Optimal Level of Product Availability Chapter 13 of Chopra

Outline Determining optimal level of product availability Single order in a season Continuously stocked items Ordering under capacity constraints Levers to improve supply chain profitability Notes:

Example: Apparel Industry How much to order? Parkas at L.L. Bean Notes: Mention that L.L. Grain is a mail order company deciding on the number of units of a Fall jacket to order. An estimate of demand using past information and expertise of buyers is given here. What should the appropriate order quantity be? In general distribution may not be known. Discuss methodology used in the Matching supply and demand article as a possibility in deciding on demand uncertainty (distribution). Expected demand is 1,026 parkas.

Parkas at L.L. Bean Cost per parka = c = $45 Sale price per parka = p = $100 Discount price per parka = $50 Holding and transportation cost = $10 Salvage value per parka = s = 50-10=$40 Profit from selling parka = p-c = 100-45 = $55 Cost of overstocking = c-s = 45-40 = $5 Notes: What information is required to make the ordering decision? Stress cost of understocking and overstocking. How to evaluate these costs for this example?

Optimal level of product availability p = sale price; s = outlet or salvage price; c = purchase price CSL = Probability that demand will be at or below reorder point Raising the order size if the order size is already optimal Expected Marginal Benefit = =P(Demand is above stock)*(Profit from sales)=(1-CSL)(p - c) Expected Marginal Cost = =P(Demand is below stock)*(Loss from discounting)=CSL(c - s) Define Co= c-s; Cu=p-c (1-CSL)Cu = CSL Co CSL= Cu / (Cu + Co) Notes:

Order Quantity for a Single Order Co = Cost of overstocking = $5 Cu = Cost of understocking = $55 Q* = Optimal order size Notes: Explain how formula is derived using decision tree.

Optimal Order Quantity 0.917 Notes: Optimal Order Quantity = 13(‘00)

Parkas at L.L. Bean Notes: Discuss marginal benefit and marginal cost of each jacket. We keep increasing order size as long as expected benefit exceeds expected cost.

Revisit L.L. Bean as a Newsvendor Problem Total cost by ordering Q units: C(Q) = overstocking cost + understocking cost Marginal cost of raising Q* - Marginal cost of decreasing Q* = 0 Show Excel to compute expected single-period cost curve.

The Newsvendor Approach Assumptions: 1. single period 2. random demand with known distribution 3. linear overage/shortage costs 4. minimum expected cost criterion Examples: newspapers or other items with rapid obsolescence Christmas trees or other seasonal items capacity for short-life products

Newsvendor Model Notation

Newsvendor Model Cost Function: Note: for any given day, we will be either over or short, not both. But in expectation, overage and shortage can both be positive.

Newsvendor Model (cont.) Optimal Solution: taking derivative of Y(Q) with respect to Q, setting equal to zero, and solving yields: Notes: Critical Ratio is probability stock covers demand 1 G(x) Q*

Newsvendor Example – T Shirts Scenario: Demand for T-shirts is exponential with mean 1000 (i.e., G(x) = P(X  x) = 1- e-x/1000). (Note - this is an odd demand distribution; Poisson or Normal would probably be better modeling choices.) Cost of shirts is $10. Selling price is $15. Unsold shirts can be sold off at $8. Model Parameters: cs = 15 – 10 = $5 co = 10 – 8 = $2

Newsvendor Example – T Shirts (cont.) Solution: Sensitivity: If co = $10 (i.e., shirts must be discarded) then

Newsvendor Model with Normal Demand Suppose demand is normally distributed with mean  and standard deviation . Then the critical ratio formula reduces to: (z) z Note: Q* increases in both  and  if z is positive (i.e., if ratio is greater than 0.5).

Multiple Period Problems Difficulty: Technically, Newsvendor model is for a single period. Extensions: But Newsvendor model can be applied to multiple period situations, provided: demand during each period is iid, distributed according to G(x) there is no setup cost associated with placing an order stockouts are either lost or backordered Key: make sure co and cs appropriately represent overage and shortage cost.

Example Scenario: Problem: how should they set order amounts? GAP orders a particular clothing item every Friday mean weekly demand is 100, std dev is 25 wholesale cost is $10, retail is $25 holding cost has been set at $0.5 per week (to reflect obsolescence, damage, etc.) Problem: how should they set order amounts?

Example (cont.) Newsvendor Parameters: Solution: c0 = $0.5 cs = $15 Every Friday, they should order-up-to 146, that is, if there are x on hand, then order 146-x.

Newsvendor Takeaways Inventory is a hedge against demand uncertainty. Amount of protection depends on “overage” and “shortage” costs, as well as distribution of demand. If shortage cost exceeds overage cost, optimal order quantity generally increases in both the mean and standard deviation of demand.

Impact of Improving Forecasts EX: Demand is Normally distributed with a mean of R = 350 and standard deviation of R = 150 Purchase price = $100 , Retail price = $250 Disposal value = $85 , Holding cost for season = $5 How many units should be ordered as R changes? Price=p=250; Salvage value=s=85-5=80; Cost=c=100 Understocking cost=p-c=250-100=$150, Overstocking cost=c-s=100-80=$20 Critical ratio=150/(150+20)=0.88 Optimal order quantity=Norminv(0.88,350,150)=526 units Expected profit? Expected profit differs from the expected cost by a constant. Note: Cost of understocking = 250-100 = $150 Cost of overstocking = 100+5-85 = $20 p = 150/(150+20) = 0.88 Order size = 426

Computing the Expected Profit with Normal Demands

Impact of Improving Forecasts Where is the trade off? Expected overstock vs. Expected understock. Expected profit vs. ?????

Cost or Profit; Does it matter?

Learning Objectives Optimal order quantities are obtained by trading off cost of lost sales and cost of excess stock Levers for improving profitability Increase salvage value and decrease cost of stockout Improved forecasting Quick response with multiple orders Postponement Notes: