Use the minimum-cost method to find a bfs for Problems 4, 7, and 8:
PROBLEMS 4
Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel (regardless of type) and the costs at each plant are shown in Table.
|
|
Cost ($)
|
|
Time
|
Plant
|
Steel 1
|
Steel 2
|
Steel 3
|
(minutes)
|
1
|
60
|
40
|
28
|
20
|
2
|
50
|
30
|
30
|
16
|
3
|
43
|
20
|
20
|
15
|
Each week, 100 tons of each type of steel (1, 2, and 3) must be produced. Each plant is open 40 hours per week.
a Formulate a balanced transportation problem to minimize the cost of meeting Steelco’s weekly requirements.
b Suppose the time required to produce 1 ton of steel depends on the type of steel as well as on the plant at which it is produced (see Table). Could a transportation problem still be formulated?
|
|
Time (minutes)
|
|
Plant
|
Steel 1
|
Steel 2
|
Steel 3
|
1
|
15
|
12
|
15
|
2
|
15
|
15
|
20
|
3
|
10
|
10
|
15
|
PROBLEMS 7
The U.S. government is auctioning off oil leases at two sites: 1 and 2. At each site, 100,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules state that no bidder can receive more than 40% of the land being auctioned. Cliff has bid $1,000/acre for site 1 land and $2,000/acre for site 2 land. Blake has bid $900/acre for site 1 land and $2,200/acre for site 2 land. Alexis has bid $1,100/acre for site 1 land and $1,900/acre for site 2 land. Formulate a balanced transportation model to maximize the government’s revenue.
PROBLEMS 8
The Ayatola Oil Company controls two oil fields. Field 1 can produce up to 40 million barrels of oil per day, and field 2 can produce up to 50 million barrels of oil per day. At field 1, it costs $3 to extract and refine a barrel of oil; at field 2, the cost is $2. Ayatola sells oil to two countries: England and Japan. The shipping cost per barrel is shown in Table.
|
To ($)
|
From ($)
|
England
|
Japan
|
Field 1
|
1
|
2
|
Field 2
|
2
|
1
|
Each day, England is willing to buy up to 40 million barrels (at $6 per barrel), and Japan is willing to buy up to 30 million barrels (at $6.50 per barrel). Formulate a balanced transportation problem to maximize Ayatola’s profits.