Question

A) Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $16,000/day to operate, and it yields 60 oz of gold and 3000 oz of silver each of x days. The Horseshoe Mine costs $18,000/day to operate, and it yields 80 oz of gold and 1250 oz of silver each of y days. Company management has set a target of at least 700 oz of gold and 17,000 oz of silver. How many days should each mine be operated so that the target can be met at a minimum cost C in dollars?

Minimize

C

=

_____ subject to the constraints

gold	=	_______
silver	=	_________
				x ≥ 0
				y ≥ 0

B) Formulate but do not solve the following exercise as a linear programming problem.

Kane Manufacturing has a division that produces two models of fireplace grates, x units of model A and y units of model B. To produce each model A requires 3 lb of cast iron and 7 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $1.50, and the profit for each model B grate is $1.00. 1400 lb of cast iron and 1200 min of labor are available for the production of grates per day.

Because of a backlog of orders on model A grates, the manager of Kane Manufacturing has decided to produce at least 150 of these grates a day. Operating under this additional constraint, how many grates of each model should Kane produce to maximize profit P in dollars?

Maximize		P	= _________		subject to the constraints
cast iron	=		__________
labor	=		__________
model A	=		__________
				y ≥ 0

C) Formulate but do not solve the following exercise as a linear programming problem.

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type A vessel, x, has 60 deluxe cabins and 165 standard cabins, whereas a type B vessel, y, has 80 deluxe cabins and 110 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 350 deluxe and 670 standard cabins for their 15-day cruise in May. It costs $44,000 to operate a type A vessel and $52,000 to operate a type B vessel for that period. How many of each type of vessel should be used in order to keep the operating costs, C (in dollars), to a minimum?

Minimize

C

=

________ subject to the constraints

deluxe cabins	= ________
standard cabins	=_________
				x ≥ 0
				y ≥ 0

Answer 1

1.

60x+80y>=700

3x+4y=35

3000x+1250y>=17000

12x+5y=68

x,y>=0

C=16000x+18000y

Plot the equation

The points of the region are (0,13.6),(11.667,0),(2.9,6.5)

Value of C at these points

C(0,13.6)=244800

C(11.667,0)=186672

C(2.9,6.5)=163400(min)

x=2.9,y=6.5,P=163400

2.

3x+4y<=1400

7x+3y<=1200

x+y>=150

P=1.5x+y

Plot the eqution

Point of region(-800,950),( 31.5,326.31),(187.5,-37.5)

P is max at (31.5,326.31)

x=31.5

y=326.31

3.

60x+80y>=350

165x+110y>=670

C=44000x+52000y

Plot the eqution

Points of region (5.8,0),(0,6),(2.3,2.7)

C is min at (2.3,2.7)=241600