solutions - ENGR62/MS&E111 Introduction to Optimization...

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Spring 2005 Introduction to Optimization June 2, 2005 Prof. Ben Van Roy Final Exam: Due Friday June 3rd, 2005, at 5:00PM Instructions Sign the honor code statement and return this hand-out when you submit your solutions. Turn in your exam to Prof. Van Roy’s office, Terman 315. The exam will be graded out of total of 100 points. Questions 1–4 are each worth 25 points. The extra credit problem is worth an additional 5 points. Partial credit will be awarded for work that is handed in that is relevant and partially solves a problem. Honor Code In recognition of and in the spirit of the Honor Code, I certify that I will neither give nor receive unpermitted aid on this examination and that I will report, to the best of my ability, all Honor Code violations observed by me. Signature: Name:
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1. Ice Cream Production (25 points) Chin’s Ice Cream Co. is producing three flavors of ice cream: vanilla, mint chip, and chocolate. These products are manufactured using common ingredients: milk, eggs, and pre-made powder with unit prices $5, $3, and $10, respectively. The three products also rely on common manufacturing processes: mixing and freezing. The cost is $1.5 per hour per gallon for mixing, and $1 per hour per gallon for freezing. Production of a gallon of vanilla ice cream requires 3 units of milk, 2 units of eggs, 1 unit of pre- made powder, 2 hours of mixing, and 5 hours of freezing. Production of a gallon of mint chip ice cream requires 4 units of milk, 1.5 units of eggs, 0.5 units of pre-made power, 2 hours of mixing, and 4 hours of freezing. Production of a gallon of chocolate ice cream requires 3.5 units of milk, 1 unit of eggs, 1.5 units of pre-made powder, 1 hour of mixing, and 3 hours of freezing. Each of the three flavors sells for $80 per gallon. A total of 1000 units of milk, 500 units of eggs, and 250 units of pre-made powder are delivered to the warehouse each day. The factory is equipped to mix up to 20 gallons per hour and freeze up to 40 gallons per hour, for up to 24 hours per day. A contractual arrangement with a restaurant chain requires that Chin’s produces at least 50 gallons of vanilla ice cream per day. The company’s objective is to maximize profit. a) Formulate a linear program that can be used to determine how much vanilla, mint chip, and chocolate ice cream Chin’s should produce each day. Solve the linear program and provide the numerical quantities. b) Formulate the dual of the linear program formulated in part (a). c) Provide the optimal dual objective value. Explain in no more than sixty words why this can be done by inspection, without solving the dual. d) Identify dual variables that are equal to zero at an optimal solution to the dual. Explain in no more than sixty words why this can be done by inspection, without solving the dual.
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Solution a) The formulation: max 80( x 1 + x 2 + x 3 ) - (15 + 6 + 10 + 3 + 5) x 1 - (20 + 4 . 5 + 5 + 3 + 4) x 2 - (17 . 5 + 3 + 15 + 1 . 5 + 3) x 3 s . t . 3 x 1 + 4 x 2
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This note was uploaded on 10/12/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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solutions - ENGR62/MS&E111 Introduction to Optimization...

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