Exam 19: Linear Programming

A manager must decide on the mix of products to produce for the coming week. Product A requires three minutes per unit for molding, two minutes per unit for painting, and one minute for packing. Product B requires two minutes per unit for molding, four minutes for painting, and three minutes per unit for packing. There will be 600 minutes available for molding, 600 minutes for painting, and 420 minutes for packing. Both products have contributions of $1.50 per unit. a. Algebraically state the objective and constraints of this problem. b. Plot the constraints on the grid below and identify the feasible region.

South Coast Papers wants to mix two lubricating oils (A and B) for its machines in order to minimize cost. It needs no less than 3,000 gallons in order to run its machines during the next month. It has a maximum oil storage capacity of 4,000 gallons. There are 2,000 gallons of Oil A and 4,000 of Oil B available. The mixed fuel must have a viscosity rating of no less than 40. When mixing fuels, the amount of oil obtained is exactly equal to the sum of the amounts put in. The viscosity rating is the weighted average of the individual viscosities, weighted in proportion to their volumes. The following is known: Oil A has a viscosity of 45 and costs 60 cents per gallon; Oil B has a viscosity of 37.5 and costs 40 cents per gallon. State the objective and the constraints of this problem. Plot all constraints and highlight the feasible region. Use your (by now, well-developed) intuition to suggest a feasible (but not necessarily optimal) solution. Be certain to show that your solution meets all constraints.

Lost Maples Winery makes three varieties of contemporary Constellation Country wines: Austin (a fine red), Ste. George (a table white), and Los Alamos (a hearty pink Zinfandel). The raw materials, labour, and contribution per case of each of these wines is summarized below. Grapes Variety A A bushels Variety B bushels Sugar KGS Labour (man- hours) Contrib. per case Austin Formation 5 0 2.5 3.25 \ 28.50 Ste. Genevieve 0 4 1.5 2.5 \ 31 Los Alamos 2 1 2 2 \ 23 The winery has 4000 bushels of Variety A grapes, 2800 bushels of Variety B grapes, 1500 pounds of sugar, and 1600 man-hours of labour available during the next week. The firm operates to achieve maximum contribution. Refer to the POM for Windows panels showing the solution to this problem. Answer the following questions. a. For maximum contribution, how much of each wine should be produced? b. How much contribution will be made by selling the output? c. If you could increase any constraint by one unit, which one would it be and why?

What is the simplex method?

Using the iso-profit line solution method to solve a maximization problem requires that we

Constraints are needed to solve linear programming problems by hand, but not by computer.

In sensitivity analysis, a zero shadow price (or dual value) for a resource ordinarily means that the resource has not been used up.

Solving a linear programming problem with the iso-profit line solution method requires that we move the iso-profit line to each corner of the feasible region until the optimum is identified.

________ is an analysis that projects how much a solution might change if there were changes in the variables or input data.

For a linear programming problem with the constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, two of its corner points are (0, 0) and (0, 25).

A shadow price (or dual value) reflects which of the following in a maximization problem?

Linear programming helps operations managers make decisions necessary to make effective use of resources such as machinery, labour, money, time, and raw materials.

For the constraints given below, which point is in the feasible region of this maximization problem? (1) 14x + 6y < 42 (2) x - y < 3 (3) x, y ≥ 0

A craftsman builds two kinds of birdhouses, one for wrens (X1), and one for bluebirds (X2). Each wren birdhouse takes four hours of labour and four units of lumber. Each bluebird house requires two hours of labour and twelve units of lumber. The craftsman has available 60 hours of labour and 120 units of lumber. Wren houses profit $6 each and bluebird houses profit $15 each. Use the software output that follows to interpret the problem solution. Include a statement of the solution quantities (how many of which product), a statement of the maximum profit achieved by your product mix, and a statement of "resources unused" and "shadow prices." Bird Houses Solution Variable Value Reduced Original Val Lower Bound Upper Bound X1 12. 0. 6. 5. 30. X2 6. 0. 15. 3. 18. Constraint Dual Value sack/Surplus Original Val Lower Bound Upper Bound Constraint 1 0.3 0 60. 20. 120. Constraint 2 1.2 0. 120. 60. 360.

A firm makes two products, Y and Z. Each unit of Y costs $10 and sells for $40. Each unit of Z costs $5 and sells for $25. If the firm's goal were to maximize profit, the appropriate objective function would be

________ is a mathematical technique designed to help operations managers plan and make decisions relative to the trade-offs necessary to allocate resources.

A linear programming problem has three constraints: 2X + 10Y ≤ 1004X + 6Y ≤ 1206X + 3Y ≥ 90 What is the largest quantity of X that can be made without violating any of these constraints?

A company has the following usage of inputs, constraints on inputs and profit margins to make their three products. Product Wiring Drilling Assembly Profit X202 0.7 0.4 0.8 9 Y303 0.2 0.5 0.5 11 Z404 0.5 0.3 0.2 14 There are maximum amounts of the following: 1500 hours for wiring, 1400 hours for drilling and 1100 hours for Assembly. Make sure that the units produced are integers. Solve for the above problem. a) How many units of each product will be made? b) What is the maximum profit? c) Which input has the most amount of slack?

A linear programming problem contains a restriction that reads "the quantity of Q must be at least as large as the sum of R, S, and T." Formulate this as a constraint ready for use in problem solving software.

Phil Bert's Nuthouse is preparing a new product, a blend of mixed nuts. The product must be at most 50% peanuts, must have more almonds than cashews, and must be at least 10% pecans. The blend will be sold in one-pound bags. Phil's goal is to mix the nuts in such a manner that all conditions are satisfied and the cost per bag is minimized. Peanuts cost $1 per pound. Cashews cost $3 per pound. Almonds cost $5 per pound and pecans cost $6 per pound. Identify the decision variables of this problem. Write out the objective and the set of constraints for the problem. Do not solve.