1.1.The manager of a small firm is considering whether to produce a new product that would require leasing some special equipment at a cost of $20,000 per month. In addition to this leasing cost, a production cost of $10 would be incurred for each unit of the product produced. Each unit sold would generate $20 in revenue.

Develop a mathematical expression for the monthly profit that would be generated by this product in terms of the number of units produced and sold per month. Then determine how large this number needs to be each month to make it profitable to produce the product.

1.2.Management of the Toys R4U Company needs to decide whether to introduce a certain new novelty toy for the upcoming Christmas season, after which it would be discontinued. The total cost required to produce and market this toy would be $500,000 plus $15 per toy produced. The company would receive revenue of $35 for each toy sold.

a.Assuming that every unit of this toy that is produced is sold, Write expression for the profit in terms of the number produced and sold. Then find the break-even point that this number must exceed to make it worthwhile to introduce this toy.

b.Now assume that the number that can be sold might be less than the number produced. Write expression for the profit in terms of these two numbers.

c.Formulate a spreadsheet that will give the profit in partbfor any values of the two numbers.

d.Write mathematical expression for the constraint that the number produced should not exceed the number that can be sold.

1.3.A reliable sales forecast has been obtained indicating that the Special Products Company (seeSection 1.2) would be able to sell 30,000 iWatches, which appears to be enough to justify introducing this new product. However, management is concerned that this conclusion might change if more accurate estimates were available for the research-and-development cost, the marginal production cost, and the unit revenue. Therefore, before a final decision is made, management wants what-if analysis done on these estimates.

Use the spreadsheet fromFigure 1.3(see this chapter's Excel files atwww.mhhe.com/Hillier6e) and trial-and-error to perform what-if analysis byindependentlyinvestigating each of the following questions.

a.How large can the research-and-development cost be before the watches cease to be profitable?

b.How large can the marginal production cost be before the watches cease to be profitable?

c.How small can the unit revenue be before the watches cease to be profitable?

1.4. Reconsider the problem facing the management of the Special Products Company as presented inSection 1.2.

A more detailed investigation now has provided better estimates of the data for the problem. The research-and-development cost still is estimated to be $10 million, but the new estimate of the marginal production cost is $1,300. The revenue from each watch sold now is estimated to be $1,700.

a.Use a graphical procedure to find the new break-even point.

b.Use an algebraic procedure to find the new break-even point.

c.State the mathematical model for this problem with the new data.

d.Incorporate this mathematical model into a spreadsheet with a sales forecast of 30,000. Use this spreadsheet model to find the new break-even point, and then determine the production quantity and the estimated total profit indicated by the model.

e.Suppose that management fears that the sales forecast may be overly optimistic and so does not want to consider producing more than 20,000 watches. Use the spreadsheet from partdto determine what the production quantity should be and the estimated total profit that would result.

2.2.Reconsider the Wyndor Glass Co. case study introduced inSection 2.1. Suppose that the estimates of the unit profits for the two new products now have been revised to $600 for the doors and $300 for the windows.

a.Formulate and solve the revised linear programming model for this problem on a spreadsheet.

b.Formulate this same model algebraically.

c.Use the graphical method to solve this revised model.

2.3.Reconsider the Wyndor Glass Co. case study introduced inSection 2.1. Suppose that Bill Tasto (Wyndor's vice president for manufacturing) now has found a way to provide a little additional production time in Plant 2 to the new products.

a.Use the graphical method to find the new optimal solution and the resulting total profit ifoneadditional hour per week is provided.

b.Repeat partaiftwoadditional hours per week are provided instead.

c.Repeat partaifthreeadditional hours per week are provided instead.

d.Use these results to determine how much each additional hour per week would be worth in terms of increasing the total profit from the two new products.

2.4.Use Solver to do Problem 2.3.

2.5.The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.

All the assumptions of linear programming hold.

a.Formulate and solve a linear programming model for this problem on a spreadsheet.

b.Formulate this same model algebraically.

2.7.You are given the following linear programming model in algebraic form, wherex₁andx₂are the decision variables andZis the value of the overall measure of performance.

MaximizeZ=x₁+ 2x₂

subject to

Constraint on resource 1:x₁+x₂ 5 (amount available)

Constraint on resource 2:x₁+ 3x₂ 9 (amount available)

and

x₁ 0x₂ 0

a.Identify the objective function, the functional constraints, and the nonnegativity constraints in this model.

b.Incorporate this model into a spreadsheet.

c.Is (x₁,x₂) = (3, 1) a feasible solution?

d.Is (x₁,x₂) = (1, 3) a feasible solution?

e.Use Solver to solve this model.