

80.0 173.4 183.4 56106.2 0.0 80.4 0.0 0.0 894.0 16.0 0.0 0.0 0.0 0.0 0.0 0.0 17985.0 0.0 4046.2 686.8 0

90.0 0.0 0.0 108949.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 973.0 0.0 803.2 305.8 0

100.0 1.2 5.0 1255.2 5.0 11.0 0.0 1.4 53.4 0.0 3.0 0.0 0.0 0.4 0.4 0.0 364.2 64.6 0 826.8 144

110.0 0.0 22636.2 170.8 15.2 0.0 0.0 145.4 63.2 195.8 0.0 0.0 0.0 0.0 0.0 0.0 625.2 16.0 2668.8 3882.8 9378.8

120.0 214.0 1901.2 1834.4 0.0 910.0 242.4 0.0 0.0 26.4 0.0 0.0 0.0 0.0 0.0 0.0 3522.4 0.0 7015 2709.6 98

130.0 0.0 15.2 0.0 0.0 207.4 0.0 23.4 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0 0

140.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1095.6 0.0 3.4 0 0

153.2 2.2 9.4 49.6 0.0 0.0 0.0 21.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 236.8 2.6 298.4 73.8 0

160.0 17.6 0.0 3.6 0.0 0.0 0.0 4.8 2.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1613.0 132.0 35 171.8 1546.6

1713168.2 23145.4 107296.4 0.0 1396.6 5304.6 0.0 57831.0 0.0 12383.4 19568.0 0.0 0.0 929.2 874.4 363.0 0.0 0.0 0 0 0

18636.8 21201.2 1947.4 100568.6 65.0 922.0 0.0 15346.6 33615.4 26656.8 2298.6 335.0 0.0 0.0 97.8 7.4 0.0 0.0 0 0 0

195489.0 48.4 48328.2 0.0 0.0 0.0 516.0 0.0 0.0 0.0 1331.6 0.0 0.0 682.8 10.0 0.6 0.0 0.0 0 0 0

201759.8 0 7 0 0 0 0 0 0 0 0 13.2 0 0 394.2 0 0.4 0 0 0 0

2111235.4 48.4 0 0 0 0 0 0 0 0 0 0 0 220.2 0 0 0 0 0 0 0

Appendix 2.

Combined Report for dk 2008 kred

Decision Solution Unit Cost or Total ReducedBasis Allowable Allowable

VariableValue Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 X1 0 35 956.4000 0 6 531.9750 at bound-M M

