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Uncertain Activity Times in Project Management

This lecture covers the three-time estimate approach in project management, where activity completion times are assumed to follow a Beta distribution. It discusses the mean completion time and the optimistic and pessimistic completion time estimates.

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Uncertain Activity Times in Project Management

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  1. Lecture 5 Project Management Chapter 17

  2. Probabilistic/Uncertain Activity Times • In the three-time estimate approach, the time to complete an activity is assumed to follow a Beta distribution. • An activity’s mean completion time is: t = (a + 4m + b)/6 • a = the optimistic completion time estimate • b = the pessimistic completion time estimate • m = the most likely completion time estimate

  3. Example

  4. Management Scientist Solution

  5. Lecture 5 Inventory Management Chapter 11

  6. Types of Inventories • Raw materials & purchased parts • Partially completed goods called work in progress • Finished-goods inventories • (manufacturingfirms) or merchandise (retail stores) • Replacement parts, tools, & supplies • Goods-in-transit to warehouses or customers

  7. Functions of Inventory • To meet anticipated demand • To smooth production requirements • To decouple operations • To protect against stock-outs • To take advantage of order cycles • To help hedge against price increases • To permit operations • To take advantage of quantity discounts

  8. Objective of Inventory Control • To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds • Level of customer service • Costs of ordering and carrying inventory

  9. Key Inventory Terms • Lead time: time interval between ordering and receiving the order • Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year • Ordering costs: costs of ordering and receiving inventory • Shortage costs: costs when demand exceeds supply

  10. Inventory Classification SystemsABC Analysis • Divides inventory into three classes based on annual dollar volume • Class A - high annual dollar volume • Class B - medium annual dollar volume • Class C - low annual dollar volume • Used to establish policies that focus on the few critical parts and not the many trivial ones • No “hard-and-fast” rule to classify into different categories

  11. ABC Analysis Example $232,057

  12. Economic Order Quantity Models • Economic order quantity (EOQ) model • Economic production model (EPQ) • Quantity discount model

  13. The Inventory Cycle Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Time Place order Receive order Receive order Receive order Place order Lead time

  14. Annual carrying cost Annual ordering cost Total cost = + Q D S H TC = + 2 Q Total Cost Formula (11-1)

  15. Cost Minimization Goal The Total-Cost Curve is U-Shaped Annual Cost Ordering Costs Order Quantity (Q) QO (optimal order quantity)

  16. Deriving the EOQ & Minimum Total Cost The total cost curve reaches its minimum where the carrying and ordering costs are equal. Formula (11-2) Number of orders per year = D/Q0 Length of order cycle = Q0/D Formula (11-3)

  17. Inventory Management – In-class Example • Number 2 pencils at the campus book-store are sold at a fairly steady rate of 60 per week. It cost the bookstore $12 to initiate an order to its supplier and holding costs are $0.005 per pencil per year. • Determine • (a) The optimal number of pencils for the bookstore to purchase to minimize total annual inventory cost, • (b) Number of orders per year, • (c) The length of each order cycle, • (d) Annual holding cost, • (e) Annual ordering cost, and • (f) Total annual inventory cost. • (g) If the order lead time is 4 months, determine the reorder point. • Illustrate the inventory profile graphically. • What additional cost would the book-store incur if it orders in batches of 1000?

  18. Management Scientist Solutions (a) (d) (e) (f) (g) (b) (c)

  19. Assumptions of EOQ Model • Only one product is involved • Annual demand requirements known • Demand is even throughout the year • Lead time does not vary • Each order is received in a single delivery • There are no quantity discounts

  20. Economic Production Quantity (EPQ) • Economic production quantity (EPQ) model: variant of basic EOQ model • Production done in batches or lots • Replenishment order not received in one lump sum unlike basic EOQ model • Inventory is replenished gradually as the order is produced • hence requires the production rate to be greater than the demand rate • This model's variable costs are • annual holding cost, and • annual set-up cost (equivalent to ordering cost). • For the optimal lot size, • annual holding and set-up costs are equal.

