This is your. is a 100-minute exam (1hr. 40 min.). There are 4 questions (25 minutes per question). To avoid the temptation to cheat, you must abide by these rules then sign below after you understand the rules and agree to them: Turn off your cell phones. You cannot leave the room during the exam, not even to use the restroom. The only things you can have in your possession are pens or pencils and a simple non-graphing, non-programmable, non-text calculator. All other possessions (including phones, computers, or papers) are prohibited and must be placed in the designated corner of the room. Possession of any prohibited item (including phones, computers, or papers) during the exam (even if you don t use them but keep them in your pocket) earns you a zero on this exam, and you will be reported to the Academic Integrity Committee for further action. Print name here: Sign name here: Each individual question on the following exam is graded on a 4-point scale. After all individual questions are graded, I sum the individual scores, and then compute that total as a percentage of the total of all points possible. I then apply a standard grading scale to determine your letter grade: 90-100% A; 80-89% B; 70-79% C; 60-70% D; 0-59% F Finally, curving points may be added to letter grades for the entire class (at my discretion), and the resulting curved letter grade will be recorded on a standard 4-point numerical scale. Tip: Explain your answers. And pace yourself. When there is only ½ hour left, spend at least 5 minutes outlining an answer to each remaining question. 1
There are no computer questions on this exam. On each question, you may only use blank or graph paper, pencils, a ruler, and a calculator. You may not use a computer or notes. 2
Rounding Off Question 1. Jacuzzi produces two types of hot tubs: Fuzion and Torino. There are 5 pumps, 36 hours of labor, and 60 feet of tubing available to make the tubs per week. Here are the input requirements, and unit profits: Fuzion Torino Pumps 1 1 Labor 9 hours 4 hours Tubing 8 feet 15 feet Unit Profit $6 $9 a. Develop a linear programming model for this problem to determine how much should be produced. b. Graphically solve the linear-programming problem from Part a if you require that production units be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that production units be integers (instead, production units are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 3
Part a: Let F = Fuzion units produced per week. Let Y = Torino units produced per week. Part c: Max 6 F + 9 T s.t. F + T < 5 (pumps) 9 F + 4 T < 36 (labor hours) 8 F + 15 T < 60 (tubing) F, T 0 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 A graph of the feasible set and isovalue lines (dashed lines above) reveals the optimum occurs where the first and the third constraint bind. Solving the binding form of those two constraints yields the optimal solution: F = 2.143, T = 2.857 4
Part b: Max 6 F + 9 T s.t. F + T < 5 (pumps) 9 F + 4 T < 36 (labor hours) 8 F + 15 T < 60 (tubing) F, T 0 Part c: 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 A graph of the feasible set dots and isovalue lines (dashed lines above) reveals the optimum occurs at either (0,4) or (1,3) or (3,2). It turns out there are two optima: (0,4) and (3,2). Part d: The integer solutions in Part b are not the result of rounding off the continuous solution in Part c. 5
Transshipment Problems with New Nodes Question 2. Apple supplies distribution centers in California, Colorado, Florida, and New York with smart phones. Apple orders computers from manufacturers in Arizona and China. Each day s demand for phones are 400 for California, 200 for Colorado, 300 for Florida, and 500 for New York. Arizona can supply up to 800 computers and China can supply up to 1200 computers. Apple can ship from Chicago, or from Las Vegas, or from Newark (or from any combination). If any units are shipped from Chicago, there is a fixed cost of 100 per day. If any units are shipped from Las Vegas, there is a fixed cost of 70 per day. If any units are shipped from Newark, there is a fixed cost of 80 per day. Unit costs from the manufacturers to the shipment centers are: Chicago Las Vegas Newark Arizona 2 5 4 China 9 8 6 The unit costs to distribution centers are: California Colorado Florida New York Chicago 3 4 6 5 Las Vegas 5 6 4 7 Newark 2 5 8 3 Formulate the linear program that minimizes the cost of meeting all demands at minimum cost. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 6
List origin, transshipment, and destination nodes in this transshipment problem: 1 = Arizona, 2 = China 3 = Chicago, 4 = Las Vegas, 5 = Newark, 6 = California, 7 = Colorado, 8 = Florida, 9 = New York. Define binary variables for transshipment nodes, Y3 = 1 if the Chicago facility is used; 0, if not Y4 = 1 if the Las Vegas facility is used; 0, if not Y5 = 1 if the Newark facility is used; 0, if not Define integer shipment variables just as in any transshipment problem, Xij = the units shipped from node i (i = 1, 2, 3, 4, 5) to node j (j = 3, 4, 5, 6, 7) on a particular day. The objective is minimize total cost. From cost data, shipping costs are 2X13 + 5X14 + 4X15 + 9X23 + 8X24 + 6X25 + 3X36 + 4X37 + 6X38 + 5X39 + 5X46 + 6X47 + 4X48 + 7X49 + 2X56 + 5X57 + 8X58 + 3X59 + From cost data, facility use fixed costs are 100Y3 + 70Y4 + 80Y5 7