2 X2 0 195 374.8000 0 176 042.6000 at bound19 332.1500 M

3 X3 0 937 496.8000 0 251 192.0000 at bound-M M

4 X4 0 435 219.2000 0 435 042.8000 at bound176.3852 M

5 X5 0 12 522.6000 0 12 149.8600 at bound372.7430 M

6 X6 86.6612 10 749.4000 931 556.1000 0 basic 239.4355 3 160 911.0000

7 X7 0 4 862.2000 0 -978 008.6000 at bound-M M

8 X8 0 79 406.6000 0 -316 410.3000 at bound-M M

9 X9 0.9465 38 268.4000 36 222.7700 0 basic 0 578 914.4000

10 X10 0 39 916.6000 0 16 742.4400 at bound-M M

11 X11 1 766.7440 26 403.0000 46 647 330.0000 0 basic 59.9941 266 185.2000

12 X12 0 2 992.2000 0 -37 589.5400 at bound-M M

13 X13 0 987.0000 0 -7 970.8160 at bound-M M

14 X14 0 1 841.8000 0 1 841.8000 at bound0 M

15 X15 0 2 232.0000 0 2 214.5750 at bound17.4247 M

16 X16 0 393.8000 0 -603.8490 at bound-M M

17 X17 16.2581 215 706.6000 3 506 979.0000 0 basic -13 384.9400 322 438 900.0000

18 X18 0 6 246.0000 0 -13 954 120.0000 at bound-M M

19 X19 0 405 435.8000 0 404 087.6000 at bound1 348.2280 M

20 X20 0 131 692.6000 0 131 692.6000 at bound0.0000 M

21 X21 0 205 957.4000 0 205 957.2000 at bound-M M

Objective Function (Min.) = 51 122 090.0000

Left Hand Right Hand Slack Shadow Allowable Allowable

Constraint Side Direction Side or Surplus Price Min. RHS Max. RHS

1 C1 265 809.3000 >= 83 898.0000 181 901.6000 0 -M 265 799.6000

2 C2 381 309.2000 >= 193 718.2000 187 236.8000 0.0012 -7 091 605 000.0000 380 955.0000

3 C3 42 091 950.0000 >= 698 558.2000 41 393 570.0000 0 -M 42 092 130.0000

4 C4 588 451.1000 >= 586 583.2000 0 0.3512 483 457.6000 M

5 C5 62 559.7400 >= 18 239.0000 44 334.2700 0 -M 62 573.2700

6 C6 86 242.7100 >= 86 237.2000 0 43.1910 47 479.2700 942 494 800.0000

7 C7 348 603.4000 >= 348 604.0000 0 2.6127 268 327.0000 544 470.4000

8 C8 1 204 525.0000 >= 80 171.4000 1 124 159.0000 0 -M 1 204 330.0000

9 C9 111 658.2000 >= 111 031.4000 0 416.8197 -626.7969 149 190.3000

10 C10 548 472.2000 >= 2 735.6000 545 257.9000 0 -M 547 993.5000

11 C11 321 275.6000 >= 39 798.2000 281 466.3000 0 -M 321 264.5000

12 C12 75 984.5500 >= 18 473.4000 57 506.9600 0 -M 75 980.3600

13 C13 2 322.5200 >= 246.6000 2 077.0340 0 -M 2 323.6340

14 C14 15 904.3100 >= 1 099.0000 14 806.0200 0 -M 15 905.0200

15 C15 14 216.0800 >= 697.0000 13 519.2400 0 -M 14 216.2400

16 C16 6 179.0060 >= 3 526.8000 2 652.6490 0.0000 -101 238 500 000 000.0000 6 179.4490

17 C17 2 972 657.0000 >= 242 260.2000 2 730 582.0000 0 -M 2 972 842.0000

18 C18 256 186.9000 >= 203 698.6000 52 492.6600 0 -M 256 191.3000

19 C19 7 933 958.0000 >= 56 406.6000 7 877 899.0000 0 -M 7 934 306.0000

20 C20 7 864 172.0000 >= 2 174.6000 7 862 088.0000 0 -M 7 864 262.0000

21 C21 16 677 980.0000 >= 11 504.0000 16 666 650.0000 0 -M 16 678 150.0000

Appendix 3. Dual problem

Combined Report for k98



1 X1 0 83 898.0000 0 -103 239.1000 at bound-M 187 137.1000

2 X2 0 193 718.2000 0 -1 005 504.0000 at bound-M 1 199 223.0000

3 X3 0 698 558.2000 0 -605 664.3000 at bound-M 1 304 223.0000

