Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls
Question:
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 |
Of the three percentage of effort constraints, Shipped by Truck, Shipped by Rail, and Shipped by Air, which should be examined for potential cost reduction? Explain?
Expert Answer:
Thi question is for the LP formulation which can be done as below Xij total ... View the full answer
