Question

In: Statistics and Probability

Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power...

Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation's major residential markets, the annual demand in each market (in megawatts or MW), and the cost to supply electricity to each market from each power generation plant (in $/MW).

City Distribution Costs ($/MW) Demand (MW)
Los AngelesL TulsaT SeattleS
Seattle1 356.25 593.75 59.38 950.00
Portland2 356.25 593.75 178.13 831.25
San Francisco3 178.13 475.00 296.88 2,375.00
Boise4 356.25 475.00 296.88 593.75
Reno5 237.50 475.00 356.25 950.00
Bozeman6 415.63 415.63 296.88 593.75
Laramie7 356.25 415.63 356.25 1,187.50
Park City8 356.25 356.25 475.00 712.50
Flagstaff9 178.13 475.00 593.75 1,187.50
Durango10 356.25 296.88 593.75 1,543.75

(a)

If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem?

(i)

Which cities should be supplied by which power plants? (Enter your optimal solution as a comma-separated list of ordered triples for the amount of power each power plant supplies to each city. Report electrical power in MW.)

(L1, T1, S1), (L2, T2, S2), …, (L10, T10, S10) =  

  

(ii)

What is the total annual power distribution cost for this solution? (Round your answer to the nearest whole dollar.)

$

(b)

If at most 4,000 MW of power can be supplied by any one of the power plants, what is the optimal solution? (Enter your optimal solution as a comma-separated list of ordered triples for the amount of power each power plant supplies to each city. Report electrical power in MW. Round your answers to two decimal places.)

(L1, T1, S1), (L2, T2, S2), …, (L10, T10, S10) =  

  

What is the annual increase in power distribution cost that results from adding these constraints to the original formulation? (Round your answer to the nearest whole dollar.)

$

Solutions

Expert Solution

Given the following data

City Los Angeles(($/MW) Tulsa T($/MW) Seattle($/MW) Demand(MW)
Seattle 1 356.25 593.75 59.38 950
Portland 2 356.25 593.75 178.13 831.25
San Fransisco 178.13 475 296.88 2375
Boise 4 356.25 475 296.88 593.75
Reno5 237.5 475 356.25 950
Bozeman 6 415.63 415.63 296.88 593.75
Laramie 7 356.25 415.63 356.25 1187.5
Park City 8 356.25 356.25 475 712.5
Flagstaff 9 178.13 475 593.75 1187.5
Durango 10 356.25 296.88 593.75 1543.75

a)

If there are no restrictions on the amount of power that can be supplied by any of the power plants. Then The optimal solution will be based on the minimum cost of supply among the three power station.

i.e The station from where the cost of supply is least will be selected.

City Los Angeles(($/MW) Tulsa T($/MW) Seattle($/MW) Demand(MW) Minimum Cost Electricity supply($/MW)
Seattle 1 356.25 593.75 59.38 950 59.38
Portland 2 356.25 593.75 178.13 831.25 178.13
San Fransisco 178.13 475 296.88 2375 178.13
Boise 4 356.25 475 296.88 593.75 296.88
Reno5 237.5 475 356.25 950 237.5
Bozeman 6 415.63 415.63 296.88 593.75 296.88
Laramie 7 356.25 415.63 356.25 1187.5 356.25
Park City 8 356.25 356.25 475 712.5 356.25
Flagstaff 9 178.13 475 593.75 1187.5 178.13
Durango 10 356.25 296.88 593.75 1543.75 296.88

Hence the optimal solution

City Minimum Cost Electricity supply($/MW) Power Generation Plant
Seattle 1 59.38 SeattleS
Portland 2 178.13 SeattleS
San Fransisco 178.13 Los AngelesL
Boise 4 296.88 SeattleS
Reno5 237.5 Los AngelesL
Bozeman 6 296.88 SeattleS
Laramie 7 356.25 Los AngelesL/SeattleS
Park City 8 356.25 Los AngelesL/TulsaT
Flagstaff 9 178.13 Los AngelesL
Durango 10 296.88 TulsaT

i) City Supply paatern if the sypply is not a constraint

{(0,0,950),(0,0,831.25),(2375,0,0),(0,0,593.75),(950,0,0),(0,0,593.75),(593.75,0,593.75),(356.25,356.25,0),(1187.5,0,0),(0,1543.75,0)}

II) total annual power distribution cost for this solution= Cost of Electricity Supply * Demand

=950*59.38+831.25*178.13+......+1543.75*296.88

=$ 2552423

b)

If the power supply constraint is =4000MW

This is the optimization problem where we will try to minimize the total cost

Minimization function = Cost of supply from ith Plant * Power supplied to the jth City

i=1,2,3

j= 1to 10

Constraints

Demand for each city is given

So total Supply from each plant for a city is fixed

i.e Supply for Seattle = SLos-Seatle +STulsa-Seatle+SSeattle-Seatle= 950

Supply for Port land = SLos-Portland +STulsa-Portland+SSeattle-Portland= 831.25

Similarly we can list all the supply to the demand

Also

Sum of all supplies to the city from each plant <=4000

i.e

SLos-Seatle+SLos-Portland+.........<=4000

Supply cost table

City Los Angeles(($/MW) Tulsa T($/MW) Seattle($/MW) Demand(MW)
Seattle 1 356.25 593.75 59.38 950
Portland 2 356.25 593.75 178.13 831.25
San Fransisco 178.13 475 296.88 2375
Boise 4 356.25 475 296.88 593.75
Reno5 237.5 475 356.25 950
Bozeman 6 415.63 415.63 296.88 593.75
Laramie 7 356.25 415.63 356.25 1187.5
Park City 8 356.25 356.25 475 712.5
Flagstaff 9 178.13 475 593.75 1187.5
Durango 10 356.25 296.88 593.75 1543.75

Optimized Supply Table

City Los Angeles((MW) Tulsa T(MW) Seattle(MW) Sum
Seattle 1 0 0 950 950
Portland 2 0 0 831.25 831.25
San Fransisco 2066.721127 0 308.2788735 2375.000001
Boise 4 0 0 593.75 593.75
Reno5 745.7788727 0 204.2211273 950
Bozeman 6 3.55271E-15 7.10543E-15 593.75 593.75
Laramie 7 0 668.7500008 518.7499992 1187.5
Park City 8 0 712.5 0 712.5
Flagstaff 9 1187.5 0 0 1187.5
Durango 10 0 1543.75 0 1543.75
Total Sum 4000 2925.000001 4000

Cost Table

Seattle 1 0 0 56411
Portland 2 0 0 148070.5625
San Fransisco 368145.0344 0 91521.83197
Boise 4 0 0 176272.5
Reno5 177122.4823 0 72753.77661
Bozeman 6 1.47661E-12 2.95323E-12 176272.5
Laramie 7 0 277952.5628 184804.6872
Park City 8 0 253828.125 0
Flagstaff 9 211529.375 0 0
Durango 10 0 458308.5 0
Total Cost 2652992.938

Optimal Solution

{(0,0,950),(0,0,831.25),(2066.72,0,308.27),(0,0,593.75),(745.78,0,204.22),(0,0,593.75),(0,668.75,518.75),(0,712.5,0),(1187.5,0,0),(0,1543.75,0)}

ii)

Now the Total cost = $2652993

Earlier Cost= $2552423

Difference=$100570

Increase=(New-Earlier)*100/Earlier

3.94% equivalent to 4%

orchestra answered 2 years ago
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