COURSE/SUBJECT: Quantitative Methods Please show ALL workings/explanations

Quantitative Methods

Show ALL workings/explanation

QUESTION 1

Consider the activities, durations, and predecessor relationships in the following network. Draw the network and answer the questions that follow.

Activity Description	Immediate Predecessor(s)	Optimistic (Weeks)	Most Likely (Weeks)	Pessimistic (Weeks)
	---			10
				20
			12	16

	D, C			22


				10
	E, G, H			12

a. What is the expected time for activity B?

b. What is the variance for activity B?

c. Based on the calculation of estimated times, what is the critical path?

d. What is the estimated time of the critical path?

e. What is the activity variance along the critical path?

f. What is the probability of completion of the project before week 42?

QUESTION 2

Karel Smith is the purchasing manager for the headquarters of a large insurance company with a central inventory operation. Karel’s fastest-moving inventory item has a demand of 6,000 units per year. The cost of each unit is $100, and the inventory carrying cost is $10 per unit per year. The average ordering cost is $30 per order. It takes about 5 days for an order to arrive, and the demand for 1 week is 120 units. (This is a corporate operation, and there are 250 working days per year.)

a) What is the EOQ?

b) What is the average inventory if the EOQ is used?

c) What is the optimal number of orders per year?

d) What is the optimal number of days in between any two orders?

e) What is the annual cost of ordering and holding inventory?

f ) What is the total annual inventory cost, including the cost of the 6,000 units?

B). As an inventory manager, you must decide on the order quantity for an item. Its annual demand is 679 units. Ordering costs are $7 each time an order is placed, and the holding cost is 10% of the unit cost. Your supplier provided the following price schedule.

Quantity	Price per Unit
1 - 100	$5.65
101 - 350	$4.95
351 or more	$4.55

What ordering-quantity policy do you recommend?

QUESTION 3

A). Favors Distribution Company purchases small imported trinkets in bulk, packages them, and sells them to retail stores. The managers are conducting an inventory control study of all their items. The following data are for one such item, which is not seasonal.

Use a trend projection to estimate the relationship between time and sales (state the equation).
Calculate forecasts for the first four months of the next year.

										10	11	12
Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Sales	51	55	54	57	50	68	66	59	67	69	75	77

B). Consider the following annual sales data for 2001-2008:

Year	Sales
2001
2002
2003	10
2004
2005	14
2006	18
2007	17
2008	20

Use the linear regression method and determine the estimated sales equation.
Calculate the coefficient of determination.
Calculate the correlation coefficient.

QUESTION 4

A). A crew of mechanics at the Highway Department Garage repair vehicles that break down at an average of λ = 8 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 11 vehicles per day with a repair time distribution that approximates an exponential distribution. The crew cost is approximately $300 per day. The cost associated with lost productivity from the breakdown is estimated at $150 per vehicle per day (or any fraction thereof). Which is cheaper, the existing system with one service crew, or a revised system with two service crews?

B). A crew of mechanics at the Highway Department Garage repair vehicles that break down at an average of λ = 7 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 11 vehicles per day with a repair time distribution that approximates an exponential distribution.

1. What is the utilization rate for this service system?
2. What is the average time before the facility can return a breakdown to service?
3. How much of that time is spent waiting for service?
4. How many vehicles are likely to be waiting for service at any one time?

C). A dental clinic at which only one dentist works is open only two days a week. During those two days, the traffic arrivals follow a Poisson distribution with patients arriving at the rate of three per hour. The doctor serves patients at the rate of one every 15 minutes.

(a) What is the probability that the clinic is empty (except for the dentist and staff)?

(b) What is the probability that there are one or more patients in the system?

(d) What is the probability that there are four or more patients in the system?