1 hour 15 minutes. ( 9 pages + 1 page extra graph paper + 1 page fun reading ) Question Total Score (a) 15. (b) 20. (c) 20. (d) 20. (e) 15.

Transcription

1 Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Fall 2006 (October 25): Midterm 1 hour 15 minutes ( 9 pages + 1 page extra graph paper + 1 page fun reading ) Question Total Score (a) 15 (b) 20 (c) 20 (d) 20 (e) 15 (f) 25 (g) 15 (h) 10 (i) Hermione: " "I hope you're pleased with yourselves. We could have been all killed -- or worse, expelled." Your situation is not as dire. Have a good test! ~ E.C. Name: 1

2 Bertie Bott s every flavor jellybeans is a favorite amongst Harry Potter and his friends, who frequent Honeydukes in Hogsmead to purchase these tasty treats. "When they say every flavor, they mean every flavor..." ~ Ron Weasley Picture excerpted from harrypotter.warnerbros.com/bertiebotts "Bertie Bott's Every Flavor Beans A Risk With Every Mouthful!" ~ Quidditch World Cup ad Bertie Bott s would like to make some decisions on the quantity of Brussel Sprouts, Marmalade, Earwax, and Sardine flavor jellybeans to produce, each of which requires special potions and labor to create. Each batch of Brussel Sprouts, Marmalade, Earwax, and Sardine jellybeans sell for 4, 2, 3, and 5 sickles respectively. Bertie Bott s has no problems selling every jelly bean it makes. Each batch of Brussel Sprouts jellybeans requires 2 units of potions and 8 units of labor. Each batch of Marmalade jellybeans requires 3 units of potions and 1 unit of labor. Each batch of Earwax jellybeans requires 4 unit of potions and 1 unit of labor. Each batch of Sardine jellybeans requires 2 units of potions and 5 units of labor to produce. Assume that exactly 300 units of potions and exactly 300 units of labor have to be consumed in the production. (a) Formulate a linear programming model that will help Bertie Bott s determine how it can maximize profits subject to some constraints. [ 15 points ] E.C. 2

3 (b) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial basic feasible solution. Also identify the initial entering basic variable and the leaving basic variable. [20 points] E.C. 3

4 (c) Using the two-phase method, construct the complete first simplex tableau for phase I and identify the corresponding initial basic feasible solution. Also, identify the initial entering basic variable and the leaving basic variable. [20 points] E.C. 4

5 (d) You are given the final tableau of Phase I below. Construct the initial tableau for Phase II, and identify the corresponding initial basic feasible solution. Also identify the initial entering basic variable and the leaving basic variable. [20 points] Z x 1 x 2 x 3 x 4 x 5 x 6 RHS Z x /15 1 1/5 4/15-1/15 60 x /30 0 3/5-1/30 2/15 30 E.C. 5

6 (e) Construct the dual of the problem in part (a). [ 15 points ] E.C. 6

7 (f) Use the graphical method to find the solution to the dual. Identify the solution and the binding constraints and label them with on the graph. [ 25 points ] If you need to re-do your graph, you can find extra graph paper on page 10. E.C. 7

8 (g) Now, augment the dual problem with only surplus variables, and use your findings in part (f) to propose an optimal solution, with values for all the surplus variables, for the problem. [ 15 points ] (h) Given the solution to the dual you have derived in part (f), use the complementary slackness property to identify the basic and non-basic variables in the optimal solution to the original problem in part (a). [ 10 points ] E.C. 8

9 (i) Now, return to the initial tableau for Phase II given in part (d). You can either continue from your results from part (d) or use your results in part (h) to find the optimal solution for the primal problem. [ 20 points ] E.C. 9

10 Extra graph paper E.C. 10

11 In case you have extra time for some reading, here s a little information on Bertie Bott s, shamelessly lifted from the Harry Potter Lexicon ( Bertie Bott's Every Flavor Beans One of the most famous sweets in the wizarding world is Bertie Bott's Every Flavor Beans. These beans are essentially the same as Muggle jelly beans, but with flavors that Muggles would never expect. Bertie Bott's Every Flavor Beans was creating by inventor Bertie Bott ( present) quite by mistake. His original purpose was to create tasty candies from food but he accidentally included a pair of dirty socks in his experiment. Bertie recognized the sales potential of a candy that presented a "a risk with every mouthful!" According to Ron, there are ordinary flavors like chocolate and peppermint and marmalade, but also more adventuresome ones like spinach and liver and tripe. George said he'd had a booger-flavored one. Dumbledore said that, in his youth, he ate a vomitflavored one and was put off Bertie Bott's Beans for quite a while. Eventually he tried what he thought might be a "nice toffee," only to discover that the flavor was actually ear wax. Bertie Bott's Every Flavor Beans are a popular sweet for kids at Hogwarts. They buy them from the lunch trolley on the Hogwarts Express and from Honeydukes in Hogsmeade. They give them for Christmas gifts and as gifts when someone is ill and in the hospital wing. Bertie Bott's Beans are sold in bags. flavors of Bertie Bott's Every Flavor Beans NOTE: These are the only flavors mentioned in the books. * baked beans * booger (at least George Weasley said so) * chocolate * coconut * coffee * curry * ear wax * grass * liver * marmalade * pepper * peppermint * sardine * spinach * sprouts * strawberry * toast * tripe * vomit In the US and Canada, the Muggle version of Bertie Bott's Every Flavour Beans are made by the Jelly Belly company. They are sold in bags and also in boxes. The flavors include sardine, pepper, grass, horseradish, vomit, booger, earwax, and dirt. New flavors have been added recently: earthworm, spaghetti, spinach, and soap. Personally, I like the pepper and grass flavors the best, although spaghetti is interesting too. E.C. 11