A New York State wine maker owns two wineries, one in Niagara Falls (w1) and one in the Finger Lakes (w2). The wine maker also owns three grape farms that supply all grapes needed for his two wineries. Grape farm 1 is located in Tompkins County (g1), grape farm 2 is located in Seneca County (g2), and grape farm 3 is located in Niagara County (g3). Define any shipment of grapes from grape farm i to winery j as giwj. The unit transportation costs from each grape farm to each winery, as well as the annual supply of grapes are given below:
|
From/To
|
w1 w2
($/ton of grapes)
|
Total Supply (tons)
|
|
g1
|
$400
|
$150
|
105
|
|
g2
|
$250
|
$70
|
70
|
|
g3
|
$25
|
$220
|
50
|
The wine maker has a contract to sell 25,000 bottles of wine to a distributor in Buffalo (d1), 15,000 bottles to a distributor in New York City (d2), and 70,000 bottles to a distributor in Albany (d3). Define any shipment of wine from winery j to distributor k as wj dk (*make j and k subscripts). The unit transportation costs and from each win- ery to each distributor are:
|
From/To
|
d1
|
d2
($/bottle)
|
d3
|
|
w1
|
$0.25
|
$1.10
|
$1.00
|
|
w2
|
$0.70
|
$0.75
|
$0.65
|
|
Total Demand
|
25,000
|
15,000
|
70,000
|
One ton of grapes will make 500 bottles of wine. Assume that the wine maker's sole objec- tive is to minimize total transportation costs and that he has infinite capacity at each winery.
a. Write this LP problem in general form using the notation outlined on the previous page, i.e., giwj denotes grape shipments from grape farm i to winery j, wjdk denotes wine shipments from winery j to distributor k.
The following is the optimal solution for the problem.
Primal Problem Solution
|
Variable
|
Status
|
Value
|
Return/Unit
|
Value/Unit
|
Net Return
|
|
g1w1
|
Nonbasis
|
0.00
|
400.00
|
330.00
|
70.00
|
|
g1w2
|
Basis
|
100.00
|
150.00
|
150.00
|
0.00
|
|
g2w1
|
Basis
|
0.00
|
250.00
|
250.00
|
0.00
|
|
g2w2
|
Basis
|
70.00
|
70.00
|
70.00
|
0.00
|
|
g3w1
|
Basis
|
50.00
|
25.00
|
25.00
|
0.00
|
|
g3w2
|
Nonbasis
|
0.00
|
220.00
|
-155.00
|
375.00
|
|
w1d1
|
Basis
|
25,000.00
|
0.25
|
0.25
|
0.00
|
|
w1d2
|
Nonbasis
|
0.00
|
1.10
|
0.39
|
0.71
|
|
w1d3
|
Nonbasis
|
0.00
|
1.00
|
0.29
|
0.71
|
|
w2d1
|
Nonbasis
|
0.00
|
0.70
|
0.61
|
0.09
|
|
w2d2
|
Basis
|
15,000.00
|
0.75
|
0.75
|
0.00
|
|
w2d3
|
Basis
|
70,000.00
|
0.65
|
0.65
|
0.00
|
Dual Problem Solution
|
Constraint
|
Status
|
Dual Value
|
RHS Value
|
Usage
|
Slack
|
|
g1sup
|
Nonbinding
|
0.00
|
105.00
|
100.00
|
5.00
|
|
g2sup
|
Binding
|
-80.00
|
70.00
|
70.00
|
0.00
|
|
g3sup
|
Binding
|
-305.00
|
50.00
|
50.00
|
0.00
|
|
w1tran
|
Binding
|
0.66
|
0.00
|
0.00
|
0.00
|
|
w2tran
|
Binding
|
0.30
|
0.00
|
0.00
|
0.00
|
|
d1dem
|
Binding
|
0.91
|
25,000.00
|
25,000.00
|
0.00
|
|
d2dem
|
Binding
|
1.05
|
15,000.00
|
15,000.00
|
0.00
|
|
d3dem
|
Binding
|
0.95
|
70,000.00
|
70,000.00
|
0.00
|
Objective Row Ranges
|
Variable
|
Status
|
Value
|
Return/Unit
|
Minimum
|
Maximum
|
|
g1w1
|
Nonbasis
|
0.00
|
400.00
|
330.00
|
NONE
|
|
g1w2
|
Basis
|
100.00
|
150.00
|
70.00
|
220.00
|
|
g2w1
|
Basis
|
0.00
|
250.00
|
-55.00
|
295.00
|
|
g2w2
|
Basis
|
70.00
|
70.00
|
25.00
|
150.00
|
|
g3w1
|
Basis
|
50.00
|
25.00
|
NONE
|
330.00
|
|
g3w2
|
Nonbasis
|
0.00
|
220.00
|
-155.00
|
NONE
|
|
w1d1
|
Basis
|
25,000.00
|
0.25
|
-0.66
|
0.34
|
|
w1d2
|
Nonbasis
|
0.00
|
1.10
|
0.39
|
NONE
|
|
w1d3
|
Nonbasis
|
0.00
|
1.00
|
0.29
|
NONE
|
|
w2d1
|
Nonbasis
|
0.00
|
0.70
|
0.61
|
NONE
|
|
w2d2
|
Basis
|
15,000.00
|
0.75
|
-0.30
|
1.46
|
|
w2d3
|
Basis
|
70,000.00
|
0.65
|
-0.30
|
1.36
|
Right-Hand-Side Ranges
|
Constraint
|
Status
|
Dualvalue
|
RHS Value
|
Minimum
|
Maximum
|
|
g1sup
|
Nonbinding
|
0.00
|
105.00
|
100.00
|
NONE
|
|
g2sup
|
Binding
|
-80.00
|
70.00
|
65.00
|
170.00
|
|
g3sup
|
Binding
|
-305.00
|
50.00
|
45.00
|
50.00
|
|
w1tran
|
Binding
|
0.66
|
0.00
|
0.00
|
2,500.00
|
|
w2tran
|
Binding
|
0.30
|
0.00
|
-50,000.00
|
2,500.00
|
|
d1dem
|
Binding
|
0.91
|
25,000.00
|
25,000.00
|
27,500.00
|
|
d2dem
|
Binding
|
1.05
|
15,000.00
|
0.00
|
17,500.00
|
|
d3dem
|
Binding
|
0.95
|
70,000.00
|
20,000.00
|
72,500.00
|
b. Draw a network diagram of the optimal shipments. Include in the diagram the optimal quantities, unit transportation costs, and fixed supplies and demands.
c. The SP on the supply of grapes from grape farm 2 is -80. Explain how that num- ber is derived.
d. Suppose that winery 1 has a capacity of handling 180 tons of grapes, and winery 2 has a capacity of 80 tons of grapes. Show how you would modify your model to account for these capacities.
e. From the wine maker's point of view, which distributor is in the most efficient location? Why?
f. How much would transportation costs change if additional grapes were grown on the first grape farm (g1)? Why?