If set to one, the initial and final QP tableaus are .... 10. -.5. -2. 10. -1. 10. -1. /0. 2 2
. S11 s12. S21. S22. 011. 012. 021. 022. 2 .1 1. 1. ...... IF(M.02.0) GO TO 5090.
SEBEND:
pa caq,)0
A Computer Algorithm for the Solution of Symmetric Multicommodity Spatial Equilibrium Problems Utilizing Benders Decomposition
Special Report 708 Agricultural Experiment Station Oregon State University, Corvallis
March 1984
SEBEND: A COMPUTER ALGORITHM FOR THE SOLUTION OF SYMMETRIC MULTICOMMODITY SPATIAL EQUILIBRIUM PROBLEMS UTILIZING BENDERS DECOMPOSITION
Bruce A. Professor of Agricultural Oregon State Corvallis,
McCarl and Resource Economics University Oregon
Jeff Arthur Assistant Professor of Statistics Oregon State University Corvallis, Oregon
Jeff Kennington Professor of Operations Research Southern Methodist University Dallas, Texas
Joe Polito Vice President Pritsker and Associates Albuquerque, New Mexico
CONTENTS
Problem Formulation Comments and Limitations of Problem Formulation Benders Decomposition of Spatial Equilibrium Problem
5
Algorithm
6
Comments on the Algorithm
7
Algorithmic Implementation Input Data
11 14
Set 1
14
Set 2
15
Set 3
16
Set 4
17
Example
19
Some Important Points on Data Input
22
Limitations
22
References Appendix A - Code Documentations and Listings
23 -26
Subroutines and Their Purposes
26
Key Variables
28
Code Listing
35
Appendix B - Solving the QP
100
Appendix C - Detailed Sample Problem Input and Output
104
Appendix D - Changing Problem Sizes
121
Appendix E - Transport Code Stop Conditions
122
Appendix F - Short Sample Problem Output
123
SEBEND: A Computer Algorithm for the Solution of Symmetric Multicommodity Spatial Equilibrium Problems Utilizing Benders Decomposition
Polito, McCarl and Morin [1980]; Litzenberg, McCarl and Polito, and Deckro and Morris have written on the application of Benders decomposition to spatial equilibrium problems. The first two papers contain computational results sections; however, they also contain a disclaimer stating that the algorithm utilized did not fully exploit the structure of the spatial equilibrium problem. Specifically, the algorithm used did not contain solution software exploiting the fact that the subproblem was a collection of independent transportation/network-flow problems. The algorithm also did not exploit the structure of the spatial equilibrium quadratic master problem. Consequently, the purpose of the research leading to this bulletin was to develop an algorithm which exploits the structure of the spatial equilibrium problem. This bulletin describes and documents the resultant algorithm. The algorithm developed herein utilizes a transportation/network flow algorithm (developed by Kennington and Helgeson) coupled with a quadratic program developed by Polito. The algorithm solves symmetric spatial equilibrium problems involving linear supply and demand curves. This bulletin does not contain a detailed explanation of Benders decomposition or spatial equilibrium analysis. The reader may wish to examine companion publications which cover 1) the theory of Benders decomposition and example usages (McCarl), 2) the implementation of Benders
decomposition for more generally structured problems (McCarl and Santini), and 3) the Benders decomposition approach to spatial equilibrium problems (Polito, McCarl, and Morin [1978] or Litzenberg, McCarl, and Polito). Readers unfamiliar with spatial equilibrium analysis should examine Takayama and Judge; Weinschenck et al.; McCarl and Spreen, and/or Martin.
Problem Formulation
The spatial equilibrium problem is formulated herein as a mixed quadratic programming problem. The quadratic variables arise from the price determination portion of the model, involving both supply and demand markets. The linear programming variables arise from the transportation subproblem. The overall problem with known supply and demand schedules for multiple commodities is presented algebraically below. Maximize E[c. + 0.5E d. x ik diLkx L ik
ik
- E[e. + 0.5E fi ij L
for some i,k
for some i,j
for all i, k
for all i,
LLL ijk Z ijk
C
AX
>b
where the surplus variables from (1) are complementary with X and the surplus variables from (2) are complementary with r. Converting our spatial equilibrium master problem to matrix form it becomes:
(1) (2)
101
Maximize
CX + 0.5X'DX -eY- 0.5Y'FY +
II
LL L -
L Q" < Obj + RX° - SY° q —
X is defined such that elements 1 through N are those for commodity 1, N+1, 2N are those for commodity 2, etc. Y is defined similarly such that 1 through M are those for commodity 1, M+1 through 2M for commodity 2, etc. C, D, e, F, UU, LL, UL are correspondingly defined supply and demand parameters. is a matrix of ones and zeros which forms the transportation problem involving X for a particular commodity when premultiplying X. Gy is a matrix like Gx which sums Y's when premultiplying Lq
is
a row vector of ones of the size of the number of
Benders cuts. vectors of shadow prices from the subproblem ordered are X and Y Y° are the lagged X and Y solutions which were imposed in transport subproblem.
102
Obj is the sum of the objective function value for all the transport subproblems. Q u are two positive variables the difference of which equals Q and are the new benders variable.
Rewriting the problem into the form as required by QMin we get:
Minimize
-CX - 0.5X'DX + eY + 0.5Y'FY - Q' -GxX
Qli > 0
+ G Y
> -UU
-X
> LL
X
7 U >L -RY
+ SY
- I_ (1n , 1 + I_or
> -Obj - Rx° + SY°
The complementary tableau that results is structured as in Figure 2 which shows how the data are entered into the QP algorithm. However, fixed supply or demand columns are simply adjustments on the subproblem right hand side. The quadratic programming problem is solved by the QMIN system as developed by Polito and listed above. A couple of small comments are in order regarding this. First, Memory Size: In establishing and solving QP's the user must set the values of T, KT, FACTOR, XVAL, and DVAL at Pr o p er levels. The user needs to change the T and KT along with MAXCOM. These are currently set at 2,500 but may need to be larger for larger problems. Similarly, the VALS, XVAL, and DVAL arrays may need to be larger than they currently are (100).
