MSA 640 Homework #2 Due September 17, points total / 20 points per question Show all work leading to your answers

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1 Name MSA 640 Homework #2 Due September 17, points total / 20 points per question Show all work leading to your answers 1. The annual demand for a particular type of valve is 3,500 units. The cost of each valve is $70, and the inventory carrying cost is 10% of the cost of each valve. The average ordering cost is $21 per order. Furthermore, it takes about two weeks (assume a 7-day week) for an order to arrive from the supplier, and during this time, the daily demand for valves is approximately 12. Use additional paper if necessary. N.B.: See the last page of the exam for a list of essential formulas. (a) (5 points) What is the EOQ? (144.91) (b) (3 points) What is the reorder point? (168) (c) (6 points) What is the average inventory? (72.46) What is the annual holding cost? (507.2) (d) (6 points) How many orders per year would be placed? (24.15) What is the annual ordering cost? (507.2) Page 1 of 5

2 Name 2. Consider the following linear programming problem: Maximize 10X + 12Y Subject to: 14X + 3Y 210 3X + 6Y 180 all variables 0 (a) Create a graphical LP solution on the attached graph paper, show the feasible region, and label the corner points. Show calculations for plotting the constraints and use simultaneous linear equations to determine corner point(s) when necessary. Use additional paper or the reverse of this sheet if necessary. Calculations for Plotting Constraints Calculations for Simultaneous Linear Equations Calculations for Corner Points (b) Compute values for all corner points and identify the maximum possible value for the objective function. (max value 398.4) Corner X Y Value A B C D Page 2 of 5

3 Problem

4 Name 3. Consider the following linear programming problem: Maximize 16X + 17Y Subject to: 8X + 4Y X + 6Y 780 9X + 12Y 1080 all variables 0 (a) Create a graphical LP solution on the attached graph paper, show the feasible region, and label the corner points. Show calculations for plotting the constraints and use simultaneous linear equations to determine corner point(s) when necessary. Use additional paper or the reverse of this sheet if necessary. Calculations for Plotting Constraints Calculations for Simultaneous Linear Equations Calculations for Corner Points (b) Compute values for all corner points and identify the maximum possible value for the objective function. (max value 1,566.3) Corner X Y Value A B C D E Page 3 of 5

5 Problem

Name 4. A development project has 13 major activities. The estimated times for the activities appear on the attached PERT Network. (a) Compute the ES, EF, LS, LF values and fill in the network chart.

6 Name 4. A development project has 13 major activities. The estimated times for the activities appear on the attached PERT Network. (a) Compute the ES, EF, LS, LF values and fill in the network chart. (b) What is the expected completion time for this project? (44) (c) Find the critical path for this project by computing slack times. (C E F H I K-M) (d) Assuming that the standard deviation is 3.2, what is the probability that the project will be finished in fewer than 46 weeks? Finish drawing and labeling the following normal curve to represent your solution. (0.73) Probability Calculations Page 4 of 5

7 PERT/CPM Project Network for Problem #4 Project Completion Time: Critical Path: Project Standard Deviation: 3.2 A 6 D 4 L 2 F 7 H 8 J 4 Start B 5 Finish G 4 I 5 K 3 C 8 E 9 M 4

8 Name 5. Automobiles arrive at the drive-through window at a fast food store at the rate of 7 every 10 minutes. The average service time is 10 every 12 minutes. The arrival rate is Poisson distributed and the service time is exponentially distributed. (a) (4 points) What are the values for lambda (λ) and mu (µ)? (42, 50 per hour) (b) (2 points) What is the average time a car is in the system? (0.13 hour or 7.5 minutes) (c) (2 points) What is the average number of cars in the system? (5.25) (d) (2 points) What is the average time that cars spend waiting in line to receive service? (e) (2 points) What is the average number of cars waiting in line to receive service? (f) (3 points) What is the probability that there are no cars at the food store? (0.16) (g) (2 points) What percentage of the time is the food store busy? (h) (3 points) What is the probability that there are more than 2 cars in the system? (0.59) Page 5 of 5

9 Homework 2 Essential Formulas Chapter 6 Inventory Control Models Annual Ordering Cost Annual Holding Cost Economic Order Quantity C h = I * C Reorder Point D Co Q Q C h 2 * 2 D C EOQ = Q = C EOQ = Q * = h 2 D C ROP = d L Average Inventory Q / 2 Optimal # of Orders/Year D / Q Example of Objective Function Example of Constraints Graphical Approach Corner Point Solution Chapter 7 Linear Programming 20X + 35Y I C 15X + 25Y X + 18Y 825 Determine where constraints cross axes Plot constraints Determine feasible region Determine corner points Use simultaneous equations if necessary Apply Objective Function to corner points Determine Optimal Solution o o Earliest finish time Earliest start time Latest start time Latest finish time Critical Path Probability of Project Completion Chapter 13 PERT Project Management EF = ES + time ES = largest EF of immediate predecessors LS = LF time LF = Smallest LS of following activities (1) Slack = LS ES or Slack = LF EF (2) Activities with Slack = 0 are critical activities and are on the critical path Use project completion time for the mean and the standard deviation. Z = X µ σ Chapter 14 Waiting Lines and Queuing Theory Models M/M/1 Average number in the system: Average length of the queue: Average time in the system: λ L = µ λ λ Lq = µ ( µ λ) 2 W = 1 µ λ λ Average time in the queue: Wq = µ ( µ λ) Percentage of time the system is idle: Utilization of the system: Probability number in the system > k: P 0 λ ρ = µ P n λ = 1 µ λ > k = µ k+1

10 Normal Distribution Table Z

CHAPTER 12. (The interpretation of the symbols used in the equations is given in page 3)

CHAPTER 12. (The interpretation of the symbols used in the equations is given in page 3) CHAPTER 12 The equations needed: (The interpretation of the symbols used in the equations is given in page 3) 1. ATI = t D 35 4. ws = I average weekly CU 2. I turn = CU I average 5. ds = I average daily