4 X4 0 586 583.2000 0 -4 385 523.0000 at bound-M 4 972 106.0000

5 X5 0 18 239.0000 0 -9 001.5460 at bound-M 27 240.5400

6 X6 2.2715 86 237.2000 195 886.5000 0 basic 50 746.8900 391 904.8000

7 X7 2.2259 348 604.0000 775 972.3000 0 basic 311 558.2000 3 563 342.0000

8 X8 0 80 171.4000 0 -1 119 524.0000 at bound-M 1 199 696.0000

9 X9 0 111 031.4000 0 -1 489 555.0000 at bound-M 1 600 586.0000

10 X10 0 2 735.6000 0 -1 366 565.0000 at bound-M 1 369 301.0000

11 X11 0 39 798.2000 0 -228 960.0000 at bound-M 268 758.2000

12 X12 0 18 473.4000 0 -73 462.2000 at bound-M 91 935.5900

13 X13 0 246.6000 0 -2 075.9240 at bound-M 2 322.5240

14 X14 0 1 099.0000 0 -7 114.3120 at bound-M 8 213.3130

15 X15 0 697.0000 0 -10 938.3800 at bound-M 11 635.3800

16 X16 561.0393 3 526.8000 1 978 674.0000 0 basic 1 507.1790 5 955.4440

17 X17 0 242 260.2000 0 -1 624 911.0000 at bound-M 1 867 171.0000

18 X18 0 203 698.6000 0 -24 220.7900 at bound-M 227 919.4000

19 X19 0 56 406.6000 0 -3 161 711.0000 at bound-M 3 218 118.0000

20 X20 0 2 174.6000 0 -1 001 797.0000 at bound-M 1 003 972.0000

21 X21 0 11 504.0000 0 -96 545.3900 at bound-M 108 049.4000

Objective Function (Max.) = 2 950 532.0000



1 C1 770.0264 <= 35 956.4000 35 186.3700 0 770.0273 M

2 C2 0 <= 195 374.8000 195 374.8000 0 0 M

3 C3 17 670.3900 <= 937 496.8000 919 826.4000 0 17 670.3800 M

4 C4 0 <= 435 219.2000 435 219.2000 0 0 M

5 C5 0.4543 <= 12 522.6000 12 522.1500 0 0.4541 M

6 C6 10 749.4000 <= 10 749.4000 0 86.6614 1 795.3260 17 148.0600

7 C7 263.9465 <= 4 862.2000 4 598.2540 0 263.9463 M

8 C8 182.6274 <= 79 406.6000 79 223.9800 0 182.6250 M

9 C9 0 <= 38 268.4000 38 268.4000 0 0 M

10 C10 24.9863 <= 39 916.6000 39 891.6100 0 24.9883 M

11 C11 0 <= 26 403.0000 26 403.0000 0 0 M

12 C12 2 606.6200 <= 2 992.2000 385.5805 0 2 606.6190 M

13 C13 471.1060 <= 987.0000 515.8941 0 471.1059 M

14 C14 0 <= 1 841.8000 1 841.8000 0 0 M

15 C15 0 <= 2 232.0000 2 232.0000 0 0 M

16 C16 0 <= 393.8000 393.8000 0 0 M

17 C17 215 706.6000 <= 215 706.6000 0 7.9811 199 225.7000 306 391.6000

18 C18 6 246.0000 <= 6 246.0000 0 47.6147 4 397.3250 6 589.7810

19 C19 1 485.2100 <= 405 435.8000 403 950.6000 0 1 485.2190 M

20 C20 0 <= 131 692.6000 131 692.6000 0 0 M

21 C21 0 <= 205 957.4000 205 957.4000 0 0 M

Max dbq98

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 kreditor

1 0.0 0.0 1638.6 0.0 0.0 596.8 1428.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.2 0.0 13168.2 636.8 5489.0 1759.8 11235.4 83898.0

2 913.6 0.0 5665.2 142510.6 184.2 57.8 1191.6 173.4 0.0 1.2 0.0 214.0 0.0 0.0 2.2 17.6 23145.4 21201.2 48.4 0.0 48.4 193718.2

3 2540.0 154687.4 0.0 417422.4 90.0 4098.2 176329.4 183.4 0.0 5.0 22636.2 1901.2 15.2 0.0 9.4 0.0 107296.4 1947.4 48328.2 7.0 0.0 698558.2

4 54231.6 0.0 9970.4 0.0 1016.8 2118.2 98944.4 56106.2 108949.4 1255.2 170.8 1834.4 0.0 0.0 49.6 3.6 0.0 100568.6 0.0 0.0 0.0 586583.2

5 5.0 0.0 10879.6 0.0 0.0 150.0 6.2 0.0 0.0 5.0 15.2 0.0 0.0 0.0 0.0 0.0 1396.6 65.0 0.0 0.0 0.0 18239.0

