Exam 18: Introduction to Optimization

Capital Co. is considering which of five different projects to undertake in order to maximize its net present value (NPV). Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. The NPV and required capital (in thousands) for each project are listed below. Project Net Present Value Capital Required 1 60 7 2 50 10 3 40 6 4 20 3 5 30 12 What is the proper objective function?

Car Phones, Inc. (CP), sells two models of cell phones: Model A and Model B. Records show that 3 hours of sales time are used for each Model A cell phone that is sold and 5 hours of sales time for each Model B telephone. A total of 600 hours of sales time is available for the next 4-week period. In addition, management planning policies call for minimum sales goals of 100 units for Model A and 80 units for Model B. Furthermore, the number of phones that CP makes is limited by its critical process, the plastic injection molding machine, which can produce a maximum of 1500 phones per 4-week period. CP makes a $40 profit contribution for each Model A sold and a $50 profit contribution for each Model B sold. Formulate a Linear Program that can be used to determine the optimal production amounts CP for the next 4-week period.

Consider a mathematical program where X_i represents the amount produced of item i (i = 1,2,3,4), and you want the total amount produced over all four items to equal either 100, 120, 140, or 200. If you define q_i as binary (0-1) variables (i = 1,2,3,4) and add the constraint q₁ + q₂ + q₃ + q₄ = 1, what other constraint do you need to add to the program?

A portfolio manager in charge of a bank portfolio has $10 million to invest. The securities available for purchase, as well as their respective quality ratings, maturities, and yields, are shown in the table below: The bank places the following policy limitations on the portfolio manager's actions: 1. Government and agency bonds must total at least $4 million. 2. The average quality of the portfolio cannot exceed 1.4 on the bank's quality scale. (Note that a low number on this scale means a high-quality bond.) 3. The average years to maturity of the portfolio must not exceed 5 years. Assuming that the objective of the portfolio manager is to maximize after-tax earnings and that the tax rate is 50%, formulate a linear program that can be used to determine how much money to invest in each type of bond.

Which Excel tool provides solutions to Linear Programs?

Formulate the following as a linear program. Frank's art class must provide instructions on painting (P), carving (C), and clay modeling (M). Frank has a limited material budget of $3,000. The costs per student usage for the supplies are $25, $18, and $14, respectively. Frank must provide instruction and practice opportunities for each method. What is the optimum combination of art supplies Frank should procure for the class?

The Village Butcher Shop traditionally makes its meat loaf from a combination of lean ground beef and ground pork. The ground beef contains 80 percent meat and 20 percent fat, and costs the shop 80 cents per pound; the ground pork contains 68 percent meat and 32 percent fat, and costs 60 cents per pound. Formulate a linear program to determine the amount each kind of meat that the shop should use in each pound of meat loaf if it wants to minimize its cost and to keep the fat content of the meat loaf to no more than 25 percent.

A feasible solution to an optimization problem is a particular combination of decision variables that satisfies some of the constraints.

A constraint in Excel Solver consists of what three pieces of information?

Suppose that a firm makes gears (G) and axles (A). Each gear requires 15 minutes of labor and each axle requires 30 minutes of labor. One thousand hours of labor are available. How should the labor constraint(s) be written?

Once you've written the algebraic formulation of the problem, the next setup involved in Solving Optimization Problems is:

It is possible to solve a 2-variable problem with a __________________ to find an optimal solution.

What are the allowable constraint relationship types in optimization problems?

Formulate the following as a linear programming problem. Jan lives in the country and is having a garage sale tomorrow. She has decided to sell some fresh baked goods in conjunction with the sale. She has excellent recipes for apple pie and for apple turnovers. She has 20.83 pounds of apples. Each pie will require 1.6 pounds of apples and each turnover will require 0.22 pounds of apples. She has discovered that she is short of both flour and sugar and does not want to make another trip to the store tonight. She has 5.7 pounds of flour and 200 grams of sugar. Each pie will require 0.6 pounds of flour and 11 grams of sugar. Each turnover will require .02 pounds of flour and 3 grams of sugar. She will sell pies for $6 each and turnovers for $0.75 each. She has decided to make 8 pies at most, because she is concerned about how well they will sell.

Consider the mathematical program below. Which of the following choices represents an upper bound to the problem? Max 10 - X² Subject to X ≥ 3

Capital Co. is considering which of five different projects to undertake in order to maximize its net present value (NPV). Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. The NPV and required capital (in thousands) for each project are listed below. Project Net Present Value Capital Required 1 60 7 2 50 10 3 40 6 4 20 3 5 30 12 Which of the following represents the constraint stating that project 1 can be undertaken only if project 5 is undertaken?

Large-scale optimization software can handle how many decision variables and constraints?

Formulate the following as a linear programming problem. The Ski Company makes skis and snowboards. They are currently planning their operations for October-December, and have just an exclusive signed contracts to sell up to 4,000 pairs of skis and as many snowboards as they can make, to a company that owns several sporting goods stores. The profit that Ski will make is $53 for each pair of skis and $67 for each snowboard. Every pair of skis will require 7 hours from Department A and 12 hours from Department B. Every snowboard will require 5 hours from Department A and 19 hours from Department B. The total hours planned for October-December is 44,535 hours in Department A and 93,217 hours in Department B.

What are the four steps to setting up an optimization problem with Excel Solver?

What does LP stand for?