Newsvendor Inventory Problem

Newsvendor Inventory Problem Consider a newspaper vendor selling papers on the corner. Each morning, the vendor must decide how many papers to buy at the wholesale price. The vendor then sells the papers during the day at a retail price higher than the wholesale price. At the end of the day, any unused papers can no longer be sold and are scrapped (or possibly sold back to the wholesaler at a price less than the wholesale price). How many papers should the vendor buy? Characteristics of the problem 1. 2. 3. 4. 5. Single period problem – ordering decision made every period Demand is uncertain Order is placed before demand materializes We can only order once per period There is a cost for ordering too much and a cost for ordering too few items. Examples 1. Fashion products 2. Perishable products 3. … Model Basics Inputs • Wholesale cost – the per unit cost for the vendor to purchase the item, denoted by w • Retail price – the per unit selling price, denoted by p • Salvage value – the per-unit amount the vendor can receive if a unit cannot be sold; for instance , at the end of the period, an item might be sold back to the wholesaler at a discount, or sold on a secondary market; the salvage price is denoted by s. • Demand distribution – Let F(x) denote the cumulative distribution function of demand. That is F(x) is the probability that the demand will be less than or equal to x. 1 of 8 Supply Chains: D Lab Model Intuition Suppose that we have decided to order Q units. Consider the question: should we order another unit? That is, is it economical to order the Q+1st unit? We define the incremental profit from the Q+1st unit as follows: Δ (Q ) = E ⎣⎡ profit (Q + 1)⎦⎤ − E ⎣⎡ profit (Q )⎤⎦ . It is economical to order the Q+1st unit if and only if Δ (Q ) ≥ 0. For this analysis, we assume that the Q+1st unit is reserved to serve the Q+1st demand. Δ (Q ) has three components: • • • A cost of c, namely the cost to purchase the Q+1st unit. A revenue of p, if the Q+1st unit is sold. A salvage value of s, if the Q+1st unit is not sold. Thus in expectation, we can write: Δ (Q ) = − −cc + p × Pr ⎡sell Q +1st unit ⎤⎦ + s × Pr ⎡⎣do not sell Q + 1st unit ⎤⎦ ⎣ ( = −c + p × Pr ⎡⎣ sell Q +1st unit ⎤⎦ + s × 1− Pr ⎡sell Q + 1st unit ⎤⎦ ⎣ ( ) = ( p − c ) × Pr ⎡sell Q +1st unit ⎤⎦ − ( c − s ) × 1− Pr ⎡⎣ sell Q + 1st unit ⎤⎦ ⎣ ) Now we find: (c − s ) Δ (Q ) ≥ 0 ⇒ Pr ⎣⎡sell Q + 1st unit ⎦⎤ ≥ ( p − c ) + (c − s ) The implication for this analysis is that we should increase Q until this condition does not hold; thus, we choose the first Q such that (c − s ) . Pr ⎣⎡sell Q + 1st unit ⎦⎤ < ( p − c ) + (c − s ) Now what is Pr ⎡sell Q +1st unit ⎦⎤? This is the probability that demand is at ⎣ least Q+1 units. The cumulative distribution function is defined as F ( x ) = Pr [demand ≤ x ]. Hence, we have 2 of 8 Supply Chains: D Lab Pr ⎡sell Q +1st unit ⎦⎤ =1− Pr [demand ≤ Q] =1 − F (Q ). ⎣ Alternatively, if we have a discrete probability distribution, we can write: Pr ⎡⎣ sell Q +1st unit ⎤⎦ = ∞ ∑ j = Q +1 Q Pr [demand = j ] = 1− ∑ Pr [demand = j ] j=0 Alternative Interpretation • We often term c - s to be the overage cost – the incremental per-unit cost for any items that cannot be sold. • We then term p - c to be the underage cost– the incremental per-unit cost for not meeting demand. The above result can then be expressed, more generally, as: We choose the first Q such that overage _ cost . Pr ⎡sell Q +1st unit ⎤⎦ < ⎣ underage _ cost + overage _ cost Sometimes this result will be expressed equivalently as: We choose the first Q such that underage _ cost st Pr ⎡do ⎣ not sell Q +1 unit ⎦⎤ = F (Q ) ≥ underage _ cost + overage _ cost . Useful Result If demand is normally distributed, then we can find Q in one of two ways: • • Directly from Excel: ⎛ ⎞ Underage Cost Q = NORMINV ⎜ µ,σ, ⎟ Underage Cost + Overage Cost ⎠ ⎝ Using Excel to find z* first: 3 of 8 Supply Chains: D Lab ⎛ ⎞ Underage Cost z* = NORMSINV ⎜ ⎟ ⎝ Underage Cost + Overage Cost ⎠ and then letting Q = µ + (z*)(σ) Discrete Demand Example – Theodore’s Gift Shop Problem Statement Theodore’s gift shop places orders for Christmas items during a trade show in July. One item to be ordered is a dated sterling silver tree ornament. The ornament will sell for $80. The best estimate for demand is: Demand 5 6 7 8 Probability 0.20 0.25 0.30 0.25 The ornaments cost $55 when ordered in July. Ornaments unsold by Christmas are marked down to half price and always sell during January. How many ornaments should be ordered? Calculations • Overage cost = $55 per ornament- $40 per ornament = $15 per ornament • Underage cost = $80 per ornament - $55 per ornament = $25 per ornament • We choose the first Q such that overage _ cost $15 Pr ⎣⎡ sell Q +1st unit ⎤⎦ < = = 0.375 underage _ cost + overage _ cost $25 + $15 4 of 8 Supply Chains: D Lab • Table of cumulative probability distribution Demand 5 6 7 8 • Cumulative Probability=F(x) 0.20 0.45 0.75 1.00 1-F(x) 0.80 0.55 0.25 0 To find Q, we compare: Pr ⎡sell