6 240.2 0.0 2957.4 0.0 0.2 0.0 116.2 80.4 0.0 11.0 0.0 910.0 207.4 0.0 0.0 0.0 5304.6 922.0 0.0 0.0 0.0 86237.2

7 0.0 0.0 81.2 0.0 0.0 4022.6 0.0 0.0 0.0 0.0 0.0 242.4 0.0 0.0 0.0 0.0 0.0 0.0 516.0 0.0 0.0 348604.0

8 1658.6 0.0 2510.6 0.0 177.0 85.6 1601.2 0.0 0.0 1.4 145.4 0.0 23.4 0.0 21.0 4.8 57831.0 15346.6 0.0 0.0 0.0 80171.4

9 0.0 46.4 1331.0 0.0 0.0 0.0 2262.6 894.0 0.0 53.4 63.2 0.0 0.0 0.0 0.0 2.4 0.0 33615.4 0.0 0.0 0.0 111031.4

10 36.6 4.0 414.2 0.0 91.0 14.0 77.8 16.0 0.0 0.0 195.8 26.4 0.6 0.0 0.0 0.0 12383.4 26656.8 0.0 0.0 0.0 2735.6

11 0.0 0.0 2134.2 0.0 0.0 36.2 1031.4 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 19568.0 2298.6 1331.6 0.0 0.0 39798.2

12 8.0 0.0 1639.2 0.0 0.0 876.8 120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 335.0 0.0 13.2 0.0 18473.4

13 0.0 0.0 960.2 0.0 0.0 26.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 246.6

14 0.0 0.0 0.0 0.0 0.0 9.2 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0 929.2 0.0 682.8 0.0 220.2 1099.0

15 290.0 0.0 565.2 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0 874.4 97.8 10.0 394.2 0.0 697.0

16 0.4 0.0 19.2 0.0 0.0 3.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 363.0 7.4 0.6 0.0 0.0 3526.8

17 5421.4 0.0 156796.0 0.0 5528.0 21545.6 0.0 17985.0 973.0 364.2 625.2 3522.4 0.0 1095.6 236.8 1613.0 0.0 0.0 0.0 0.4 0.0 242260.2

18 68.4 0.0 3332.4 0.0 0.0 2630.0 0.0 0.0 0.0 64.6 16.0 0.0 0.0 0.0 2.6 132.0 0.0 0.0 0.0 0.0 0.0 203698.6

19 5489.0 13078.6 289615.4 14068.2 6340.0 37134.4 24840.2 4046.2 803.2 0.0 2668.8 7015.0 0.0 3.4 298.4 35.0 0.0 0.0 0.0 0.0 0.0 56406.6

20 1759.8 2179.8 75400.4 12582.0 4072.6 11585.0 15455.6 686.8 305.8 826.8 3882.8 2709.6 0.0 0.0 73.8 171.8 0.0 0.0 0.0 0.0 0.0 2174.6

21 11235.4 23722.0 132647.8 0.0 739.2 1246.8 25198.8 0.0 0.0 144.0 9378.8 98.0 0.0 0.0 0.0 1546.6 0.0 0.0 0.0 0.0 0.0 11504.0

debitor 35956.4 195374.8 937496.8 435219.2 12522.6 10749.4 4862.2 79406.6 38268.4 39916.6 26403 2992.2 987 1841.8 2232 393.8 215706.6 6246 405435.8 131692.6 205957.4 2789661.2

Kbq98max

0.0 913.6 2540.0 54231.6 5.0 240.2 0.0 1658.6 0.0 36.6 0.0 8.0 0.0 0.0 290.0 0.4 5421.4 68.4 5489.0 1759.8 11235.4 1

35956.4

0.0 0.0154687.

4 0.0 0.0 0.0 0.0 0.0 46.4 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13078.6 2179.8 23722.0 2

195374.8

1638.6 5665.2 0.0 9970.410879.

6 2957.4 81.2 2510.6 1331.0 414.2 2134.2 1639.2960.

