TIME: 3 hours (180 minutes) PAGES: 20 (including this cover page)

INSTRUCTIONS: 1. Total Mark Value: 100 Number of questions: 7 2. 1 page (single-sided-letter-sized) notes are permitted. 3. Stand-alone-non-programmable calculators are permitted. 4. Budget your time carefully. 5. Answers are to be given in the space provided. However, should you require additional space for a complete answer, use the blank page attached for this purpose. 6. For all problems where calculation space is provided, show your reasoning and work. Unsubstantiated answers will usually receive no marks. 7. You are to stop writing immediately upon being told that the exam is over. The exam time includes the time to write your name. If you fail to stop immediately, a penalty will be applied. TOTAL MARKS Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 TOTAL 16 15 13 16 14 6 20 100 MARKS RECEIVED

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Question 1 Graphical solution (16 marks) For a linear programming model given below: Decision variables x1 Units of product 1 to produce. x2 – Units of product 2 to produce. Objective function Maximize 4.0x1 + 3.6x2 Constraints Constraint 1: 11x1 + 5x2 > 55 Constraint 2: 3x1 + 4x2 < 36 Constraint 3: 4x1 – 9x2 < 0 Nonnegativity: x1, x2 >= 0 Solve this linear programming model by using the graphical approach (Graph paper is provided on the next page). For your graphical solution, Label the axes. Draw and label each constraint. Show your procedure of drawing Constraint 3 only. For each constraint line, determine and label which side is feasible. Briefly explain how to determine the feasible side for Constraint 3 only. Shade and label the feasible region. Identify all feasible corner points and determine the coordinates of each feasible corner point. Show only your calculations for the corner point determined by Constraints 1 and 2. Determine the optimal solution and objective function value. For all calculations in this question, please keep two significant decimal places. Calculation or explanation area:

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The optimal solution is x1 = _______ and x2 =______ with an objective function value of ______. Which constraints are binding? _____________________ Briefly explain how to obtain the optimal solution.

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Question 2 Linear programming formulation (15 marks) Steelco manufactures two types of steel at three different steel mills. During a given month, each steel mill has 200 hours of blast furnace time available. Because of differences in the furnaces at each mill, the time and cost to produce a ton of steel differ for each mill, as listed below: Production cost $/ton Mill1 Mill2 Mill3 Steel 1 $10 $12 $14 Steel 2 $11 $9 $10

Blast furnace minutes/ton Steel 1 Steel 2 Mill1 20 22 Mill2 24 18 Mill3 28 30 Each month, Steelco must manufacture at least 500 tons of steel 1 and 600 tons of steel 2. Determine how Steelco can minimize the cost of manufacturing the desired steel. Formulate a linear programming model for this problem AND DO NOT SOLVE. Decision variables (2 marks):

Objective function (2 marks):

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Constraints (11 marks):

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Question 3: Linear programming formulation (13 Marks) A wood mill cuts 20-foot pieces of wood into several different lengths: 8-foot, 10-foot, and 12foot. The mill has 350 pieces of 20-foot stock on hand and receives orders for the various sizes. The objective is to fill the orders with as little waste as possible. For example, if two 8-foot lengths are cut from a 20-foot piece, there will be a waste of 4 feet as the leftover amount. Currently, the mill has the following orders that are to be filled from the stock of 350 pieces of 20-foot wood on hand: Size in feet Number ordered 8 276 10 100 12 250 Formulate this problem as a linear programming model. DO NOT SOLVE. Decision variables (2 marks):

Objective function (2 marks):

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Constraints (9 marks)

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Question 4: Linear programming formulation (16 marks) Bill Cooper is an...