The Newsvendor Model: Lecture 10 Risks from stockout and markdown The Newsvendor model Application to postponement Review for inventory management

Risks from Stockout and Markdown MBPF designed a fancy garage FG to sell in the Christmas season Each costs $3000 in materials and sales for $5500. Unsold FG will be salvaged for $2800 each All raw materials have to be purchased in advance Based on market research, MBPF estimated the demand of FG to be between 10 and 23 and the probabilities are given in table 1 What should be the amount of raw materials to purchase for producing FG?

Table 1: The Demand Distribution Probability 10 0.01 11 0.02 12 0.04 13 0.08 14 0.09 15 0.11 16 0.16 17 0.20 18 19 0.10 20 21 22 23 Total 1.00 Table 1: The Demand Distribution

U2’s Spring T-Shirt U2 has a new premier T-Shirt for Spring05 in 4 colors Hong Kong retail market has a 3 month season slide 23 The standard production method is to dye the fabric first and then make shirts with different colors. The production cost is low but leadtime is long, at 3 months. So U2 needs to place order in December The production and in-bound logistic cost is $30/shirt, and U2 will sell the shirt at $90/shirt U2 does not sell its premier shirts at discount in Hong Kong market. After the season, U2 wholesales the shirts to a mainland company at $25/shirt

Marginal Cost and Marginal Benefit Suppose MBPF starts with a potential order quantity of Q and considers adding an additional unit Q - If this unit is sold, there is a benefit (profit) B = B is called marginal benefit or underage cost - If this unit cannot be sold, there is a cost C = C is called marginal cost or overage cost For U2, Underage cost B = /shirt and Overage cost C = /shirt

Fashion Goods MBPF and U2’s have the so called “fashion goods” or newsvendor problem Short selling season Limited ordering opportunity Uncertain demands Newspapers, magazines, fish, meat, produce, bread, milk, high fashion …

One Ordering Chance MBPF and U2 have only one chance to order (long) before the selling season Too late to order when the selling starts No more demand information before the sales There is no way to predict demands accurately MBPF keeps past sales record which can be useful U2 also can forecast, but what are past sales data?

The Ordering Risks Suppose MBPF or U2 orders Q and demand is D If D > Q, there will be stockout The cost (risk) = B max {D –Q, 0} If D ≤ Q, there will be overstocks The cost (risk) = C max {Q – D, 0} The (potential) stockout and markdown costs In some industries, such as fashion industry, the total stockout and markdown cost is higher than the total manufacturing cost!

How many papers should the newsboy buy? The Clever Newsboy ? Extra Read all about it gfdhg gfhjsg fgdhgs fgashjg ghdgsh dsagh lzkfjh yteeorkjrorthjoi oier; oerzjlztrj ezrlkjzfsgj ;zrurtgfdgdggfg dsg fgsdf gf fgs gfd How many papers should the newsboy buy?

The Newsvendor Model We do not know for sure if it can be sold or not. Thus, we have to work with the expected marginal benefit and expected marginal cost Expected marginal benefit = B·Prob.{ Demand > Q} Expected marginal cost = C·Prob. { Demand ≤ Q}

Marginal Analysis Detailed numerical calculations in MBPFinventory.xls show, as Q increases: - The expected marginal benefit decreases; - The expected marginal cost increases; and Q= 19 is the largest value of Q at which the marginal benefit is still greater than the marginal cost Given an order quantity Q, increase it by one unit if and only if the expected benefit of being able to sell it exceeds the expected cost of having that unit left over

The Critical Ratio Suppose Q can be continuous. Then, there is a Q at which the expected marginal benefit and cost are equal We call B/(B+C)= β the critical ratio What does (1) say? The optimal order quantity Q* is smallest integer greater than the Q obtained from (1) (1)

Critical Ratio Solutions For MBPF Inc. B =, C = From MBPFinventory.xls, Q should be

Newsvendor with Continuous Demands The demand in the selling cycle can be characterized by a continuous random variable D with mean μ, standard deviation σ, and distribution function F (x) The optimal order quantity Q* is such that (2)

Normally Distributed Demands Consider normal demands N(μ, σ 2) with distribution F (Q) We then have By this equation, we see that the critical ratio is the probability that the standard normal demand Ds ≤(Q – μ)/σ. Prob.(demand≤Q) µ Q

Solution For Normal Demands Set (Q – μ)/σ= zβ. Recall that there is a one-to-one correspondence between zβ and β, and they are completely tabulated in the normal table We then have this simple solution: Q* = μ + zβσ (3)

Solving Discrete Problems by Normal Approximation Consider the product FG of MBPF Inc. We use the normal distribution to approximate the demand distribution. From MBPFinventory.xls: µ = 16.26 and  = 2.48 From the normal table, we have z0.926 = Then Q* = Also from NORMINV(0.926, 16.26, 2.48)

