Mountain Laurel Vineyards produces three kinds of wine—Mountain
Blanc, Mountain Red, and Mountain Blush; and sells in local market.
The company has 17 tons of grapes available to produce wine this
season. A cask of Blanc requires 0.21 ton of grapes, a cask of Red
requires 0.24 ton, and a cask of Blush requires 0.18 ton. The
vineyard has enough storage space in its aging room to store 80
casks of wine. The vineyard has 1200 hours of production capacity,
and it requires 12 hours to produce a cask of Blanc, 14.5 hours to
produce a cask of Red, and 16 hours to produce a cask of Blush.
From past sales the vineyard knows that in the local market the
demand for the Blush will be no more than half of the sales of the
other two wines combined. The profit for a cask of Blanc is $7,500,
the profit for a cask of Red is $8,200, and the profit for a cask
of Blush is $10,500.
The vineyard constructed a LP model to maximize its profit as
follows.
LP-1:
max Z = 7500x1+8200x2+10500x3
subject to:
0.21x1+0.24x2+0.18x3≤17
x1+x2+x3≤80
12x1+14.5x2+16x3≤1200
-12x1+x2+x3≤0
x1,x2,x3≥0
The optimal solution is 20 casks of Blanc, 33.33 casks of Red,
and 26.67 casks of Blush,
with profit of $703,333.
Mountain Laurel Vineyards gets good reputation for their high
quality of wines. A wine seller in another city requests to order
35 casks of Mountain Red. In order to gain larger market, the
vineyard accepted the deal and signed the contract.
For this problem and the market opportunity, Mountain Laurel
Vineyards conducted research regarding how to expand their wine
production system. From sensitivity report of LP-1, they can see
expand what resources can benefit more. However, any increased
resource has certain cost. They found the following facts:
Although their grape farm only produces 17 tons of grape; the
vineyards possibly to purchase more grape from other farms nearby
at the cost of $2000 per ton, and each purchased ton of grape will
cost 3 labor hour. Finally, the total amount of purchased grape
cannot be excess 7 tons.
Although the vineyards only has storage capacity of 80 casks,
they can rent more storage at the cost of $800 per cask; maximum
storage of 30 casks can be rent.
The growing production may require more labor hours beyond the
original 1200 hours. To hire a local help will cost $50 per hour,
and the maximum of 500 hours they can increase.
The profit for a cask of Mountain Red increases from $8,200 to
$8,500.
TASKS:
From systems perspective, regarding Mountain Laurel Vineyards
as a system, what is its function? What are inputs to the system?
What are outputs? What are “control signals” or decision variables?
(5%)
From perspective of rational decision, should the Mountain
Laurel Vineyards simply change their original production and sales
plan and provide 35 casks Red for the contracted wine seller in
another city? Why? (5%)
Assume Mountain Laurel Vineyards want to maximize its profit by
expanding their production system. Carefully review the above
stated changes; extend the former linear programming model LP-1
into a new LP model. Write the LP mathematic model as LP-1 does.
(5%)
(Hints: The demand of Blanc applies to
local market only!)