FIGURE 2 QP Tableau
Point 1
Point
2
Bender Variable
Supply Activities
Demand Activities 4,‘Poiht N
Point 1
Point 2
...Point M
Q'
Q"
Supply-Demand Balances
I
Demand Activities
Bounds on Supply and Demand
+1-1
Nero .
I
Supply Activities
I
F
Cuts
RHS
R
> c
+1-1
I
D
Benders
_
.
+1-1 +1-1
-S
> -e
1
11111
>
-1-1-1-1-]
> -1
Bender Variables
Supply-Demand Balances
-I
_
>
• •
. .
0
> +LL 5- -UU _1 Bounds,on Supply and Demand
1
> _
LL > -UU —
-1 1 -1 1 .
Benders Cuts
-R
S
-1 -1 -1 -1
1 1 1 1
> -Obj -RX* + SY*
104 APPENDIX C Detailed Example Problem Input and Output
22 2 Sil S12 521 S22
all
012 021 022
11 21 12 22
0 5 3 0
.1111 0110 1. 1. 2.
.5
.5 3. 50. 53. 55.
1. .4 —1.
55. 00 00 00 0 0
200 0 0 0 0 i 0 Q0 2 1 5 00 12. /0 0 0 22 0 20 200 0 0 0. 0
-.5 -1.
.5
2.
10. 10.
1.3 —.5
-2.
10.
10, 10. 10.
105
RUN OF BENDERS DECOMPOSITION SPATIAL EQUILIBRIUM :COE NUMBER OF SUPPLY POINTS 2 NUMBER OF DEMAND POINTS 2 NUMBER OF COMMODITIES 2 .1000000 SOLUTION TOLERANCE READING SUPPLY INFORMATION
POINT I COMMODITY i CURVE NAME S11 0.00 LOWER BOUND STARTING POINT 10.00 UPPER BOUND 1.00 1.00 SLOPES CURVE PARAMETERS INTERCEPT
0.00 .5u
POINT i COMMODITY 2 CURVE NAME S12 13.01 UPPER BOUND STARTING POINT 0.00 LOWER BOUND .50 2.00 SLOPES CURVE PARAMETERS INTERCEPT
0.04 2.00
POINT
2 COMMODITY
STARTING POINT CURVE PARAMETERS POINT
1 CURVE NAME S21 10000 UPPER BOUND
INTERCEPT
0.00. .40
2-CURVE NAME S22
2 COMMODITY
STARTING POINT CURVE PARAMETERS
0.00 LOWER BOUND 1.00
.50 SLOPES
0.00 0.00 LOWER BOUND18.00 UPPER BOUND .40 3.00 SLOPES V.30 INTERCEPT
READING DEMAND INFORMATION POINT 1 COMMODITY 1 CURVE NAME (311. 0.00 LOWER BOUND STARTING POINT 10.00 UPPER BOUND 50.00 SLOPES •1*-00 INTERCEPT CURVE PARAMETERS
0.013 •.50
POINT 2 CURVE NAME 012 1 COMMODITY 0.00 LOWER BOUND STARTING POINT 10.00 UPPER BOUND -.50 53.00 SLOPES CURVE PARAMETERS INTERCEPT
0.00 -2.00
POINT
2 COMMODITY
STARTING POINT CURVE PARAMETERS POINT
1 CURVE NAME 021
0.00 LOWER BOUND 10.00 UPPER BOUND -1.00 INTERCEPT 55.00 SLOPES
2 COMMODITY
STARTING POINT CURVE PARAMETERS
2 CURVE NAME 022
0.00 LOWER BOUND 13.03 UPPER BOUND 0.00 INTERCEPT 55.00 SLO P ES
READING TRANSPORT DATA COMMODITY ORIGIN 1 1 2 1 11 1 2 1 1 2 2 2 2 1. 2 2
0.00 0.00
COST DESTINATION 0.00 1. 1 5.00 3.00 2 0.00 2 1 0.00 5.00 i 10.00 2 2 0.00
LOWER LIM UPPER LIM 0.00 0.00 0.0J 0.01 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 G.00 G.00 0.116 0.00
0.00
-1J.00
106.
..0PSET
** GMT
2 CONSTRAINTS 10 VARIABLES ZERO TOLERANCE= 0.1500000E•05 2500 BLANK COMMON IS RECWIREO LENGTH OF BLANK COMMON IS
..QPSET
MATRIX ENTRIES FOR COLUMN •1.00000e 1 VALUE ROW
..QPSET
MATRIX ENTRIES FOR COLUMN 1 VALUE= —.5000000 ROW -2.000000 2 VALUE ROW
..QPSET
MATRIX ENTRIES FOR COLUMN ROW 3 VALUE = •1.000000
..QPSET
MATRIX ENTRIES FOR COLUMN 4 VALUE= —1.000000 ROW
..QPSET
MATRIX ENTRIES FOR COLUMN 5 VALUE= •1.000000 ROW
..QPSET
MATRIX ENTRIES FOR COLUMN 5 VALUE= —.5000000 ROW B. VALUE = —2.000000 ROW
..QPSET
MATRIX ENTRIES FOR COLUMN 7 VALUE= - 1.000000 ROW
0.01'5E7
MATRIX ROW ROW
..QPSET
..QPSET
ENTRIES FOR COLUMN 7 VALUE= —.4000000 •16300000 8 VALUE =
RHS ENTRIES VALUE= 1 ROW 2 VALUE= ROW VALUE 3 ROW VALUE= * ROW 5 VALUE= ROW VALUE= 6 ROW VALUE= 7 ROW VALUE = ROW '8 ROW 9 VALUE= VALUE = 110 ROW MATRIX ROW ROW ROW ROW
7
•50.00000 •53.00000 •55.00000 - 55.00000 1.000000 2.000000 .5000000 3.000000 -1.000000
tl.occaoc
ENTRIES FOR COLUMN 11 1 VALUE= •1.000000 3 VALUE= •1.000000 5 VALUE 1.000000 7 VALUE= 1.000000
174
107
..QPSET
..QPSET
MAT R IX ROW ROW ROW ROW
ENTRIES FOR COLUMN L2 2 VALUE ....1.000000 •1.000000 4 VALUE 1.031006 6 VALUE= 8 VALVE = 14003000
