Use graphical methods to solve each linear programming problem.

$$\begin{array}{lr} \text{Minimize} & z=x+3 y \\ \text{subject to:} & x+y \leq 10 \\ & 5 x+2 y \geq 20 \\ & -x+2 y \geq 0 \\ & x \geq 0 \\ & y \geq 0. \end{array}$$

A large Southeast city is requesting federal funding for a park-and-ride project. One of the requirements in the request application is a network plan for the design phase of the project. Sophie Kim, the chief engineer, wants you to develop a project network plan to meet this requirement. She has gathered the activity time estimates and their dependencies shown here. Show your project network with the activity early, late, and slack times. Mark the critical path.

A teacher records the scores on a project and wants to group the data in intervals of 5 points. Should the teacher use a stem-and-leaf plot or a histogram? Explain.

Find the indicated maximum and minimum values by the linear programming method of this section.The constraints are shown below the objective function.

$$\begin{array}{l}{\text { Maximum } P :} \\ {P=3 x+4 y} \\ {2 x+y \geq 2} \\ {x+2 y \geq 2} \\ {x+y \leq 2}\end{array}$$

(Project: Drawing Shapes with Characters) Use techniques similar to those developed in given examples to produce a program that graphs a wide range of shapes.

A large eastern city is requesting federal funding for a park-and-ride project. One of the requirements in the request application is a network plan for the design phase of the project. Catherine Walker, the chief engineer, wants you to develop a project network plan to meet this requirement. She has gathered the activity time estimates and their dependencies shown here. Show your project network with the activity early, late, and slack times. Mark the critical path.

What type of chart compares the current project schedule with the original baseline schedule so that deviations from the original plan can be easily noticed

(Game: scissor, rock, paper) Programming Exercise $3.17$ gives a program that plays the scissor-rock-paper game. Revise the program to let the user continuously play until either the user or the computer wins three times more than their opponent.

Argile Textiles is evaluating a new product, a silk/wool blended fabric. Assume that you were recently hired as assistant to the director of capital budgeting, and you must evaluate the proposed project.

The fabric would be produced in an unused building located adjacent to Argile's Southern Pines, North Carolina, plant; Argile owns the building, which is fully depreciated. The required equipment would cost $200,000, plus an additional $40,000 for shipping and installation. With the new project, inventories would rise by $25,000, and accounts payable would increase by $5,000. All of these costs would be incurred at$t=0$. By a special ruling, the machinery could be depreciated under the MACRS system as 3-year property.

The project is expected to operate for four years, and then be terminated. The cash inflows are assumed to begin one year after the project is undertaken, or at$t=1$, and to continue to$t=4$. At the end of the project's life$(t=4)$, the equipment is expected to have a salvage value of $25,000.

Unit sales are expected to total 100,000 five-yard rolls per year, and the expected sales price is $2 per roll. Cash operating costs for the project (total operating costs excluding depreciation) are expected to amount to 60 percent of dollar sales. Argile's marginal tax rate is 40 percent, and its required rate of return is 10 percent. Tentatively, the silk/wool blend fabric project is assumed to be of equal risk to Argile's other assets.

You have been asked to evaluate this project and to make an accept/reject recommendation. To guide you in your analysis, your boss has asked you to answer the following set of questions.

(1) What is the regular payback period and the discounted payback period for the project?

(2) What is the rationale for the payback? According to the payback criterion, should Argile accept the project if the firm's maximum acceptable payback is two years?

(3) Explain the main difference between the regular payback and the discounted payback.

(4) What are the main disadvantages of the regular payback method? Is the payback method of any real usefulness in capital budgeting decisions?

How can the firm structure management compensation to minimize agency problems? What is the current view with regard to the execution of many compensation plans?

(Some familiarity with linear programming is assumed for this exercise.) Suppose you have a linear programming problem with two unknowns and 20 constraints. You decide that graphing the feasible region would take a lot of work, but then you recall that corner points are obtained by solving a system of two equations in two unknowns obtained from two of the constraints. Thus, you decide that it might pay instead to locate all the possible corner pointsby solving all possible combinations of two equations and then checking whether each solution is a feasible point. a. How many systems of two equations in two unknowns will you be required to solve? b. Generalize this to n constraints.

J. Wold, project manager of Print Software, Inc., wants you to prepare a project network; compute the early, late, and slack activity times; determine the planned project duration, and identify the critical path. His assistant has collected the following information for the Color Printer Drivers Software Project:

(Count vowels and consonants) Assume that the letters A, E, I, O, and U are vowels. Write a program that prompts the user to enter a string, and displays the number of vowels and consonants in the string.

Enter a string: Programming is fun 
The number of vowels is 5
The number of consonants is 11

Describe the calculation s used to project a due date of $60$ days from a date of purchase, assuming the $60$ days are within the same year