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Student Name______________________________ ID______________________________ Section_____________________________ BUAD-311 Operations Management Mid-term Exam I Fall 2007 Open Book, Open Notes, No Lap Tops State all your assumptions clearly Make sure that you have 6 pages (Including the cover page) Good Luck VERSION A 1 Section A: Multiple Choices [4 pts each] 1. Comparing make-to-order systems and make-to-stock systems, which of the following statement is false? a) Make-to-order systems are more responsive to the customer's needs. b) Make-to-order systems have less inventory spoilage. c) Make-to-stock systems can fulfill customer demand in a shorter waiting time. d) Make-to-stock systems can have a higher annual utilization rate for seasonal items. e) None of the above. e) 2. Which of the following will not reduce the patient's waiting time in a clinic. a) Make appointments b) Hire more full-time doctors/nurses c) Hire more part-time doctors/nurses during the rush hours d) Build excess inventory on medical supplies e) None of the above d) 3. A company has five machines and two products. Product X will be processed on Machine A, and then D, then E. Product Y will be processed on Machine B, then C, then D, then E. The demands for both products are 50 units per week. The capacities (units/week) of the machines are marked in the graph on the right. Which machine is the bottleneck? a) A b) B c) C d) D e) E d) 4. Which of the following is not one of the advantages of flexible manufactories? a) Higher utilization rate b) Higher responsiveness in changing demand c) Lower training cost d) Equipment last longer e) None of the above c) A: 60 D: 90 C: 70 E: 100 B: 80 2 5. Given the feasible region as the shaded area in the following graph and the objective function maximizing (x y), which of the following statement is true? y 8 4 (4,4) (2,2) 4 8 x a) b) c) d) e) e) x = -1 and y = 4 is a feasible solution. x 3y >= 0 is one of the constraints. x + y >= 8 is one of the constraints. The optimal solution is unbounded. The optimal solution is not unique. 6. Consider the following linear program: Maximize 2x + 4y Subject to: 4x + 6y <= 120 2x + 6y <= 72 y <= 10 x >=0, y>= 0. The optimum solution is x = 24 and y = 4. The sensitivity report created by solver follows. Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease x 24 0 2 0.666666667 0.666666667 y 4 0 4 2 1 Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease 4x + 6y <= 120 120 0.333333333 120 24 36 2x + 6y <= 72 72 0.333333333 72 18 12 y <= 10 4 0 10 1E+30 6 Suppose the coefficient of y in the objective function is 8 instead of 4. Which of the following statement is true? a) The optimal solution remains (24, 4). b) The optimal solution will have a y value smaller than 4. c) We do not have enough information from the sensitivity report to determine the optimal solution. d) The optimal objective value will increase by 16. e) None of the above. c) 3 Section B: Problems 1. [24 pts] 21 Choices Ice Cream decides to open a new store on USC campus. The company estimates that during the rush hour, a customer will show up every two minutes. To make an order of its famous customized ice cream, it takes a worker two minutes to cut up the ingredients, one minute to mix the ingredients and ice cream with a mixer, one minute to accept the payment using a POS (point of sale) system. a) [12 pts] To satisfy the rush hour demand, how many workers should the new store hire, how many mixers and POS systems should the store purchase? Worker: 4 minutes per order. 4/2 = 2 workers are required (4pts) Mixer: 1 minute per order. = 0.5, 1 mixer is sufficient (4pts) POS system: 1 minute per order. = 0.5, 1 POS system is sufficient (4pts) b) [12 pts] The USC Food Safety Regulations requires that no cashiers (defined as the one accepts the payment) can prepare the food. How many workers should the new store hire? Cashier: 1 minute per order. = 0.5, 1 cashier is sufficient (4pts) Food Preparers: 3 minute per order. 