Q +1st unit ⎦⎤ =1− Pr [demand ≤ Q] =1− F (Q ). ⎣ To the critical fractile 0.375 In this case, Q = 7 ornaments. (as the probability that we will see the 8th unit is only 0.25) • With Q = 7 ornaments, what is the total expected profit? Demand Probability Revenue Expected Revenue 5 0.20 5(80)+2(40)=480 (0.20)(480)=96 6 0.25 6(80)+1(40)=520 (0.25)(520)=130 7 0.30 7(80)+0(40)=560 (0.30)(560)=168 8 0.25 7(80)+0(40)=560 (0.25)(560)=140 Total Expected Revenue = 534 Total expected profit = total expected revenue – cost = 534 – 7(55) = 149 • As a point of reference: For Q = 6, expected revenue = 472, cost = 330 and expected profit = 142 For Q = 8, expected revenue = 584, cost = 440 and expected profit = 144 5 of 8 Supply Chains: D Lab Continuous Demand Example – Johnson Shoe Company Problem Statement • The Johnson Shoe Company buys shoes for $40 per pair and sells them for $60 per pair. If there are surplus shoes left at the end of the season, all shoes are expected to be sold at the sale price of $30 per pair. How many shoes should the Johnson Shoe Company buy? Calculations • Overage cost = $40 per pair - $30 per pair = $10 per pair • Underage cost = $60 per pair - $40 per pair = $20 per pair • We choose the first Q such that overage _ cost $10 Pr ⎣⎡ sell Q +1st unit ⎤⎦ < = = 0.333 underage _ cost + overage _ cost $20 + $10 • Suppose demand is normally distributed with a mean of 500 units and a standard deviation of 100 units/season. We then want to find the first value of Q such that 1− F (Q ) < 0.33. That is, we need to find the z-score such that 1/3 of the area under the curve is beyond z. Equivalently we find z such that Φ ( z ) = 0.67 where Φ ( ) is the cumulative distribution function for a standard normal distribution. Looking up Φ ( z ) = 0.67 F(z) yields a z of 0.44. Q = mean + (0.44)*(standard deviation) = 500+ (.44)(100) = 544 units 6 of 8 Supply Chains: D Lab Multi-item Style Goods Problem with Capacity Constraint • Suppose that instead of buying a single item, you had to place orders for several items with the condition that the total order quantity across all items can not exceed your capacity C. Example • Consider a newsstand that sells three scholarly journals: Management Science (MS), Operations Research (OR), and Manufacturing & Service Operations Management (M&SOM). For simplicity, let’s assume that the journals have the same costs and revenues. Journals cost $1.00 to buy and they are sold at a retail price of $4.00. Journals left unsold at the end of the season can be returned to the publisher for $0.50. • This implies the underage cost is $3.00 ($4.00 - $1.00) and the overage cost is $0.50 ($1.00 - $0.50). Therefore, the critical fractile is 0.14. • We assume normally distributed demand with parameters shown in table; then we can calculate an order quantity for each, as shown in Table Journal MS OR MSOM • Mean 80 50 20 Sigma 40 30 15 CF 0.14 0.14 0.14 z-score Q* 1.07 123 1.07 82 1.07 36 Total 241 Now let’s assume that there is a capacity limit of 200 total magazines. How would you determine how many to order? Most people would guess a linear allocation based on the journal’s percentage composition of the total optimal order size. That is, ⎛ Q * ⎞ Linear Order Size = ⎜ ⎟ ( 200 ) ⎝ 241 ⎠ Journal MS OR MSOM • Mean 80 50 20 Sigma Q* 40 123 30 82 15 36 Total 241 % of Total 51% 34% 15% Linear Order Size 102 68 30 200 While the linear allocation approach is intuitively appealing, it is incorrect because the resulting order quantities now have different z-scores. This implies that each of the journals now has a different probability of fully satisfying the demand (termed fill rate here). Recall that before the 7 of 8 Supply Chains: D Lab capacity constraint was imposed, each journal’s order quantity was set to ensure a 86% fill rate. The following table shows the new fill rates and zscore values for each journal: Journal MS OR MSOM • Mean 80 50 20 % of Total 51% 34% 15% Sigma Q* 40 123 30 82 15 36 Total 241 Linear (L) Order Size 102 68 30 200 L’s zscore .55 .60 055 L’s Fill Rate 71% 73% 75% The correct approach is to find the single z-value that equalizes the fill rate for all three products while not exceeding the capacity constraint. This can be written as follows: Find the z such that ∑i (µi + zσi ) = C where i is an index denoting the products. In our example, this can be written out as: 80 + 40z + 50 + 30z + 20 + 15z = 200 85z = 50 z = 0.59 This yields the following ordering quantities Journal MS OR MSOM Mean 80 50 20 Sigma 40 30 15 Total z-score Order Amount 103.5 67.6 28.8 200 0.59 You can find the z-score by using the Goal Seek function of excel. That is, make the Order Amount cells equal to µi + zσi and the Total cell equal to the sum of the three order amounts. Then set Total equal to 200 by changing the z-score cell. • If the products do not have the same costs and retail prices, then the problem is much harder. You will want to develop some decision rules to determine what exactly to do. 8 of 8 Supply Chains: D Lab