Lecture 5 Decision Analysis Chapter 14

Decision Environments Certainty - Environment in which relevant parameters have known values Risk - Environment in which certain future events have probabilistic outcomes Uncertainty - Environment in which it is impossible to assess the likelihood of various future events

Decision Making under Uncertainty Maximin - Choose the alternative with the best of the worst possible payoffs Maximax - Choose the alternative with the best possible payoff Minimax Regret - Choose the alternative that has the least of the worst regrets

Payoff Table: An Example Possible Future Demand Low Moderate High Small facility $10 Medium facility 7 12 Large facility - 4 2 16 Values represent payoffs (profits)

Maximax Solution Note: choose the “minimize the payoff” option if the numbers in the previous slide represent costs

Maximin Solution

Minimax Regret Solution

Decision Making Under Risk - Decision Trees State of nature 1 B Payoff 1 State of nature 2 Payoff 2 Payoff 3 2 Choose A’1 Choose A’2 Payoff 6 Payoff 4 Payoff 5 Choose A’3 Choose A’4 Choose A’ 1 Decision Point Chance Event

Decision Making with Probabilities Expected Value Approach Useful if probabilistic information regarding the states of nature is available Expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring Decision yielding the best expected return is chosen.

Example: Burger Prince Burger Prince Restaurant is considering opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120 with a probability of 0.4, 0.2, and 0.4 respectively The payoff (profit) table for the three models is as follows. s1 = 80 s2 = 100 s3 = 120 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Choose the alternative that maximizes expected payoff

Decision Tree Payoffs 10,000 2 15,000 d1 14,000 8,000 d2 1 3 18,000 d3 .4 10,000 s2 .2 2 15,000 s3 .4 d1 14,000 .4 s1 8,000 d2 1 3 s2 .2 18,000 d3 s3 .4 12,000 s1 .4 6,000 4 s2 .2 16,000 s3 .4 21,000

Management Scientist Solutions EVPI = expected payoff under certainty – expected payoff under risk

Lecture 5 Inventory Management Chapter 11

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

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

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

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

Inventory Classification Systems ABC 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

ABC Analysis Example $232,057 Item Stock Number Percent of Number of Items Stocked Annual Volume (units) x Unit Cost = Annual Dollar Volume Percent of Annual Dollar Volume Class #10286 20% 1,000 $ 90.00 $ 90,000 38.8% 72% A #11526 500 154.00 77,000 33.2% #12760 1,550 17.00 $ 26,350 11.3% B #10867 30% 350 42.86 15,001 6.4% 23% #10500 12.50 12,500 5.4% #12572 600 $ 14.17 $ 8,502 3.7% C #14075 2,000 .60 1,200 .5% #01036 50% 100 8.50 850 .4% 5% #01307 .42 504 .2% #10572 250 150 .1% $232,057

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

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

Total Cost Annual carrying cost ordering Total cost = + Q 2 Ch D Co TC = Formula (11-4)

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

Deriving the EOQ & Minimum Total Cost The total cost curve reaches its minimum where the carrying and ordering costs are equal. Formula (11-5) Number of orders per year = D/Q0 Length of order cycle = Q0/D

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. 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?

Management Scientist Solutions (d) (e) (f) (g) (b) (c)

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

EOQ with Quantity Discounts EOQ with quantity discounts model 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

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 11.28 of book) (holding + ordering + purchase)

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

Management Scientist Solutions