Exam 2: Introduction to Optimization and Linear Programming

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.

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.

Solve the following LP problem graphically using level curves. MiN: $\quad 5 \mathrm { X } _ { 1 } + 7 \mathrm { X } _ { 2 }$ Subject to: $\quad 4 X _ { 1 } + 1 X _ { 2 } \geq 16$ $6 \mathrm { X } _ { 1 } + 5 \mathrm { X } _ { 2 } \geq 60$ $5 \mathrm { X } _ { 1 } + 8 \mathrm { X } _ { 2 } \geq 80$ $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 } \geq 0$

A company uses 4 pounds of resource 1 to make each unit of X₁ and 3 pounds of resource 1 to make each unit of X₂. There are only 150 pounds of resource 1 available. Which of the following constraints reflects the relationship between X₁, X₂ and resource 1?

Which of the following special conditions in an LP model represent potential errors in the mathematical formulation?

What are the three common elements of an optimization problem?

The third step in formulating a linear programming problem is

Solve the following LP problem graphically by enumerating the corner points. MiN: $\quad 8 \mathrm { X } _ { 1 } + 5 \mathrm { X } _ { 2 }$ Subject to: $\quad 6 X _ { 1 } + 7 X _ { 2 } \geq 84$ $\mathrm { X } _ { 1 } \geq 4$ $X _ { 2 } \geq 6$ $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 } \geq 0$

A production optimization problem has 4 decision variables and resource 1 limits how many of the 4 products can be produced. Which of the following constraints reflects this fact?

The desire to maximize profits is an example of a(n)

What most motivates a business to be concerned with efficient use of their resources?

A production optimization problem has 4 decision variables and a requirement that at least b₁ units of material 1 are consumed. Which of the following constraints reflects this fact?

A diet is being developed which must contain at least 100 mg of vitamin C. Two fruits are used in this diet. Bananas contain 30 mg of vitamin C and Apples contain 20 mg of vitamin C. The diet must contain at least 100 mg of vitamin C. Which of the following constraints reflects the relationship between Bananas, Apples and vitamin C?

A company makes two products, X₁ and X₂. They require at least 20 of each be produced. Which set of lower bound constraints reflect this requirement?

The constraints of an LP model define the

A redundant constraint is one which

The constraint for resource 1 is 5 X₁ + 4 X₂ $\le$ 200. If X₁ = 20 and X₂ = 5, how much of resource 1 is unused?

The number of units to ship from Chicago to Memphis is an example of a(n)

Project 2.1 Joey Koons runs a small custom computer parts company. As a sideline he offers customized and pre-built computer system packages. In preparation for the upcoming school year, he has decided to offer two custom computer packages tailored for what he believes are current student needs. System A provides a strong computing capability at a reasonable cost while System B provides a much more powerful computing capability, but at a higher cost. Joey has a fairly robust parts inventory but is concerned about his stock of those components that are common to each proposed system. A portion of his inventory, the item cost, and inventory level is provided in the table below. The requirements for each system are provided in the following table: System System B Processor 366 700 Memory 64 96 Hard Drive 6 20 Monitor 15 15 " Graphics Card Stock Stock CD-ROM 40 72 Sound Card Stock Stock Speakers Stock 60W Modem Stock Stock Mouse Stock Stock Keyboard Stock Stock Each system requires assembly, testing and packaging. The requirements per system built and resources available are summarized in the table below. System A System B Total Hours Available Assembly (hours) 2.25 2.50 200 Testing (hours) 1.25 2.00 150 Packaging (hours) 0.50 0.50 75 Joey is uncertain about product demand. In the past he has put together similar types of computer packages but his sales results vary. As a result is unwilling to commit all his in-house labor force to building the computer packages. He is confident he can sell all he can build and is not overly concerned with lost sales due to stock-outs. Based on his market survey, he has completed his advertising flyer and will offer System A for $ 1250 and will offer system B for $ 2325. Joey now needs to let his workers know how many of each system to build and he wants that mix to maximize his profits. Formulate an LP for Dave's problem. Solve the model using the graphical method. What is Dave's preferred product mix? What profit does Dave expect to make from this product mix?

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?