Question
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week | Trucking Limits | Railway Limits | Air Cargo Limits |
1 | 45 | 60 | 15 |
2 | 50 | 55 | 10 |
3 | 55 | 45 | 5 |
Costs ($ per 1000 tons) | $200 | $140 | $400 |
The following is the LP model for this logistics problem.
Let | Xij = amount shipped by mode i in week j |
where i = 1(Truck), 2(Rail), 3(Air) | |
and j = 1, 2, 3 | |
Let | WLij = weekly limit of mode i in week j (as provided in above table) |
MIN: | 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33) |
Subject to: | |
Xij WL ij for all i and j | Weekly limits by mode |
X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 250 | Total at end of three weeks |
X11 + X21 + X31 + X12 + X22 + X32 200 | Total at end of two weeks |
X11 + X21 + X31 120 | Total at end of first week |
X11 + X12 + X13 0.45*250 | Truck mix requirement |
X21 + X22 + X23 0.40*250 | Rail mix requirement |
X31 + X32 + X33 0.15*250 | Air mix limit |
Xij 0 for all i and j |
Final | Reduced | Objective | Allowable | Allowable | ||
Cell | Name | Value | Cost | Coefficient | Increase | Decrease |
$D$6 | Week 1 by Truck | 45 | 0 | 200 | 360 | 1E+30 |
$E$6 | Week 1 by Rail | 60 | 0 | 140 | 360 | 1E+30 |
$F$6 | Week 1 by Air | 15 | 0 | 500 | 1E+30 | 360 |
$D$7 | Week 2 by Truck | 50 | 0 | 200 | 0 | 1E+30 |
$E$7 | Week 2 by Rail | 55 | 0 | 140 | 0 | 1E+30 |
$F$7 | Week 2 by Air | 0 | 360 | 500 | 1E+30 | 360 |
$D$8 | Week 3 by Truck | 13 | 0 | 200 | 1E+30 | 0 |
$E$8 | Week 3 by Rail | 12 | 0 | 140 | 60 | 0 |
$F$8 | Week 3 by Air | 0 | 360 | 500 | 1E+30 | 360 |
Constraints | ||||||
Final | Shadow | Constraint | Allowable | Allowable | ||
Cell | Name | Value | Price | R.H. Side | Increase | Decrease |
$D$18 | Week 1 by Truck | 45 | 360 | 45 | 13 | 0 |
$E$18 | Week 1 by Rail | 60 | 360 | 60 | 15 | 0 |
$F$18 | Week 1 by Air | 15 | 0 | 15 | 1E+30 | 0 |
$D$19 | Week 2 by Truck | 50 | 0 | 50 | 13 | 25 |
$E$19 | Week 2 by Rail | 55 | 0 | 55 | 12 | 25 |
$F$19 | Week 2 by Air | 0 | 0 | 10 | 1E+30 | 10 |
$D$20 | Week 3 by Truck | 13 | 0 | 55 | 1E+30 | 42 |
$E$20 | Week 3 by Rail | 12 | 0 | 45 | 1E+30 | 33 |
$F$20 | Week 3 by Air | 0 | 0 | 5 | 1E+30 | 5 |
$D$9 | Shipped by Truck | 108 | 60 | 108 | 12 | 13 |
$E$9 | Shipped by Rail | 127 | 0 | 100 | 27 | 1E+30 |
$F$13 | Total Shipped Tons | 250 | 140 | 250 | 33 | 0 |
$F$9 | Shipped by Air | 15 | 0 | 37.5 | 1E+30 | 22.5 |
$G$6 | Week 1 Totals | 120 | 360 | 120 | 0 | 15 |
$G$7 | Week 2 Totals | 225 | 0 | 200 | 25 | 1E+30 |
$G$8 | Week 3 Totals | 250 | 0 | 250 | 0 | 1E+30 |
The Week 1 by Truck and Week 1 by Rail constraints each have a shadow price of 360. What do these values imply?
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