2 0.0 565.2 19.2156796.

0 3332.4289615.

475400.

4132647.

8 3

937496.8

0.0142510.

6417422.

4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14068.212582.

0 0.0 4

435219.2

0.0 184.2 90.0 1016.8 0.0 0.2 0.0 177.0 0.0 91.0 0.0 0.0 0.0 0.0 0.0 0.0 5528.0 0.0 6340.0 4072.6 739.2 5

12522.6

596.8 57.8 4098.2 2118.2 150.0 0.0 4022.6 85.6 0.0 14.0 36.2 876.8 26.8 9.2 0.0 3.2 21545.6 2630.0 37134.411585.

0 1246.8 6

10749.4

1428.6 1191.6176329.

4 98944.4 6.2 116.2 0.0 1601.2 2262.6 77.8 1031.4 120.0 0.0 0.0 0.0 0.0 0.0 0.0 24840.215455.

6 25198.8 7

4862.2

0.0 173.4 183.4 56106.2 0.0 80.4 0.0 0.0 894.0 16.0 0.0 0.0 0.0 0.0 0.0 0.0 17985.0 0.0 4046.2 686.8 0.0 8

79406.6

0.0 0.0 0.0108949.

4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 973.0 0.0 803.2 305.8 0.0 9

38268.4

0.0 1.2 5.0 1255.2 5.0 11.0 0.0 1.4 53.4 0.0 3.0 0.0 0.0 0.4 0.4 0.0 364.2 64.6 0.0 826.8 144.0 10

39916.6

0.0 0.0 22636.2 170.8 15.2 0.0 0.0 145.4 63.2 195.8 0.0 0.0 0.0 0.0 0.0 0.0 625.2 16.0 2668.8 3882.8 9378.8 11

26403

0.0 214.0 1901.2 1834.4 0.0 910.0 242.4 0.0 0.0 26.4 0.0 0.0 0.0 0.0 0.0 0.0 3522.4 0.0 7015.0 2709.6 98.0 12

2992.2

0.0 0.0 15.2 0.0 0.0 207.4 0.0 23.4 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13

987

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1095.6 0.0 3.4 0.0 0.0 14

1841.8

3.2 2.2 9.4 49.6 0.0 0.0 0.0 21.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 236.8 2.6 298.4 73.8 0.0 15

2232

0.0 17.6 0.0 3.6 0.0 0.0 0.0 4.8 2.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1613.0 132.0 35.0 171.8 1546.6 16

393.8

13168.2 23145.4

107296.4 0.0 1396.6 5304.6 0.0

57831.0 0.0

12383.4

19568.0 0.0 0.0 929.2 874.4 363.0 0.0 0.0 0.0 0.0 0.0 17

215706.6

636.8 21201.2 1947.4100568.

6 65.0 922.0 0.015346.

6 33615.426656.

8 2298.6 335.0 0.0 0.0 97.8 7.4 0.0 0.0 0.0 0.0 0.0 18

6246

5489.0 48.4 48328.2 0.0 0.0 0.0 516.0 0.0 0.0 0.0 1331.6 0.0 0.0 682.8 10.0 0.6 0.0 0.0 0.0 0.0 0.0 19

405435.8

1759.8 0.0 7.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13.2 0.0 0.0 394.2 0.0 0.4 0.0 0.0 0.0 0.0 20

131692.6

11235.4 48.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 220.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21

205957.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2183898.

0193718.

2698558.

2586583.

218239.

086237.

2348604.

080171.

4111031.

4 2735.639798.

218473.

4246.

6 1099.0 697.03526.

8242260.

2203698.

