The Humber Transport Company has expanded its shipping capacity by purchasing 75 trailer trucks from a competitor that went bankrupt. The company subsequently located 25 of the purchased trucks at each of its shipping warehouses in Mississauga, Windsor, and Kingston. The company makes shipments from each of these warehouses to terminals in Montreal, New York, and Chicago. Each truck is capable of making one shipment per week. The terminal managers have each indicated their capacity for extra shipments. The manager in Montreal needs to accommodate 20 additional trucks per week, the manager in New York needs to accommodate 15 additional trucks per week, and the manager in Chicago needs to accommodate 30 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates.
The Humber Transport Company has expanded its shipping capacity by purchasing 75 trailer trucks from a competitor that went bankrupt. The company subsequently located 25 of the purchased trucks at each of its shipping warehouses in Mississauga, Windsor, and Kingston. The company makes shipments from each of these warehouses to terminals in Montreal, New York, and Chicago. Each truck is capable of making one shipment per week. The terminal managers have each indicated their capacity for extra shipments. The manager in Montreal needs to accommodate 20 additional trucks per week, the manager in New York needs to accommodate 15 additional trucks per week, and the manager in Chicago needs to accommodate 30 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates.
| Terminal | |||||||
| Warehouse | A | B | C | ||||
| Montreal | New York | Chicago | |||||
| 1. | Mississauga | x1Ax1A | $4100$4100 | x1Bx1B | $3600$3600 | x1Cx1C | $4500$4500 |
| 2. | Windsor | x2Ax2A | $2200$2200 | x2Bx2B | $2900$2900 | x2Cx2C | $3500$3500 |
| 3. | Kingston | x3Ax3A | $2800$2800 | x3Bx3B | $3000$3000 | x3Cx3C | $4300$4300 |
The company wants to know how many trucks to assign to each route (i.e., warehouse to terminal) to maximize profit. In the table above, xijxij represents the number of trucks assigned to the route from warehouse ii to terminal jj, where
- i=i= 1 == Mississauga, i=i= 2 == Windsor, i=i= 3 == Kingston
- j=j= A == Montreal, j=j= B == New York, j=j= C == Chicago
Formulate a linear programming model for this problem.
(a) The objective function for this problem is:
- minimize
Z=4100x1A+2200x1B+2800x1C+3600x2A+Z=4100x1A+2200x1B+2800x1C+3600x2A+
+2900x2B+3000x2C+4500x3A+3500x3B+4300x3C +2900x2B+3000x2C+4500x3A+3500x3B+4300x3C - maximize
Z=4100x1A+3600x1B+4500x1C+2200x2A+Z=4100x1A+3600x1B+4500x1C+2200x2A+
+2900x2B+3500x2C+2800x3A+3000x3B+4300x3C +2900x2B+3500x2C+2800x3A+3000x3B+4300x3C - minimize
Z=4100x1A+3600x1B+4500x1C+2200x2A+Z=4100x1A+3600x1B+4500x1C+2200x2A+
+2900x2B+3500x2C+2800x3A+3000x3B+4300x3C +2900x2B+3500x2C+2800x3A+3000x3B+4300x3C - maximize
Z=4100x1A+2200x1B+2800x1C+3600x2A+Z=4100x1A+2200x1B+2800x1C+3600x2A+
+2900x2B+3000x2C+4500x3A+3500x3B+4300x3C +2900x2B+3000x2C+4500x3A+3500x3B+4300x3C - None of the above.
(b) Please choose the correct constraint:
- x1A+x1B+x1C≤75x1A+x1B+x1C≤75
- x1A+x2B+x3C≤25x1A+x2B+x3C≤25
- x1A+x2B+x3C≥25x1A+x2B+x3C≥25
- x1A+x1B+x1C≤25x1A+x1B+x1C≤25
- x1A+x1B+x1C≥25x1A+x1B+x1C≥25
- x1A+x1B+x1C≥75x1A+x1B+x1C≥75
- None of the above.
(c) Please choose the correct constraint:
- x1A+x2B+x3C≥25x1A+x2B+x3C≥25
- x2A+x2B+x2C≥75x2A+x2B+x2C≥75
- x2A+x2B+x2C≥25x2A+x2B+x2C≥25
- x2A+x2B+x2C≤25x2A+x2B+x2C≤25
- x1A+x2B+x3C≤25x1A+x2B+x3C≤25
- x2A+x2B+x2C≤75x2A+x2B+x2C≤75
- None of the above.
(d) Please choose the correct constraint:
- x1A+x2B+x3C≤25x1A+x2B+x3C≤25
- x1A+x2B+x3C≥25x1A+x2B+x3C≥25
- x3A+x3B+x3C≥25x3A+x3B+x3C≥25
- x3A+x3B+x3C≥75x3A+x3B+x3C≥75
- x3A+x3B+x3C≤25x3A+x3B+x3C≤25
- x1A+x2B+x3C≤75x1A+x2B+x3C≤75
- None of the above.
(e) Please choose the correct constraint:
- x1A+x2A+x3A≥20x1A+x2A+x3A≥20
- x1A+x2A+x3A≤20x1A+x2A+x3A≤20
- x1A+x2A+x3A=9100x1A+x2A+x3A=9100
- x1A+x2A+x3A≥9100x1A+x2A+x3A≥9100
- x1A+x2A+x3A=20x1A+x2A+x3A=20
- x1A+x2A+x3A≤9100x1A+x2A+x3A≤9100
- None of the above.
(f) Please choose the correct constraint in the standard form:
- x1B+x2B+x3B≤9500x1B+x2B+x3B≤9500
- x1B+x2B+x3B=15x1B+x2B+x3B=15
- x1B+x2B+x3B≤15x1B+x2B+x3B≤15
- x1B+x2B+x3B≥15x1B+x2B+x3B≥15
- x1B+x2B+x3B=9500x1B+x2B+x3B=9500
- x1B+x2B+x3B≥9500x1B+x2B+x3B≥9500
- None of the above.
(g) Please choose the correct constraint in the standard form:
- x1C+x2C+x3C=30x1C+x2C+x3C=30
- x1C+x2C+x3C≥12300x1C+x2C+x3C≥12300
- x1C+x2C+x3C≤12300x1C+x2C+x3C≤12300
- x1C+x2C+x3C≤30x1C+x2C+x3C≤30
- x1C+x2C+x3C=12300x1C+x2C+x3C=12300
- x1C+x2C+x3C≥30x1C+x2C+x3C≥30
- None of the above.
(h) Non-negativity for this LP problem means that:
- at least one xij≥0xij≥0
- all xij≥0xij≥0
- x1A+x2A+x3A≥0x1A+x2A+x3A≥0
- x1A×x2B×x3C ≥0x1A×x2B×x3C ≥0
- x1B+x2B+x3B≥0x1B+x2B+x3B≥0
- x1C+x2C+x3C≥0x1C+x2C+x3C≥0
- None of the above.
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