  21. EPQ = EOQ with Incremental Inventory Replenishment

  22. EPQ Model Assumptions • Demand occurs at a constant rate of D items per year. • Production rate is p items per year (and p > u ). • u = usage rate • Set-up cost: $S per run. • Holding cost: $H per item in inventory per year. • Purchase cost per unit is constant (no quantity discount). • Set-up time (lead time) is constant. • Planned shortages are not permitted.

  23. EPQ Model Formulae • Optimal production lot-size (formula 11-5 of book) • Run time: Q */p • Time between set-ups (cycle time): Q */u years

  24. Example: Non-Slip Tile Co. • Non-Slip Tile Company (NST) has been using production runs of 100,000 tiles, 10 times per year to meet the demand of 1,000,000 tile annually. • The set-up cost is $5,000 per run • Holding cost is estimated at 10% of the manufacturing cost of $1 per tile. • The production capacity of the machine is 500,000 tiles per month. • The factor is open 365 days per year. • Determine • Optimal production lot size • Annual holding and setup costs • Number of setups per year • Loss/profit that NST is incurring annually by using their present production schedule

  25. Management Scientist Solutions • Optimal TC = $28,868 • Current TC = .04167(100,000) + 5,000,000,000/100,000 = $54,167 • LOSS = 54,167 - 28,868 = $25,299

  26. Economic Production Quantity Assumptions • Only one item is involved  • Annual demand is known  • Usage rate is constant  • Usage occurs continually • Production occurs periodically • Production rate is constant • Lead time does not vary  • No quantity discounts 

  27. EOQ with Quantity Discounts • The EOQ with quantity discounts model is applicable where a supplier offers a lower purchase cost when an item is ordered in larger quantities. • This model's variable costs are • Annual holding, • Ordering cost, and • Purchase costs • For the optimal order quantity, the annual holding and ordering costs are not necessarily equal.

  28. EOQ with Quantity Discounts • Formulae • Optimal order quantity: the procedure for determining Q * will be demonstrated • Number of orders per year: D/Q * • Time between orders (cycle time): Q */D years • Total annual cost: (formula 11.9 of book) (holding + ordering + purchase)

  29. Example – EOQ with Quantity Discount • Walgreens carries Fuji 400X instant print film • The film normally costs Walgreens $3.20 per roll • Walgreens sells each roll for $5.25 • Walgreens's average sales are 21 rolls per week • Walgreens’s annual inventory holding cost rate is 25% • It costs Walgreens $20 to place an order with Fujifilm, USA • Fujifilm offers the following discount scheme to Walgreens • 7% discount on orders of 400 rolls or more • 10% discount for 900 rolls or more, and • 15% discount for 2000 rolls or more • Determine Walgreen’s optimal order quantity

  30. Management Scientist Solutions

  31. Operations Strategy • Too much inventory • Tends to hide problems • Easier to live with problems than to eliminate them • Costly to maintain • Wise strategy • Reduce lot sizes • Reduce safety stock

  32. The Balance Sheet – Dell Computer Co.

  33. (in millions, except per share amount) Fiscal Year Ended 28-Jan-00 29-Jan-99 Net revenue $25,265 $18,243 Cost of revenue 20,047 14,137 Gross margin 5,218 4,106 Operating expenses: Selling, general and administrative 2,387 1,788 Research, development, and engineering 568 272 Total operating expenses 2,955 2,060 Operating income 2,263 2,046 Other income 188 38 Income before income taxes 2,451 2,084 Provision for income taxes 785 624 Net income $1,666 $1,460 Earnings per common share: Basic $0.66 $0.58 Diluted $0.61 $0.53 Weighted average shares outstanding: Basic 2,536 2,531 Diluted 2,728 2,772 Retained Earnings: Balances at beginning of period 606 607 Net income 1,666 1,460 Repurchase of common stocks (1,012) (1,461) Balances at end of period $1,260 $606 Income Statement – Dell Computer Co.

  34. Debt Ratio • What It Measures: The extent to which a form uses debt financing • How You Compute: The ratio of total debt to total assets

  35. Inventory Turnover Ratio • What It Measures: How effectively a firm is managing its inventories. • How You Compute: This ratio is computed by dividing sales by inventories Inventory turnover ratio =

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