Chapter 2—Introduction to Optimization and Linear Programming

56. Solve the following LP problem graphically using level curves.

MIN:	5 X1 + 7 X2
Subject to:	4 X1 + 1 X2³ 16
	6 X1 + 5 X2³ 60
	5 X1 + 8 X2³ 80
	X1, X2³ 0

57. The Happy Pet pet food company produces dog and cat food. Each food is comprised of meat, soybeans and fillers. The company earns a profit on each product but there is a limited demand for them. The pounds of ingredients required and available, profits and demand are summarized in the following table. The company wants to plan their product mix, in terms of the number of bags produced, in order to maximize profit.

Product	Profit per Bag ($)	Demand for product	Pounds of Meat per bag	Pounds of Soybeans per bag	Pounds of Filler per bag
Dog food	4	40	4	6	4
Cat food	5	30	5	3	10
Material available (pounds)	100	120	160

a.	Formulate the LP model for this problem.
b.	Solve the problem using the graphical method.

58. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced.

a.	Formulate the LP model for this problem.
b.	Solve the problem using the graphical method.

59. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces a limited demand. There are a limited number of wiring, assembly and inspection hours available next month. The data for this problem is summarized in the following table.

Computer Model	Profit per Model ($)	Maximum demand for product	Wiring Hours Required	Assembly Hours Required	Inspection Hours Required
Plain	30	80	.4	.5	.2
Fancy	40	90	.5	.4	.3
	Hours Available	50	50	22

a.	Formulate the LP model for this problem.
b.	Solve the problem using the graphical method.

60. The Big Bang explosives company produces customized blasting compounds for use in the mining industry. The two ingredients for these explosives are agent A and agent B. Big Bang just received an order for 1400 pounds of explosive. Agent A costs $5 per pound and agent B costs $6 per pound. The customer's mixture must contain at least 20% agent A and at least 50% agent B. The company wants to provide the least expensive mixture which will satisfy the customers requirements.

a.	Formulate the LP model for this problem.
b.	Solve the problem using the graphical method.

61. Jim's winery blends fine wines for local restaurants. One of his customers has requested a special blend of two burgundy wines, call them A and B. The customer wants 500 gallons of wine and it must contain at least 100 gallons of A and be at least 45% B. The customer also specified that the wine have an alcohol content of at least 12%. Wine A contains 14% alcohol while wine B contains 10%. The blend is sold for $10 per gallon. Wine A costs $4 per gallon and B costs $3 per gallon. The company wants to determine the blend that will meet the customer's requirements and maximize profit.

a.	Formulate the LP model for this problem.
b.	Solve the problem using the graphical method.
c.	How much profit will Jim make on the order?

62. Bob and Dora Sweet wish to start investing $1,000 each month. The Sweets are looking at five investment plans and wish to maximize their expected return each month. Assume interest rates remain fixed and once their investment plan is selected they do not change their mind. The investment plans offered are:

Fidelity	9.1% return per year
Optima	16.1% return per year
CaseWay	7.3% return per year
Safeway	5.6% return per year
National	12.3% return per year

Since Optima and National are riskier, the Sweets want a limit of 30% per month of their total investments placed in these two investments. Since Safeway and Fidelity are low risk, they want at least 40% of their investment total placed in these investments.

Formulate the LP model for this problem.