RHS ENTRIES
SOLVE LPS FOR EA H COMMODITY AT ITERATIO4 1 COST IS LP SOLUTION FOR COMMODITY DUAL VARIABLES -SUPPLY BY POINT
0.000
owilaa
DUAL VARIABLES DEMAND BY POINT
0 0.00
a.acio 0.000
SUPPLY PT DEMAND PT QUANITY COST LOW LIM UP LIM COMMCDITY ""**** 0.00 0.00 10.00 1 1 4104.***411.• 0.00 5.00 0.00 2 V 1 41.0** *441* 0.00 3.00 0.03 1 ******** 0.00 0.00 10.06 2 2 1 0.00 OMMODITY 2 COST IS LP SOLUTION FOR DUAL VARIABLES ... SUPPLY BY POINT DUAL VARIABLES-DEMAND BY POINT
0.003 0.000
0.000
0.000
SUPPLY PT DEMAND PT QUANITY COST LOW LIM UP LIM 0.00 4" ► "*" 0.00 V 1'a.00 11 41.4.4464%*** 0.00 5.00 2 1. 0.00 **M.*** a.00 0.00 10.00 2 2 1 .**4.4411-41.,4 0.00 0.00 2 2 10.00 2 0.00 THE TOTAL LP OBJE:TIVE FUNCTION FOR ALL COMMODITIES IS 1410.0000 0 IS NEW TRIAL LOWER BOUND AT ITERATION IS BETTER THAN CLD ....1000000E+25 1410.000 NEW LOWER BOUND
:OMMCDITY 2
THIS IS THE BEST SOLUTION FOUND ADO CUT • RHS.COEF 0.000 0.000 ADD CUT • RHS,COEF 1.000
SO FAR-STORE II 0.000 0.000 0.000 0.000
0. 0 0 0
COEFFICIENTS FOR ADDED ROU ARE 0. O. -1.000000 O. . RIGHT HAND SIDE IS SOLVE THE OP AT ITERATION 0 -RROW O. O.
O. 1400000
0.
O.
Z 1 THRU Z 10 ARE PRIMAL VARIABLES Z 11 THRU Z 13 ARE DUAL VARIABLES Z 14 THRU Z 23 ARE U MULTIPLIERS Z 24 THRU Z 26 ARE PRIMAL SLACKS CD CO
TABLEAU 0 Z 14 Z 15 Z 16 Z 17 Z 18 Z 19 Z 20 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26
BHAT 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0 -50.00 -53.00 -55.00 -55.00 1.000 2.000 .5000 3.000 -1.000 1.000 O. O.
Z 1 -1.000 -.5000 O. O. 0. O. O. O. O. O. 1.000 O.
'
Z 2 -.5000 -2,000 O. O. O. O. O. O. O. O. O. 1.000 O.
Z
3
O. O. -1.000 O. O. O. O. O. O. O. 1.000 O. O.
Z
4
O. O. O. -1.000 O. Q. O. O. O. O. O. 1.000 0.
5
6
O. Q. O. O. -1.000 -.5000 O. O. O. O. -1.000 O. O.
O. O. O. O. -.5000 -2.000 O. O. O. O. O. -1.000 O.
Z
O. O. 0. O. O. O. -1.000 -.4000 O. O. -1.000 O. O.
o.
o.
0.
0. 0.
0.
0.
2 11 -1.000 0.
0. O. 0.
0.
-1,000
0. O.
0.
O. O. 0. O. O. 0.
0.
0. 1.000
0. 1.000
0. -1.000
Z 10
14 15 16 17 18 19 20 21 22 23 24 25 26
O. 0. O.
-.4000 -1.300 0.
0. O. -1.000 O.
G. O.
O. 0. 0.
1.000
Z 13
Z 12 0.
O.
-1400
0.
0. -1.000
. 0.
0. 1.000 O.
0, O. 0. O.
0. o.
O.
o.
O.
O. -1.000 1.000
0. O. O.
O. O.
O. 0.
O.
0,
1.000
Z 3 ENTERS Z 16 LEAVES BASIS PIVOT ELEMENT-I( 3, 3) Z 4 ENTERS Z 17 LEAVES BASIS PIVOT ELEMENT-I( 4, 4) Z 11 ENTERS Z 24 LEAVES BASIS PIVOT ELEMENT-T( 11, 11) Z 12 ENTERS Z 25 LEAVES BASIS PIVOT ELEMENT-T( 12, 12) Z 7 ENTERS Z 20 LEAVES BASIS PIVOT ELEMENT-T( 7, 7) PIVOT ELEMENT-T( 6, 6) Z 6 ENTERS Z 19 LEAVES BASIS Z 8 ENTERS Z 21 LEAVES BASIS PIVOT ELEMENT-I( 8, 8) Z 2 ENTERS Z 15 LEAVES BASIS PIVOT ELEMENT-T( 2, 2) 5, 5) PIVOT ELEMENT-T( Z 5 ENTERS Z 18 LEAVES BASIS 1, 1) Z 1 ENTERS Z 14 LEAVES BASIS PIVOT ELEMENT-I( 9 ENTER, Z 26 AND Z 22 LEAVE BASIS BLOCK PIVOT Z 13 AND Z
PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT= PIVOT ELEMENT=
-1.000000 -1.000000 -1.000000 -1.000000 -2.000000 -3.000000 -1.88666?
-2.431095 -1.344840 -1.180694
0 1.0
TABLEAU Z Z Z Z Z Z Z Z Z Z Z Z Z
Z
12 1 2 3 4 5 6 7 8 9 23 11 12 13
1
Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 23 Z 11 Z 12 Z 13
NEW OP NEW OP NEW OP NEW OP
NAT 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Z 21 .1041 -.1682 .2002E-01 -.2843 -.1041 .1682 .2283 -.6207 0. O. -.2002E-01 .2843 0.
18.52 5.153 26.10 21.57 22.91
9.990 21.71 16.73 O. 0. 28.90 33.43 1.000
Z 15 .2033 -.4603 -.2683E-01 ,1810 .8238E-01 -.1111 .9411E-01 -.1682 0. 0, ,2683E-01 -.1810 0,
Z 16 .2547 -.2683E-01 -.7587 .7369E-01 -.2547 .2683E-01 -,2493 .2002E-01 O.
o.