3/2 = 1.5, 2 preparers are required (4pts) Total workers: 1+2 =3 workers are required (4pts) 4 2. [28 pts] Kristen's cookie store has been a success and she never needs to worry about demands. Now she sells lines two of cookies, standard and premium. The cookies are sold in dozens. Standard cookies retail for $6 per dozen, and premium cookies retail for $9 per dozen. For the coming month, Kristen has rent an oven which is capable of cooking 1100 dozen cookies. She also has enough ingredients for 1200 dozen standard cookies or 800 dozen premium cookies. Furthermore, she has contracted 200 hours labor. One dozen standard cookies need 10 minutes labor and one dozen premium cookies need 12 minutes labor. a) [14 pts] Kristen wonders the optimal product mix that maximizes her revenue. Write down the corresponding linear program. List the decision variables, objective function, and the constraints. Decision Variables: Objective Function: Constraints: x_1 dozens of standard cookies x_2 dozens of premium cookies maximize 6x_1+9x_2 x_1 + x_2 <= 1100 x_1/1200+x_2/800 <= 1 10x_1+12x_2 <= 200*60 = 12000 x_1, x_2 >= 0 (2pts) (2pts) (2pts) (2pts) (2pts) (2pts) (2pts) b) [14 pts] Kristen is thinking about retiring and her roommate would like to acquire Kristen's ingredients and her right to use the oven and the labor. To make the transaction, her roommate needs to name the prices for the resources. Kristen has agreed that she will accept her roommate's proposal as long as the prices make cooking both standard cookies and premium cookies unattractive. Kristen's roommate wants to find the prices that minimize her acquiring cost. Formulate her problem as a linear program. List the decision variables, objective function, and the constraints. Decision Variables: y_1 price for using oven baking a dozen y_2 price for all the ingredients y_3 price for a minute labor minimize 1100y_1+y_2+12000y_3 y_1 + y_2/1200 + 10y_3 >= 6 y_1 + y_2/800 + 12y_3 >= 9 y_1, y_2, y_3 >= 0 (2pts) (2pts) (2pts) (2pts) (2pts) (2pts) (2pts) Objective Function: Constraints: (If you have 4 constraints in a), i.e., two ingredient constraints, you get full credit for b if your formulation is the dual of a) ) 5 3. [24 pts] A local grocery store has two cashier stations, and one experienced cashier and one novice cashier. On average it takes 6 minutes for the novice cashier to serve one customer and 3 minutes for the experienced cashier to serve one customer. During a typical working day (8 hours), 120 customers will show up. The novice cashier will serve 48 customers and the experienced cashier will serve 72 customers. a) [4 pts] During the rush hours, approximate 25 customers will show up during an hour. Does the store have enough capacity for the rush hour? The novice cashier can serve 10 customers per hour. The experienced cashier can serve 20 customers per hour. They can serve 30 customers per hour in total. The store has enough capacity for the rush hour. b) [12 pts] The manager finds that during a day, both cashier stations have two customers waiting on average. What is the average waiting time in the novice cashier's line? What is the average waiting time in the experienced cashier's line? What is the average waiting time of a customer shopping at the store? Average waiting time in the novice cashier's line 2/(48/8)=1/3 hour = 20 minutes (4pts) Average waiting time in the experienced cashier's line 2/(72/8)=2/9 hour = 13.3 minutes (4pts) Average waiting time in the store (2+2)/(120/8)=4/15 hour = 16 minutes (4pts) (Another way to get the average waiting time in the store: 20*(48/(48+72))+13.3*(72/(48+72))=16 minutes ) c) [8 pts] To shorten the waiting-time, the manager does a detailed study and finds that half of the customers purchase 5 items or less, and half of the customers purchase more than 5 items. The manager decides to let the novice cashier only serves the customers purchase 5 items or less. After the change, it turns out that both cashier stations have 1.75 customers waiting on average. Assume that the novice cashier serves all of the customers purchasing 5 items or less and the experienced cashier serves all of the customers purchasing more than 5 items. What is the average waiting time in the novice cashier's line? What is the average waiting time in the experienced cashier's line? Average waiting time in the novice cashier's line 1.75/(120*0.5/8)=7/30 hour = 14 minutes Average waiting time in the experienced cashier's line 1.75/(120*0.5/8)=7/30 hour = 14 minutes (4pts) (4pts) 6

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