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