6 56406.6 2174.6 11504.0

2789661.2

Solution: min

Combined Report for kbq1998



1 X1 74.8335 83 898.0000 6 278 378.0000 0 basic 12 199.5800 2 475 040.0000

2 X2 3.0539 193 718.2000 591 604.3000 0 basic 9 401.6250 M

3 X3 0 698 558.2000 0 121 188.9000 at bound-M M

4 X4 0.8670 586 583.2000 508 553.0000 0 basic 289 363.6000 6 989 823.0000

5 X5 0 18 239.0000 0 17 839.5800 at bound399.4214 M

6 X6 0 86 237.2000 0 84 932.5500 at bound1 304.6500 M

7 X7 0 348 604.0000 0 348 604.0000 at bound-M M

8 X8 73.5586 80 171.4000 5 897 296.0000 0 basic -2 630 878.0000 206 010.4000

9 X9 0 111 031.4000 0 102 029.7000 at bound9 001.7420 M

10 X10 0 2 735.6000 0 2 650.8400 at bound84.7597 M

11 X11 0 39 798.2000 0 39 558.5500 at bound239.6529 M

12 X12 0 18 473.4000 0 17 935.6000 at bound537.7993 M

13 X13 0 246.6000 0 246.6000 at bound0.0000 M

14 X14 0 1 099.0000 0 1 067.0460 at bound31.9537 M

15 X15 0 697.0000 0 -15 395.6000 at bound-M M

16 X16 0 3 526.8000 0 3 526.8000 at bound0 M

17 X17 1.6811 242 260.2000 407 260.8000 0 basic -M M

18 X18 0 203 698.6000 0 188 625.9000 at bound15 072.6700 M

19 X19 0 56 406.6000 0 -1 118 396.0000 at bound-M M

20 X20 0 2 174.6000 0 -365 137.2000 at bound-M M

21 X21 264.6489 11 504.0000 3 044 521.0000 0 basic 0.6621 45 691.9200

Objective Function (Min.) = 16727610.0



1 C1 3 154 362.0000 >= 35 956.4000 3 118 408.0000 0 -M 3 154 365.0000

2 C2 6 278 002.0000 >= 195 374.8000 6 082 632.0000 0 -M 6 278 007.0000

3 C3 35 701 930.0000 >= 937 496.8000 34 764 460.0000 0 -M 35 701 960.0000

4 C4 435 219.2000 >= 435 219.2000 0 1.2934 0 24 908 380.0000

5 C5 219 385.5000 >= 12 522.6000 206 863.0000 0 -M 219 385.6000

6 C6 419 154.5000 >= 10 749.4000 408 405.4000 0 -M 419 154.8000

7 C7 6 982 946.0000 >= 4 862.2000 6 978 089.0000 0 -M 6 982 951.0000

8 C8 79 406.6000 >= 79 406.6000 0 5.2974 49 628.8400 824 806.3000

9 C9 96 092.1200 >= 38 268.4000 57 823.7200 0 -M 96 092.1200

10 C10 39 916.5700 >= 39 916.6000 0 79.8843 2 993.1130 M

11 C11 2 493 984.0000 >= 26 403.0000 2 467 583.0000 0 -M 2 493 986.0000

12 C12 34 100.9800 >= 2 992.2000 31 108.8000 0 -M 34 101.0000

13 C13 1 721.2710 >= 987.0000 734.2712 0 -M 1 721.2710

14 C14 1 841.8000 >= 1 841.8000 0 -716.4033 0 3 707.7720

15 C15 2 232.0000 >= 2 232.0000 0 3 812.3600 1 573.0390 556 084.3000

16 C16 412 427.6000 >= 393.8000 412 034.1000 0 -M 412 427.9000

17 C17 5 310 074.0000 >= 215 706.6000 5 094 368.0000 0 -M 5 310 074.0000

18 C18 1 328 466.0000 >= 6 246.0000 1 322 220.0000 0 -M 1 328 466.0000

19 C19 410 908.7000 >= 405 435.8000 5 472.8440 0 -M 410 908.7000

20 C20 131 692.6000 >= 131 692.6000 0 40.7424 129 938.0000 494 080.2000

21 C21 840 931.7000 >= 205 957.4000 634 974.3000 0 -M 840 931.6000

Solution:

Combined Report for dbq1998

Decision Solution Unit Cost or Total Reduced Basis Allowable Allowable

Variable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 X1 0 35 956.4000 0 -233 503.7000at bound -M 269 460.1000