Z 24 .2547 -.2683E-01 .2413 .7369E-01 -.2547 .2683E-01 -.2493 .2002E-01
Z 25 -.1682E-01. .1810 .7369E-01 .3536 .1632E-01 -.1810 .4005E-01 -.2843
-1.000
O.
O.
O. -.2413 -.7369E-01 O.
O.
14 -.8470 .2033 .2547 -.1682E-01 -.2959 .8238E-01 -.2964 .1041 O. . O. .2547 = .1682E-01 O.
Z 10
Z 22 O. O. O. O. 0. O. O. O. O. 1.000 O.
0.
0. 0. 0. 0. 0. 0.
OPTIMAL SOLUTION = -1858.376 SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT
2 1 2
0.
-.2413 -.7369E-0 0.
-.7369E-01 -.3536
Z 17 -.1682E-01 .1810 .7369E-01 -.6464 .1682E-01 -.1810 .4005E-01 -.2843 O. O. -.7369E-01 -,3536
Z 18 -.2959 .8238E-01 -.2547 .1682E-01 -.8470
.2033 .2964 -.1041 0, O. .2547 -.1682E-01
Z 19 .8238E-01 -.1111 .2683E-01 -.1810 .2033 -.4603 -.9411E-01 .1682 0.
Z 20 -.2964 .9411 E-01 -.2493 .4005E-01 .2964 -.9411E-01 -.13420 .2283 0.
0.
0.
-.2683E-01 .1810
.2493 -.4005E-01 0.
O.
O.
Z 26 0. 0. 0. 0. 0. 0.
0. 0. 1.000 0. 0.
O.
0. 0.
18.5209 26.0974 22.9077 21.7106
5.1531 21.5667 9.9897 16.7301
111
1853.3755 1 IS NEW UPPER BOUND AT ITERATION QP SOLUTIION OBJ IS •058.376 Z— a. BENDER VARIABLE a+ a. 1858.376 THE QP PORTION OF THE OBJECTIVE EQUALS SOLVE LPS FOR EACH COMMODITY AT ITERATION 1 COST IS LP SOLU T IO N FOR COMMODITY DUAL VARIABLES •SUPPLY BY POINT DUAL VARIASLES*DEMANO BY POINT
0.000 0.000
RESULTING IN Z =
0.
1; •13.16 3.0003.000
COMMODITY SUPPLY PT DEMAND PT QUANITY COST LOW LIM UP LIM ***44•0** 0.00 0.00 18-.5a
2 1 LP SOLUTION FOR COMMODITY
2 a
0.00
5.30
0 . 00
********
4.39 21.71 2 COST IS
3.00 0.00 *48.37
0.00 0.00
"4-****1' *** ►****
DUAL VARIABLES -SUPPLY BY POINT DUAL VARIABLES -DEMAND BY POINT
0:003 0. 000
10.000 111.000
UP LIM LOW LIM COST COMMODITY SUPPLY PT DEMAND PT QUANITY *******.* 0,010 0.00 5.15 2 ""*** 0.00 5.00 0.00 1 2 2 "*"*" 0.00 10.03 4.842 1 2 0.00 "4""" 0.00 16.73 THE TOTAL LP OBJECTIVE FUNCTION FOR ALL COMMODITIES IS 1;796.8489 it IS NEW TRIAL LOWER 33UNO AT ITERATION 1410.000 IS BETTER THAN CLO 1796.841 NEW LOWER BOUND THIS IS THE BEST SOLUTION FOUND SO FAR-STORE IT 0.000 0.000 ADO CUT • RHS,COEF 3.000 0.000 0.300 v.aaa ADO CUT **-RHS,COEF
-3.000 10.000
*421.53
•10.000 4'1.000
..OPROW COEFFICIENTS FOR ADDED ROW ARE O. O. -3.000000 3.000000 10.00000 -1.000000 RIGHT HAND SIDE IS -.1295835E-04
-10.00000 1.000000
..OPROW THIS ROW WILL BE DIVIDED BY 10,**( SOLVE THE OP AT ITERATION 1
0.
0.
-2)
..0MIN
Z 1 THRU Z 10 ARE PRIMAL VARIABLES Z 11 THRU Z 14 ARE DUAL VARIABLES Z 15 THRU Z 24 ARE U MULTIPLIERS Z 25 THRU Z 28 ARE PRIMAL SLACKS
TABLEAU Z Z Z Z Z Z
BHAT 1 2 3 4 5 6 7
8 9 24 11 12 13 28
1 .000 1 .000
.000 .000 1.'000 .000 .000 .000 1.000 1.000 .000 1.000 1.000 1.000 1
0 18.52 5.153 26.10 21.57 22.91 9.990 21.71 16.73 0.
Z 15 -.8470 .2033 .2547 -.1682E-01 -.2959 .8238E-01 -.2964 .1041
O.
O.
28.90 33.43 1.000 -6153.
-.2547 .1682E-01
O.
O.
-44.37
Z 16 .2033 -.4603 -,2683E-01 .1810 .8238E-01 -.1111 .9411E-01 -.1682 O. O. .2683E-01 -.1810 O. -312.9
Z 17 .2547 -.2683E-01 -.7587 .7369E-01 -.2547 .2683E-01 -.2493 .2002E-01
Z•18 -.1682E-0) .1810 .7369E-01 -.6464 .1682E-01 -.1810 .4005E-01 -.2843
Z 19 -.2959 .8238E-01 -.2547 .1682E-01 -.8470 .2033 .2964 -.1041 O.
O.
O.
O.
O.
-,2413 -.7369E-01 O. 99.16
-.7369E-01 -.3536 O. 351.9
.
O. .2547 -.1682E-01 O. 44.37
Z2 .8238E-01 -.1111 .2683E-01 -.1810 .2033 -.4603 -.9411E-01 .1682 O.
Z 21 -.2964 .9411E-01 -.2493 .4005E-01 .2964 -.9411E-01 -.0420 .2283
O.
O.
