AMPL Stochastic Programming: It is late summer in a northern city and the direct
ID: 3773543 • Letter: A
Question
AMPL Stochastic Programming:
It is late summer in a northern city and the director of streets is making decisions regarding the purchase of fuel and salt to be used in snow removal in the approaching winter. If insufficient supplies are purchased before a severe winter, substantially higher prices must be paid to purchase these supplies quickly during the winter. There are two methods of snow removal: plowing or salting. Salting is generally cheaper. However, excess salt after a warm winter must be carried until next winter. On the other hand, excess fuel is readily usable by other city departments, so the penalty for stocking too much fuel is lower. The director of streets classifies a winter as either warm or cold and attaches probabilities 0.4 and 0.6, respectively, to these two possible states of nature.
Daily cost of operating a truck, exclusive of fuel and salt, is $110 in a warm winter and $120 in a cold winter.
Truck fleet has a capacity of 5,000 truck-days during the winter season.
If only plowing is used, then a warm winter will require 3,500 truck-days of snow removal; whereas, a cold winter will require 5,100.
Salting is a more effective use of trucks than plowing. o In a warm winter, one truck salting is equivalent to 1.2 trucks plowing o In a cold winter, one truck salting is equivalent to 1.1 trucks plowing.
Note that in a cold winter, the limited truck capacity implies some salting will be necessary.
A. Develop an AMPL model and Data file
I need what I would put in a .mod file and a .dat file into the program AMPL. If you don't know what AMPL is please do not answer the question.
Explanation / Answer
BF1 = truck-days of fuel bought in period 1
BS1 = truck-days of salt bought in period 1
BFW = fuel bought in period 2, warm winter
BSW = salt bought in period 2, warm winter
XFW = excess fuel at end of warm winter
XSW = excess salt at end of warm winter
PW = truck-days plowing, warm winter
SW = truck-days spreading salt, warm winter
KW = costs incurred during a warm winter in dollars
BFC = fuel bought in period 2, cold winter
BSC = salt bought in period 2, cold winter
XFC = excess fuel at end of a cold winter
XSC = excess salt at end of a cold winter
PC = truck-days plowing, cold winter
SC = truck-days spreading salt, cold winter
KC = costs incurred during a cold winter in dollars
The Warm Winter Solution
If the winter is known to be warm, then the relevant LP is:
MIN = 70 * BF1 + 20 * BS1 + KW;
-BF1 - BFW + XFW + PW + SW = 0; !(Fuel Usage);
-BS1 - BSW + XSW + SW = 0; !(Salt Usage);
PW + SW < 5000; !(Truck Usage);
PW + 1.2 * SW > 3500; !(Snow Removal);
KW - 73*BFW - 30*BSW + 65*XFW + 15*XSC !(Cost);
- 110 * PW - 110 * SW = 0;
When solved, the solution is:
Objective value: 583333.3
Variable Value Reduced Cost
BF1 2916.667 0.0000000
BS1 2916.667 0.0000000
KW 320833.3 0.0000000
BFW 0.0000000 3.000000
XFW 0.0000000 5.000000
PW 0.0000000 13.33334
SW 2916.667 0.0000000
BSW 0.0000000 10.00000
XSW 0.0000000 5.00000
Row Slack or Surplus Dual Price
1 583333.3 1.000000
2 0.0000000 70.00000
3 0.0000000 20.00000
4 2083.333 0.0000000
5 0.0000000 -166.6667
6 0.0000000 -1.000000
The Cold Winter Solution
The corresponding LP, if we knew the winter would b
e cold, is:
MIN = 70 * BF1 + 20 * BS1 + KC;
-BF1 - BFC + XFC + PC + SC = 0;
-BS1 + SC - BSC + XSC = 0;
PC + SC <= 5000;
PC + 1.1 * SC >= 5100;
KC - 73 * BFC + 65 * XFC - 120 * PC - 120 * SC - 32 * BSC + 15 *
XSC=0;
Optimal solution found at step: 6
Objective value: 970000.0
Variable Value Reduced Cost
BF1 5000.000 0.0000000
BS1 1000.000 0.0000000
KC 600000.0 0.0000000
BFC 0.0000000 3.000000
XFC 0.0000000 5.000000
PC 4000.000 0.0000000
SC 1000.000 0.0000000
BSC 0.0000000 12.00000
XSC 0.0000000 5.000000
Row Slack or Surplus Dual Price
1 970000.0 1.000000
2 0.0000000 70.00000
3 0.0000000 20.00000
4 0.0000000 10.00000
5 0.0000000 -200.0000
6 0.0000000 -1.000000
The solution to the complete problem is:
Optimal solution found at step: 12
Objective value: 819888.3
Variable Value Reduced Cost
BF1 2916.667 0.0000000
BS1 2916.667 0.0000000
KW 320833.3 0.0000000
KC 715091.7 0.0000000
BFW 0.0000000 3.000000
XFW 0.0000000 0.2000004
PW 0.0000000 4.683331
SW 2916.667 0.0000000
BSW 0.0000000 3.580000
XSW 0.0000000 8.420000
BFC 1891.667 0.0000000
XFC 0.0000000 4.799998
PC 1891.667 0.0000000
SC 2916.667 0.0000000
BSC 0.0000000 7.620001
XSC 0.0000000 2.580000
Row Slack or Surplus Dual Price
1 819888.3 1.000000
2 0.0000000 26.20000
3 0.0000000 8.420000
4 2083.333 0.0000000
5 0.0000000 -65.51667
6 0.0000000 -0.4000000
7 0.0000000 43.80000
8 0.0000000 11.58000
9 191.6667 0.0000000
10 0.0000000 -115.8000
11 0.0000000 -0.6000000
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