2 X2 0.2423 195 374.8000 47 340.0700 0 basic 123 233.1000 261 124.8000

3 X3 0 937 496.8000 0 -6 287 942.0000 at bound -M 7 225 439.0000

4 X4 0 435 219.2000 0 -692 501.6000at bound -M 1 127 721.0000

5 X5 0 12 522.6000 0 -361 883.4000at bound -M 374 406.0000

6 X6 0 10 749.4000 0 -1 072 615.0000 at bound -M 1 083 364.0000

7 X7 0 4 862.2000 0 -1 389 348.0000 at bound -M 1 394 211.0000

8 X8 0 79 406.6000 0 -17 518.1500 at bound -M 96 924.7500

9 X9 5.3840 38 268.4000 206 036.9000 0 basic 29 044.4600 M

10 X10 0 39 916.6000 0 -35 467.8500 at bound -M 75 384.4500

11 X11 0 26 403.0000 0 -322 676.6000at bound -M 349 079.6000

12 X12 0 2 992.2000 0 -250 275.8000at bound -M 253 268.0000

13 X13 415.8014 987.0000 410 396.0000 0 basic 0 M

14 X14 216.3389 1 841.8000 398 453.0000 0 basic 744.0557 12 227.9800

15 X15 0 2 232.0000 0 -4 784.9520 at bound -M 7 016.9520

16 X16 0 393.8000 0 -17 716.4800 at bound -M 18 110.2800

17 X17 0 215 706.6000 0 -970 750.9000at bound -M 1 186 458.0000

18 X18 0 6 246.0000 0 -39 328.5500 at bound -M 45 574.5500

19 X19 0 405 435.8000 0 -236 541.2000at bound -M 641 977.1000

20 X20 1.7681 131 692.6000 232 850.7000 0 basic 0.6719 M

21 X21 4.9909 205 957.4000 1 027 916.0000 0 basic 129 673.9000 M

Objective Function (Max.) = 2 322 993.0000



1 C1 59 186.5200 <= 83 898.0000 24 711.4700 0 59 186.5200 M

2 C2 241.5604 <= 193 718.2000 193 476.6000 0 241.5625 M

3 C3 43 813.9200 <= 698 558.2000 654 744.3000 0 43 813.9400 M

4 C4 586 583.2000 <= 586 583.2000 0 0.0847 398 140.8000 774 759.3000

5 C5 0 <= 18 239.0000 18 239.0000 0 0 M

6 C6 86 237.2000 <= 86 237.2000 0 4.7589 0 710 578.9000

7 C7 0 <= 348 604.0000 348 604.0000 0 0 M

8 C8 9 729.7520 <= 80 171.4000 70 441.6500 0 9 729.7500 M

9 C9 11.2429 <= 111 031.4000 111 020.2000 0 11.2422 M

10 C10 250.4501 <= 2 735.6000 2 485.1500 0 250.4500 M

11 C11 0 <= 39 798.2000 39 798.2000 0 0 M

12 C12 23.3394 <= 18 473.4000 18 450.0600 0 23.3398 M

13 C13 0 <= 246.6000 246.6000 0 0 M

14 C14 1 099.0000 <= 1 099.0000 0 935.3198 0 1 583.3150

15 C15 697.0000 <= 697.0000 0 334.0739 0 6 232.4380

16 C16 0 <= 3 526.8000 3 526.8000 0 0 M

17 C17 242 260.2000 <= 242 260.2000 0 1.6811 5 239.3280 15 766 790.0000

18 C18 0 <= 203 698.6000 203 698.6000 0 0 M

19 C19 8 228.9740 <= 56 406.6000 48 177.6300 0 8 228.9730 M

20 C20 2 174.6000 <= 2 174.6000 0 89.6297 1 646.4260 2 703.5220

21 C21 5 747.9330 <= 11 504.0000 5 756.0670 0 5 747.9330 M

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