-.2683E-01 .1810 O. 312.9
.2493 -.4005E-01 O. 10..41
O.
Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 I 8 Z 9 124 Z 11 Z 12 Z 13 Z 28
22 Z 23 .1041 0. -.1692 O. .2002E-01 O. O. -.2843 -.1041 O. .1682 O. .2283 O. -.6207 O. O. O. O. 1.000 -.2002E-01 O. .2843 O. O. -1.000 -273.9 O. PIVOT ELEMENT-T( 14, 14)
110
O. O. O. O. O. -1.000 0.
Z 25 .2547 -.2683E-01 .2413 .7369E-01 -.2547 .2683E-01 -.2493 .2002E-01
2 26 -.1602E-01 .1810 .7369E-01 .3536 .1682E-01 -.1810 .4005E-01 -,2043
O. O.
O.
0.
-.2413
O. O.
0.
O.
-.7369E-01 -.7369E-01 -.3536 O. O. O. O. -200.8 -648.1 Z 14 ENTERS 2 28 LEAVES BASIS O.
Z 27 O. O.
0. O. O. O. O. O. 1.000 O. O. 0, O. -100.0
Z 14 -44.37 -312.9 99.16 351.9
44.37 312.9
10.41 -273.9 O. 0. 200.8 648.1 100.0 -.6525E+06 PIVOT ELEMENT= -652495.0
TABLEAU
Z Z Z Z Z
1 1 2 3 4 5 6 7 8 9 24 11 12 13 14
Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 24 Z 11 Z 12 Z 13 Z 14
NEW OP JEW OP NEW OP NEW OP
SHAT 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Z 22 .1228 -.3683E-01 -.2160E-01 -.4321 -.1228 .3683E-01 .2239 -.5057 O.
O. -.1043 .1228E-01 -.4198E-01 .4198E-03
18.94 8.104 25.16 18.25 22.49 7.039 21.61 19.31 O. O. 27.01 27,32 .5706E-01 .9429E-02
7 15 -.8439 .2246 .2480 -.4075E-01 -.2989 .6110E-01 -.2971 .1228 0. 0. -.2684 -.2725E-01 -.6801E-02 .6801E-04
Z 16 .2246 -.3102 -.7439E-01 .1223E-01 .6110E-01 -.2612 .8912E-01 -.3683E-01 0. 0. -.6949E-01 -.4918 -.4796E-01 .4796E-03
Z 17 .2480 -.7439E-01 -.7436 .1272 -.2480 .7439E-01 -,2477 -.2160f-Q1 0. 0. -.2108 .2480E-01 0520E-01 -.1520E-03
Z 18 -.4075E-01 .1223E-01 .1272 ' -.4565 .4075E-01 -.1223E-01 .4566E-01 -.4321 0.
Z 19 -.2989 .6110E-01 -.2480 .4075E-01 -.8439 .2246 .2971 -0220
0.
0. .2684 .2725E-01 .6801E-02 -.6801E-04
Z 23 O. O. O. O. O. 0. 0. O. O. 1.000 0. 0. -1.000
Z 10 0. 0. 0. 0. 0. 0. 0. 0. -1.000 0. 0. 0. 0.
Z 25 .2684 .6949E-01 .2108 -.3464E-01 -.2684 -.6949E-01 -.2525 .1043 0. 0. -.3031 -.2732 -.3078E-01 .3078E-03
Z 26 .2725E-01 .4918 -.2480E-01 .4075E-02 -.2725E-01 -.4918 .2971E-01 -.1228E-01 0. 0. -.2732 -.9973 -.9932E-01 .9932E-03
Z 27 .6801E-02 .4796E-01 7,1520E-01 -.5394E-01 -.6801E-02 -.4796E-01 -.1596E-02 .4198E-01 1.000
0.
OPTIMAL SOLUTION = -1829.367 SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT
1 2 1 2
18.9393 25.1624 22.4893 21.6124
.3464E-01 -.4075E-02 .5394E-01 -.5394E-03
0.
-.3078E-01 -.9932E-01 -.1533E-01 .1533E-03
8.1039 18.2481 7.0389 19.3131
Z 28 -.6801E-04 -.4796E-03 .1520E-03 .5394E-03 .6801E-04 .4796E-03 .1596E-04 -.4198E-03. 0. 0. .3078E-03 .9932E-03 .1533E-03 -.1533E-05
Z 20 .6110E-01 -.2612. .7439E-01 -.1223E-01 .2246 -.3102 -.8912E-01 .3683E-01 O. O. .6949E-01 .4918 .4796E-01 -.4796E-03
x21 -.2971 .8912E-01 -.2477 .4566E-01 .2971 -.8912E-01 -.8418 .2239 O. 0.
.2525 -.2971E-01 .1596E-02 -.1596E-04
115
NEW UPPER BOUND AT ITERATION 2 IS QP SOLUTIION OBJ IS ...1829.367 BENDER VARIABLE a* 0. Z- 0. THE QP PORTION OF THE OBJECTIVE EQUALS
182 9.3 674 RZSULTING IN Z 1829.367
SOLVE LPS FOR EACH COMMODITY AT ITERATION LP SOLUTION FOR COMMODITY 1 COST IS DUAL VARIABLES •SUPPLY BY POINT DUAL VARIABLES DEMAND BY POINT
2
0,000
0.000
3.000 3.000
COMMODITY SUPPLY PT DEMAND PT QUANITY COST LOW LIM UP LIM **** ** 44 18.94 0.00 0.00 1 44441. 414P4.41 1 a 0.00 0.00 5.00 444 444 44 1 2 3.55 0.00 3.00 1 2 2 21.01. 0.00 0.00 #4444444 LP SOLUTION FOR COMMODITY -5.33 2 COST IS DUAL VARIABLES -SUPPLY BY POINT DUAL VARIABLES DEMAND BY POINT
5.003 5.000
0.000 0.000
COMMODITY SUPPLY PT DEMAND PT QUANITY COST LOW LIM UP LIM 2 7444p 0.00 0.00 """" 2 """" 5.00 0.00 1 1.07 2 2 0.40 0.00 10.00 """" * a a'"a" 2 2 0.00 18.25 0.40 THE TOTAL LP OBJECTIVE FUNCTION FOR ALL COMMODITIES IS NEW TRIAL LOWER BOUND AT ITERATION 2 IS 1813.3923 NEW LOWER BOUND 1813.392 1796.849 IS BETTE2 THAN CLO THIS IS THE BESt SOLUTION FOUND SO FAR-STORE IT ADD CUT - RHS,C3EF -.000 0.000 -5.000 0.000 5.000 3.000 ADD CUT RHS.COEF 1.000
,-3.000 0.000
-15.98
0.000 ...1.000
..OPROW COEFFICIENTS FOR ADDED ROW ARE -3.000000 -5.000000 O. -1.000000 3.000000 O. RIGHT HAND SIDE IS. -.6983487E-05
0.
