I Introduction to Decision Science First Exam - Fall 2016 Name: _______________________________________ This is a TAKE-HOME Exam which consists of 5 questions on 7 pages (CHECK NOW TO ENSURE THAT YOU HAVE A COMPLETE EXAM). You are required to answer all 4 questions. Answer each question thoroughly and show ALL your calculations and printouts. Give clear explanations for your answers where necessary. Completed exams are due by 11:59 p.m. on Wednesday, October 19th. You may contact the instructor for clarification of any question. However, no questions about the correctness of your answers will be entertained. You can email any queries to me at smallm@etsu.edu or call the office (423) 439-5303 and leave a message (but e-mail is better). EACH STUDENT IS EXPECTED TO ABIDE BY THE SENTIMENTS CONTAINED IN THE FOLLOWING HONOR STATEMENT TAKE-HOME EXAM HONOR STATEMENT: I did not request or receive assistance in preparing the answers to this exam. (Clarifications that you seek from the instructor will not be counted as assistance). 1 Question 1. Felix Navidad, an independent artist, paints large sculptured seasonal-themed figurines for sale during the holiday season. This year he will focus on painting and selling Santa Claus and Reindeer figurines. Each Santa sold will generate a net profit contribution of $18 and each Reindeer will provide a contribution of $22.50. Each Santa requires 2 hours of painting and each reindeer 2.5 hours. Felix will have 180 hours to devote to painting this season. Felix will purchase the unpainted figurines and has decided, based on past experience, to spend no more than $250 and $360 respectively to purchase the unpainted Santas and reindeers. Each unpainted reindeer costs $6 and each unpainted Santa $5. Felix has an empty storage barn on his property that has sufficient space to store as many figurines as he may need to purchase. 1. Given the information covered in the course so far, determine the type of problem represented above. What assumptions did you make in coming to this determination? 2. Provide the mathematical formulation of the problem. 3. Indicate, based on what we have covered in this course, the various methods that can be used to solve this type of problem? 4. If you think that the graphical method can be used to solve this problem, use it to solve the problem. 5. What is the optimal solution to the problem (the values of the decision variables) and the objective function value at the optimal solution? 6. Describe the special features, if any, of the solution. 7. Felix has asked you for advice on implementing this solution. What advice would you give him? 8. You have some concerns about the way that Felix has determined how much he is willing to invest in purchasing Santas ($250) and reindeers ($360). Suppose his wife has just contributed an additional $390 towards the purchase of the unpainted figurines, suggest at least one modification to the investment constraints in the model that you developed in part 2 that might provide a higher profit for Felix? 9. Determine whether the modification you suggested in part 8 results in higher profits. 2 Question 2. Provide the graphical solution for the linear programming problem below: Use graph paper, choose appropriate scales, and ensure that your lines are drawn using a straightedged ruler. Min: Subject to: 50X + 50Y 17X 10X 8X 7X 7X -5X + 14Y 1190 150 320 + 5Y 175 - 11Y 70 + 6Y 150 X, Y 0 .... ..... ..... ..... ..... ..... (1) (2) (3) (4) (5) (6) a) Identify each constraint (1, 2, 3, 4, 5, and 6) and the objective function line on your graph. b) Shade the feasible region. c) What is the optimal solution? d) What is the associated objective function value? 3 Question 3. Consider the following cost minimization linear programming problem: Min 480a + 830b + 480c 3a + 6b + 5c 40 5a + 7b + 3c 40 2a + 8b + 3c 40 (a) Use Lingo or any other linear programming software that is legally available to you to obtain a solution to this problem (b) What is the optimal solution and what is the objective function value for this problem? (c) Interpret the reduced cost for A? (d) Interpret the dual prices of the three constraints. (e) Indicate which constraints are binding and why they are binding. (f) Explain fully what would happen if the objective function coefficient of A is increased by 30. (g) Explain fully what would happen if the right hand side of constraint 1 is increased by 10 and the right hand side of constraint 2 is decreased by 4 simultaneously. 4 Question 4. Consider the following linear programming problem: Min 2A + 2B s.t. A + 3B 12 3A + B 13 A- B=3 A, B 0 a. Rewrite this problem in standard form. b. At the optimal solution to this problem B = 2.25; and constraint 1 is binding, what is the value of A? c. What is the objective function value at the optimal solution? d. Find the slack/surplus for each constraint and determine which constraints are binding. 5 Question 5. A simplex tableau is given below: X1 Basis Cb 5 S1 0 3 S2 0 9 S3 0 1 Zj 0 Cj-Zj 5 X2 8 4 15 -1 0 8 X3 12 5 20 2 0 12 S1 0 1 0 0 0 0 S2 0 0 1 0 0 0 S3 0 0 0 1 0 0 RHS 80 250 20 0 a) Is this the initial tableau? Why or why not? b) If it is the initial tableau, give the original formulation of the linear programming problem. c) What is the current solution and the objective function value? d) Is the current solution the optimal solution? Explain. e) If this is not an optimal solution, determine which variable will enter the basis and which variable will leave the basis at the next iteration. Explain your answer. 6 A subsequent simplex tableau is provided X1 Basis Cb 5 S1 0 .5 S2 0 -1 X3 12 .5 Zj 6 Cj-Zj -1 f) below: X2 8 6.5 25 -.5 -6 14 X3 12 0 0 1 12 0 S1 0 1 0 0 0 0 S2 0 0 1 0 0 0 S3 0 -2.5 -10 .5 6 -6 RHS 30 50 10 120 What is the current solution and objective function value for this tableau? g) Is this the optimal solution? Why or why not? h) If it is not the optimal solution, indicate which variable will enter the basis and which variable will leave the basis at the next iteration. If it is the optimal solution determine the values of the dual prices. 7