Hedging Factor and Safety Stock Hedging factor zβ is a function of the critical ratio β β 0.1 0.30 0.50 0.75 0.95 0.99 zβ When B < C (cost of lost sale < cost of overstock), overstock is more damaging and we order (zβσ) less than the expected demand When B>C, lost sales is more damaging and we order zβσ more When B=C, the impact of overstock and lost sales are the same, the best strategy is order the expected demand zβσ is called the safety stock

Exercise: Christmas Trees Mrs. Park owns a convenience store in Toronto Each year, she sells Christmas trees from Dec. 3 to Dec. 24 She needs to order the trees in September In the season, she sells a tree for $75 After Dec. 24, an unsold tree is salvaged for $15 Her cost is $30/tree inclusive

Exercise: Christmas Trees Mrs. Park’s past sales record Sales 29 30 31 32 33 34 35 36 Prob. .05 .10 .15 .20 .20 .15 .10 .05 Please give: (1) Critical ratio; (2) Hedging factor; and (3) Safety stock Suppose Mrs. Park’s regular profit margin is $70, $30, or $10, and all else remain the same. Do the same christmas

Postponement Delay of product differentiation until closer to the time of the sale All activities prior to product differentiation require aggregate forecasts which are more accurate than individual product forecasts Point of delivery A B A A and B B dyeing fabricating

Benefits of Postponement Individual product forecasts are only needed close to the time of sale – demand is known with better accuracy (lower uncertainty) Results in a better match of supply and demand Valuable in e-commerce – time lag between when an order is placed and when customer receives the order (this delay is expected by the customer and can be used for postponement) Question: Is postponement always good? What is the main factor(s) that determines the benefits of postponement?

Computing Value of Postponement for U2 For each color (4 colors) slide 3 Mean demand μ = 2,000; σ = 1500 For each garment Sale price p = $90, Salvage value s = $25 Production cost using Option 1 (long leadtime) c = $30 Production cost using Option 2 (uncolored thread) c = $32 What is the value of postponement?

Use of The Newsvendor Model Recall the newsvendor model, We will also calculate the expected profit by

The Value of Postponement Option 1: μ= 2000 and σ= 1500 Critical ratio = Q*= Profit from each color = Total profit = Option 2: μ= 8000 and σ= postponement 3000

Value of Postponement with Dominant Product Dominant color: μ=6,200, σ= 4500 Other three colors: μ= 600, σ= 450 Critical ratio = Option 1: Q*1= profit = Q*2= profit = Total expected profit = postponement

Worst off with Postponement Option 2: μ= 8000, σ= (45002+3x4502)1/2 = Critical ratio = Q*= Profit = Postponement allows a firm to increase profits and better match supply and demand if the firm produces a large variety of products whose demands are not positively correlated and are of about the same size 4567 postponement

Review: Inventory Management How Much to Order Tradeoff between ordering and holding costs Robustness and Square-root rule Tradeoff between setup time (capacity) and inventory cost

When to Order Reorder point ROP =  + IS = RL + zβσ Assuming demand is normally distributed: For given target SL ROP =  + zβσ= NORMINV(SL, ,σ) =  +NORMSINV(SL)·σ For given ROP SL = Pr(DL  ROP) = NORMDIST(ROP, ,.σ, True) Safety stock pooling (of n identical locations) Relation between order quantity and reorder point. Suppose your fixed ordering cost is not high while holding cost is relatively high and leadtime is relatively long. What may happen if you use the ROP policy as we defined? You may have a reorder cycle shorter than the leadtime. What is wrong with this? You may order more frequently and receive order during a leadtime for another order. When an order arrives, the inventory level may still be lower than the ROP and you may not see ROP and place an new order. You may let the safety stock be depleted and run shortage. Is the ROP system wrong or do we really have such a serious problem? In order to simplify the discussion, our textbook and many other business school textbooks have not pointed out that we in fact should monitor a different state variable (inventory level) which is called the inventory position: inventory on-hand + inventory on-order – backorder. The computation of ROP is still the same. However, when an order is placed, the inventory position is raised immediately to ROP+Q. In this case, even the order you just placed has not yet arrive, the inventory position may reduces from above ROP to ROP and another order can be placed during the leadtime of the previous order. Thus there might be multiple orders outstanding at the same time. When order quantity is greater than average leadtime demand, on average, there should be only one order outstanding and the inventory position is usually equal to the on-hand inventory level. For us, you should be aware of this subtlety, but you are not required to use inventory position in homework or midterm.

Managing System Inventory Six basic reasons (functions) to hold inventory Total average inventory for one item = Q/2 + zβσ Not own pipeline = Q/2 + zβσ+RL Own pipeline Managing multiple items - ABC analysis: 80/20 rule, Pareto Chart

Newsvendor Stockout and markdown are major risks for inventory decisions The critical ratio balances the stockout cost and the markdown cost: - when B>C, we add a positive safety stock because stockout is more damaging; - when B<C, we add a negative safety stock Safety stock is used to hedge the risks Q* = μ + zβσ