5.000000
1.000000
THIS ROW WILL BE DIVIDED BY 10.* ( ..OPROW 2 SOLVE THE OP AT ITERATION
-3)
..OMIN
Z 1 THRU Z 10 ARE PRIMAL VARIABLES Z 11 THRU Z 15 ARE DUAL VARIABLES Z 16 THRU Z 25 ARE U MULTIPLIERS Z 26 THRU Z 30 ARE PRIMAL SLACKS
TABLEAU
0 2 3 4 5 6 7
8 9 25 11 12 13 14 30
BHAT 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1A00 1.000 1.000 1.000
18.94 8.104 25.16 18.25 22.49 7.039 21.61 19.31 O. 0,
27.01 27.32 .5706E-01 .9429E-02 -.1598E+05
Z 16 -.8439 .2246 .2480 -.4075E-01 -.2989 .6110E-01 -.2971 .1228
B. O. -.2684 -.2725E-01 -.6801E-02 .6801E-04 -2453.
Z 17 .2246 -.3102 -.7439E-01 .1223E-01 .6110E-01 -.2612 .8912E-01 -.3683E-01 O. O.
-.6949E-01 -.4918 -.4796E-01 .4796E-03 735.8
Z 18 .2480 -.7439E-01 -.7436 .1272 -.2480 .7439E-01 -.2477 -.2160E-01 O.
0. -.2108 .2480E-01 .1520E-01 -.1520E-03 2232.
Z 19 -.4075E-01 .1223E-01 .1272 -.4565 .4075E-01 -.1223E-01 .4566E-01 -.4321 O. O. .3464E-01 -.4075E-02 .5394E-01 -.5394E-03 -366.8
Z 2Q -.2989 .6110E-01 -.2480 .4075E-01 -.8439 .2246 .2971 -.1228 O. O. .2684 .2725E-01 .6801E-02 -.6801E-04 2453.
Z 21 .6110E-01 -.2612 .7439E-01 -.1223E-01 .2246 -.3102 -.8912E-01 .3683E-01 O.
0. .6949E-01 .4918 .4796E-01 -.4796E-03 -735.8
Z 22 -.2971 .8912E-01 -.2477 .4566E-01 .2971 -.8912E-01 -.8418 .2239 O. 0. .2525 -.2971E-01 .1596E-02 -.1596E-04 -2674.
1 2 3 4 5
Z Z Z Z Z Z 7 Z 8 Z 9 Z 25 Z 11 Z 12 Z 13 14 30
Z 24 Z 23 .1228 0. -.3683E-01 0. -.2160E-01 0. 0. -.4321 -.1228 0. .3683E-01 0. .2239 0. 0. -.5057 O. o. O. 1.000 -.1043 O. .1228E-01 0. -.4198E-01 -1.000 .4198E-03 O. 1105. O. PIVOT ELEMENT-T( 15, 15) PIVOT ELEMENT-I( 13, 13) PIVOT ELEMENT-T( 9, 9) PIVOT ELEMENT-T( 10, 10)
Z 26 .2684 .6949E-01 .2108 -.3464E-01 -.2684 -.6949E-01 -.2525 .1043
Z 27 .2725E-01 .4918 -.2480E-01 0. .4075E-42 0. -.2725E-01 0. -.4918 0. .2971E-01 0. -.1228E-01 O. -1.000 O. O. 0, 0, -.3031 O. -.2732 0. -.2732 -.9973 0. -.3078E-01 -.9932E-01 0. ,3078E-03 .1932E-03 -2085. -4755. 0. 15 ENTERS Z 30 LEAVES BASIS Z 28 ENTERS Z 13 LEAVES BASIS Z 24 ENTERS Z 9 LEAVES BASIS Z 10 ENTERS Z 25 LEAVES BASIS Z 10
0. 0. o.
Z 28 Z 29 Z 15 .6801E-02 -.6801E-04 -2453. .4796E-01 -.4796E-03 735.8 -.1520E-01 .1520E-03 2232. -,5394E-01 .5394E-03 -366.8 -.6801E-02 .6801E-04 2453. -.4796E-01 .4796E-03 -735.8 -,1596E-02 .1596E-04 -2674. .4198E-01 -.4198E-03 1105. 1.000 0. 0. O. O. O. -.3078E-01 2085. .3078E-03 -.9932E-01 .9932E-03 4755. -.1533E-01 .1533E-03 1439. .1533E-03 -.1533E-05 -4.388 -1439. 4.38p -.2207E+08 PIVOT ELEMENT= -.2207355E+08 PIVOT ELEMENT= -.1091091 PIVOT ELEMENT= -9.165138 PIVOT ELEMENT= -.1091091
TABLEAU Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
Z Z Z Z Z Z Z Z Z Z Z Z Z Z
NEW NEW NEW NEW
OP OP OP OP
4 1 2 3 4 5 6 7 8 24 10 11 12 28 14 15
2 3 4 5 6 7 8 24 10 11 12 28 14 15
BHT 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Z 23 .4587E-01 0. .4587E-01 -.4587 -.4587E-01 0. .1376 -.4587 0. -.2752 -.4587E-01 .1376 -.2752 .3211E-03 -.3211E-04
U 19.21 7.571 25.00 18.78 22.22 7.571 21.99 18.78 O.
