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Technology services;
Oil-field;
Resource management;
Transaction based technology ;
Associated systems;
Semiconductor test equipment;
2
Offices in over 100 countries;
Employs more than 50.000 people;
Had 8.75 billion revenues in 1999;
Leading power of electricity
gas
water meters
3
Electricity can not be stored;
o Monitor the demand (carefully)o Meticulously control production
Remain competitive;
Reduce cost;
Increase efficiency; Monitor the costumers’ use of energy
4
The meters
A receiver (pole)
5
Poles’ height
material
Surroundings’ buildings
rural
urbanIncrease in coverage area decrease
in battery life span6
1,2,3...m meter M;
1,2,3...p pole P;
Capacity limit K;
To minimize the costo Minimum number of the poles;o Every meter is assigned to only one pole;o Do not exceed capacity limit;
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POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
Greedy solution 1,3,4,5,6;
Optimal solution 2,3,4,8;
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Small problem 4208 meters 2393 poles;
Large problem 116 00 meters 20 631 poles
Geographic location of poles and meters;
The subset of poles that would cover all the meters was established manually;
The method is not reliable and scalable;
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Greedy procedure was implemented by MATLAB 171 poles;
Drawbacks of MATLAB procedure;
o Time consuming;
o High amount of data;
o Can’t address to capacity problem;
o Unable to answer what-if questions;
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Infinite capacity assumptionset covering problem;
Problem formulation;
Minimize ∑ Yj Yj 1, if pole j is selected
j=1 0,otherwise
subject to ∑ Yj ≥1 , for each i
{j I i € Cj}
The constraint ensures that every meter is covered by at least one pole;
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19
POLE Meters Covered
STEP 0 STEP 1
STEP 2 STEP 3 STEP 4 STEP 5
1 1,2,3 3 2 1* ― ― ―
2 2,3,9 3 2 1 0 0 0
3 5,6,7 3 2 1 1* ― ―
4 7,9,10 3 1 1 1* 1* ―
5 3,6,8 3 3* ― ― ― ―
6 1,4,7,9 4* ― ― ― ― ―
7 4,5,9 3 1 1 1 0 0
8 1,4,8 3 1 0 0 0 0
It took 2-3 days to complete MATLAB procedures;
Feasible combinations of meters and poles were sought in relying on data in MATLAB;
476 769 feasible meter-pole combination;
2393 binary variables,476 769 nonzero coefficient,4208 constraint;
In about 5 minutes CPLEX generated the result of 171 poles;
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116 600 meters,20 636 poles,1.3 million possible combination;
If meters covered by j1 is the subset of meters covered pole j2 then pole j1is excluded from the formulation;
If the set of poles covering meter i1was a subset of poles covering meter i2 meter i2 is
eliminated as it will already be covered by pole selected for i1;
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Pole(j) Meter
1 3,4
2 3,4,7
Meter(i) Pole
1 4,5
2 4,5,7
The size of formulation decreased by 50-60%;
After 3-4 hours computation time CPLEX yield the result of 1431 poles;
10-12 percent of receivers was assigned to 360 or more meters;
22
Assumption so far:
“Receivers’ coverage area is infinite”
In reel problem:
“ Each receiver can cover only a finite number of meters,
23
The small problem; - 4.208 meters
- 2.393 poles
- 2.393 binary variables,
- 4.208 constraints,
- 476.769 non-zero coeff.
24
The large problem; - 116.600 meters
- 20.636 poles
- 144.225 binary variables,
- 6.601 constraints,
- 953.538 non-zero coeff.
“K”: # of the closest meters for each pole.
Theoretical Upper Limit for “K” : 540 meters;
Lower capacity limit is more logical;
25
Each Cj the closest K;
Without capacity constraints;
Each meter to the nearest selected pole;
26
Heuristic Approach Method:
# of meters in Urban Areas
> # of meters in Rural Areas
(<100)
27
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No capacity limit
We want to increase meter load
We want to decrease number of poles
Only %6 increase
1501
Minimize ∑ Yj
subject to ∑ Xij <= KYj, for each j
{i I i € Cj}
∑ Xij >= 1, for each i
Xij € {0 , 1}, for each i and jiçin
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Yj
1, if pole j is selected
0,otherwise
Collect the coordinates;(Geographical Info System)
Determine the distances; (MATLAB) Determine feasible combinatons; (GAMS*) Solve the combinations by IP; Assign meters to the poles with capacity restriction; Plot the results on a map. (GIS); Perform the what-if analysis;
*The General Algebraic Modeling System (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations.
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Schlumberger benefited in three areas;
Save to timeo Early revenue
Save to plannero Save to labor force
Save to receivero Reliable datao Save to equipment and labor force
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Total save;
Organizational benefit;
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The integer-programming approach could be solve extensions to problem;
Different types of polesoEasy accessibility;oMultiple function use;oAssigning different weights and without a significant change;
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Different types of receivers
o N types of receiverso One type receivers on any pole
The objective here is minimize the total cost
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Thank You
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