Reference no: EM13879216

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?