9.021 27.00 27.57 9.021 .8643E-02 .1357E-03
O. 0. 0. 0. O. 0. 0. O. O. -1.000 0. 0. O. O. O.
Z 16 -.8260 .1429 .2454 .4587E701 -.3168 .1429 -.2638 .4587E-01
Z 17 .1429 -.2857 O. O. .1429 -.2857
O.
O. O. O. -.3333
1.528 -.2454 .6957E-01 1.528 -.1154E-03 .1154E-04
Z 25 1.528 0. -1.472 -.2752 -1.528 O. 1.583 -.2752 1.000 -9.165 -1.528 -3.751 -9.165 .4026E-02 .5974E-03
OPTIMAL SOLUTION= -1828.026 SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES DEMANDS BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT SOLUTION GIVES SUPPLY BY COMMODITY AT PT
O. O.
O.
.3333E-03 -.3333E-04
Z 26 .2454 O. .2454
.4587E-01 -.2454 O. -.2638 .4587E-01 O. 1.528 -.2454 -.9709E-01 1.528 .5122E-04 -.5122E-05
1 2 1 2
Z 18 .2454 O. -.7546 .4587E-01 -.2454 O. -.2638 .4587E-01 O. -1.472 -.2454 -.9709E-01 -1.472 .5122E-04 -.5122E-05
Z 27 -.6957E-01 .3333 .9709E-01 .1957 .6957E-01 -.3333 -.4205E-01 -.1376 O. 3.751 -.9709E-01 -.4865 3.751 .2908E-03 -.2908E-04
19.2108 24.9966 22.2177 21.9897
Z 19 .4587E-01 O.
.4587E-01 -.4587 -.4587E-01 O.
.1376 -.4587 O.
-.2752 -.4587E-01 -.1957 -.2752 -.3456E-03 .3456F.04
Z 13 1.528 0, -1.472 -.2752 -1.528 O.
1.583 -.2752 O.
-9.165 -1.528 -3.751 -9.165 .4026E-02 .5974E-03
7.5714 18.7844 7.5714 18.7844
Z2 -.3168 .1429 -.2454 -.4587E-01 -.8260 .1429 .2638 -.4587E-01 0, -1.528 .2454 -.6957E-01 -1.528 ,1154E-03 -.1154E-04
Z 21 .1429 -.2857 O.
Z 29 .1154E-03 -.3333E-03 -.5122E-04 .3456E-03 -.1154E-03 .3333E-03 .1797E-03 -.3211E-03 0. -.4026E-02 .5122E-04 .2908E-03 -.4026E-02 -.6363E-06 .6363E-07
Z 30 -.1154E-04 .3333E-04 .5122E-05 -.3456E-04 .1154E-04 -.3333E-04 -.1797E-04 .3211E-04 0. -.5974E-03 -.5122E-05 -.2908E-04 -.5974E-03 .6363E-07 -.6363E-08
O.
.1429 -.2857 O.
O. O. O.
O. .3333 O. -.3333E-03 .3333E-04
Z 22 -.2638 O. -.2638 .1376 .2638 O. -.7913 .1376 O. 1.583 .2638 .4205E-01 1.583 -.1797E-03 .1797E-04
119
NEW UPPER BOUND AT ITERATION 1828.0 259 3 IS 212 SOLUTIION OBJ IS ...1828.026 BENDER VARIABLE 0. Z- 9. 020 633 THE QP PORTION 0= THE OBJECTIVE EQUALS 1837.046
RESULTING IN Z =
9:0206
SOLVE LPS FOR EACH COMMODITY AT ITERATION i COST IS LP SOLUTION FOR COMMODITY DUAL VARIABLES .. SJPPLY BY POINT
0.000
DUAL VARIABLES-DEMAND BY POINT
0.000
3.000
COMMODITY SUPPLY PT DEMAND PT QUANITY COST LOW 0.00 1 19.2L 1 1 V 5.00 1 2 0.00 3.GL 3.00 1 1 2 0.00 1 2 2 21.99 LP SOLUTION FOR :OMMOOITY a COST IS 0.00 DUAL VARIABLES -SUPPLY BY POINT DUAL VARIABLES ... OEMAND
BY POINT
0.000 0.003
-9
LIM UP LIM 0 .00 ******** 0.00 4.4.44.4.*** 0.00 ******** 0.00 44.444***
0.000 0.000
COMMODITY SUPPLY PT DEMAND PT QUANITY :3ST LOW LIM UP LIM 2 7.5r 0.00 0.00 44.•44*** 5.00 0.00 44.444.*** 2 0.03 0.00 4444.44.** 0.00 10.04 2 18.73 0.00 *4.44.4.*** 2 2 0.00 THE TOTAL LP OBJECTIVE FUNCTION FOR ALL COMMODITIES IS 11828.0259 NEW TRIAL LOWER BOUND AT ITERATION 3 IS. 1813.392 NEW LOWER BOUND 1828.026 IS BETTER THAN OLD
120
THIS IS THE BEST SOLUTION FOUND SO FAR-STOR= IT THE BEST SOLUTION c OUND IS REPORT20 HERE IT SH-CULO BE WITHIN THE DIFFERENCE OF THE THE OPTIMAL OBJECTIVE FUNCTION OBJECTIVE FUNCTION VALUE
BOUNDS OF
1828.4.126
SUPPLY SUPPLY SUPPLY SUPPLY SUPPLY
PART OF SOLUTION FROM Si: GOOD a FROM S12 GOOD 1 FROM S21 GOOD GOOD 2 =ROM 322
AT AT AT AT
1 1, 2 2
IS IS IS 13
22.21773 7.571429 21.98958 18.78440
PRICE PRICE PRICE PRICE
27.00344 28.25172 30.00344 36.21560
DEMAND DEMAND JEMAND DEMAND DEMAND
FART OF SOLUTION GOOD 1 FROM Di1 GOOD 2 FROM D12 1 FROM 021 GOOD GOOD 2 FROM 022
AT AT AT AT
t 1 2 2
IS /S IS IS
19.21085 7.571428 24.9956 18.78440
PRICE PRICE PRICE PRICE
27.00544 28.25172 30.00344 36.21550
TRANSPORT FLOWS COMMODITY ORIGON 1 1 1 2 2 2 2
2 1 7
a 1' 2
DESTINATION 1 2 1 1 2 2
FLO4 19.2008 0.0000 3.0059 21.9837 7.5714 0.0000 0.0000 18.7844
121
APPENDI X D
Changing Problem Sizes
In changing problem sizes one must: 1.