Hence, the objective to minimize total costs is Min 2X13 + 5X14 + 4X15 + 9X23 + 8X24 + 6X25 + 3X36 + 4X37 + 6X38 + 5X39 + 5X46 + 6X47 + 4X48 + 7X49 + 2X56 + 5X57 + 8X58 + 3X59 + 100Y3 + 70Y4 + 80Y5 From capacity data, Arizona capacity constraint is X13 + X14+ X15 < 800 China capacity constraint is X23 + X24+ X25 < 1200 From demand data, California demand constraint is X36 + X46 + X56 > 400 Colorado demand constraint is X37 + X47+ X47 > 200 Florida demand constraint is X38 + X48 + X58 > 300 New York demand constraint is X39 + X49+ X49 > 500 Transshipment constraints are X36 + X37 + X38 + X39 < X13 + X23, through Chicago X46 + X47 + X48 + X49 < X14 + X24, through Las Vegas X56 + X57 + X58 + X59 < X15 + X25, through Newark And fixed-cost indicator constraints are X36 + X37 + X38 + X39 < 2000 Y3, through Chicago X46 + X47 + X48 + X49 < 2000 Y4, through Las Vegas X56 + X57 + X58 + X59 < 2000 Y5, through Newark 8
Capital Budgeting Question 3. Frys Electronics is planning to expand its sales operation by offering new electronic appliances. The company has identified seven new product lines it can carry. Product Line Investment Floor Space Expected Return (Percent) ($) (Square Feet) Projection TVs 6,000 125 16 Cell Phones 12,000 150 9 Plasma TVs 20,000 200 11 IPODs 14,000 40 10 DVD Players 15,000 40 8 PDAs 2,000 20 14 Computers 32,000 100 13 Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs. They will not stock both Cell Phones and PDAs. And they will stock DVD Players only if they stock Computers. Finally, the company wishes to introduce at most four new product lines. If the company has $45,000 to invest and 420 square feet of floor space available, formulate an integer linear program for Frys to maximize its overall expected return. But you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 9
Define the Decision Variables: x 1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on. Product line 1. Projection TVs Product line 2. Cell Phones Product line 3. Plasma TVs Product line 4. IPODs Product line 5. DVD Players Product line 6. PDAs Product line 7. Computers Define the Objective: Maximize total expected return. Max.16(6000)x 1 +.09(12000)x 2 +.11(20000)x 3 +.10(14000)x 4 +.08(15000)x 5 +.141(2000)x 6 +.132(32000)x 7 1. Constrain total investment by $45,000: 6000x 1 + 12000x 2 + 20000x 3 + 14000x 4 + 15000x 5 + 2000x 6 + 32000x 7 < 45,000 2. Constrain space by 420 square feet: 125x 1 +150x 2 +200x 3 +40x 4 + 40x 5 + 20x 6 + 100x 7 < 420 3. Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs: x 3 > x 1 and x 4 > x 1. Another linear way to write that constraint is x 3 + x 4 > 2x 1, and a nonlinear way is x 3 x 4 > x 1. 4. Frys will not stock both Cell Phones and PDAs: x 2 + x 6 < 1 5. Frys will stock DVD Players only if they stock Computers: x 5 < x 7 6. Frys will introduce at most four new product lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 < 4 10
Economic Analysis with Teamwork Question 4. Food on Foot is a 501(c)(3) nonprofit organization dedicated to providing the poor and homeless of Los Angeles with nutritious meals, clothing, and assistance in the transition to employment and life off the streets. Food on Foot operates a weekly meal program every Sunday in Hollywood. They are considering three alternative distribution systems. Arrivals to the distribution stand follow the Poisson distribution with an average of one homeless person arriving every 6 minutes. The cost for a homeless person waiting is $3 per hour. The first system involves one volunteer (who talks to the homeless and serves food and serves drinks) and costs $1 per hour to operate (the opportunity cost of the volunteer s time), and it has an exponential service rate, and it can serve an average of one homeless person every 3 minutes. The second system involves two volunteers working as a team (one talks to the homeless and serves food, and the other serves drinks) and costs $2 per hour to operate (the opportunity cost of the 2 volunteer s time), and it has an exponential service rate, and it can serve an average of one homeless person every 2 minutes. The third system involves three volunteers working as a team (one talks to the homeless, one serves food, and the third serves drinks) and costs $3 per hour to operate (the opportunity cost of the 3 volunteer s time), and it has an exponential service rate, and it can serve an average of one homeless person every 1.5 minutes. Which system should be chosen? You may use any or all of the following analytical formulas for an M/M/1 system to compute your answer: The probability of no units in the system: P 0 = 1- The average number of units in the waiting line: L q = 11
The average number of units in the system: L = L q + The average time a unit spends in the waiting line: W q = L q / The average time a unit spends in the system: W = 1/ The probability that an arriving unit has to wait: P w = Probability of n units in the system: P n = ( ) n P 0 12
First system: k = 1 channel, = 10 per hour, and = 20 per hour. The average number of units in the waiting line: L q = = 10*10/(20(10)) = 1/2 The average number of units in the system: L = L q + = 1/2 + 1/2 = 1 Cost to operate = $1 + $3*L = $4 per hour. Second system: k = 1 channel, = 10 per hour, and = 30 per hour. The average number of units in the waiting line: L q = = 10*10/(30(20)) = 1/6 The average number of units in the system: L = L q + = 1/6 + 1/3 = 1/2 = 0.50 Cost to operate = $2 + $3*L = $3.50 per hour. Third system: k = 1 channel, = 10 per hour, and = 40 per hour. The average number of units in the waiting line: L q = = 10*10/(40(30)) = 1/12 The average number of units in the system: L = L q + = 1/12 + 1/4 = 1/3 = 0.33 Cost to operate = $3 + $3*L = $4 per hour. Choose the second system. (Going beyond the basic answer, ask any other volunteers beyond two to put in extra time at work and donate their income. The homeless people will benefit more from the extra money than from the faster service.) 13