Set up the proper size of QMIN as discussed in Appendix B.
2.
Make sure the NETFLO arrays ARC and NODE are proper where they have been enlarged to accommodate the largest problem generated by one commodity.
3.
Adjust the program arrays as defined in Appendix A for number of suppliers, commodities, etc.
122
APPENDIX E
Transport Code Stop Conditions
The transport code NETFLO may abort. The following gives the reasons.
Stop
Meaning
4
Too many nodes; expand dimensions if possible.
7
Requirements data fora particular node have been specified on a previous data card.
8
Infeasibility; net requirements for the network are negative.
10
Too many arcs; expand dimensions if possible.
12
Node numbers on arc specification card too large or negative or zero.
13
Upper or lower bounds on arc flow improper.
14
Too many arcs; expand dimensions if possible.
15
Too many arcs; expand dimensions if possible.
16
Infeasibility; net requirements induced by lower bounds on arc flow are negative.
20
Artificials in basis.
123 APPENDIX F Sample Problem Output RUN OF BENDERS DECOMPOSITION SPATIAL EQUILIBRIUM CODE NUMBER OF SUPPLY POINTS 2 2 NUMBER OF DEMAND POINTS 2 NUMBER OF COMMODITIES SOLUTION TOLERANCE .1000000 READING SUPPLY INFORMATION POINT 1 COMMODITY i CURVE NAME 511 0.00 LOWER BOUND STARTING POINT 10.00 UPPER BOUND 1.00 1.00 SLOPES CURVE PARAMETERS INTERCEPT 1 COMMODITY POINT STARTING POINT
CURVE PARAMETERS POINT
2 COMMODITY
2 CURVE NAME S12 10.00 UPPER BOUND
INTERCEPT
0.00 LOWER BOUND
2.00 SLOPES
.50
0.00
.50 0.0a 2.00
1 CURVE NAME S21
0.00 LOWER BOUND 10.00 UPPER BOUND 1.00 .50 SLOPES INTERCEPT
0.00 .40
2 COMMODITY 2 CURVE NAME S22 10.00 UPPER BOUND 0.00 LOWER BOUND STARTING POINT 3.00 SLOPES CURVE PARAMETERS INTERCEPT .40
0.00 1.30
STARTING POINT CURVE PARAMETERS
POINT
READING DEMAND INFORMATION POINT
1 COMMODITY
STARTING POINT CURVE PARAMETERS POINT
1 COMMODITY
STARTING POINT CURVE PARAMETERS
1 CURVE NAME 011 0.80 LOWER BOUND 10.00 UPPER BOUND -1.00 INTERCEPT 50.00 SLOPES 2 CURVE NAME 012
10.00-UPPER- BOUND INTERCEPT
INTERCEPT
0.00 LOWER BOUND
53.00 SLOPES
2 C.OMMODITY POINT 1 CURVE NAME 021 STARTING POINT 10.00 UPPER BOUND CURVE PARAMETERS
0.00 -.50
-.50
0.00 LOWER BOUND
55.00 SLOPES
-1.00
POINT 2 COMMODITY 2 CURVE NAME 022 0.00 LOWER BOUND STARTING POINT 10.00 UPPER BOUND 0. 0 0 55.00 SLOPES CURVE PARAMETERS INTERCEPT
0.00 5.00 3.00 0.00 0.00 5.00 10.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00 3.00 0.03
0.00 -2.00
0.00 0.00
0.00 -1.00
124
1410.0400 NEW TRIAL LOWER BOUND AT ITERATION 0 IS NEW LOWER BOUND 1410.000 IS BETTER THAN OLD -.1000000E+25 1858.3755 NEW UPPER BOUND AT ITERATION 1 IS NEW TRIAL LOWER BOUND AT ITERATION 1796.8489 1 IS NEW LOWER BOUND 1796.849 IS BETTER THAN OLD 1410.000 NEW UPPER BOUND AT ITERATION 1829.3674 2 IS 1813.3923 NEW TRIAL LOWER BOUND AT ITERATION , 2 IS IS BETTER THAN OLD 1796.849 NEW LOWER BOUND 1813.392 NEW UPPER BOUND AT ITERATION 1828.0259 3 IS 1828.0259 NEW TRIAL LOWER BOUND AT ITERATION 3 IS NEW LOWER BOUND 1813.392 IS BETTER THAN OLD 1828.026 TERMINATION ACHIEVED AFTER 3 ITERATIONS FINAL UPPER BOUND FINAL LOWER BOUND IS 1828.026
1828.0 26
THE BEST SOLUTION FOUND IS REPORTED HERE IT SHOULD BE WITHIN THE DIFFERENCE OF THE BOUNDS OF THE OPTIMAL OBJECTIVE FUNCTION OBJECTIVE FUNCTION VALUE SUPPLY PART OF SOLUTION SUPPLY GOOD 1 FROM S11 SUPPLY GOOD 2 FROM S12 SUPPLY GOOD 1 FROM S21 SUPPLY GOOD 2 FROM S22 DEMAND DEMAND DEMAND DEMAND DEMAND
PART OF SOLUTION I. FROM Dil GOOD GOOD 2 FROM 012 1 FROM 021 GOOD GOOD 2 FROM D22
TRANSPORT FLOWS COMMODITY ORIGON 1 1 2 1 1 2 2 2 2 2 1
1828.026
AT AT
IS 1 IS 2 IS 2 IS
7.571429 21. 98968 1 8. 78440
PRICE PRICE PRICE PRICE
27.0 03,44. 28. 25172 33.0 0344 36.21560
AT AT AT AT
1 1 2 2
19.'21015 7.571428 24.99656 18.78440
PRICE PRICE PRICE PRICE
27.8 t344 28.25172 30.0 0344 36.21560
AT AT
DESTINATION 1 1 2 2 1 1 2
IS IS
IS IS
FLOW 19.2108 0.0000 3.0069 21.9897 7.5714 0.0000 0.0000 18.7844
22. 21773
