Transcript MS401-05-Newsboy - Sabancı Üniversitesi
MS 401
Production and Service Systems Operations
Spring 2009-2010
Inventory Control – III Stochastic Demand: Newsboy Model
Slide Set #5
Murat Kaya , Sabancı Üniversitesi
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Newsboy’s Problem
• A newsboy is selling the magazine Atlas • The newsboy buys each copy for $3.00 and sells it for $4.00
• At end of the week he can return each unsold copy for $2.5
• Question: How many copies to buy at the beginning of the week?
observed demand in the past 52-week 15 19 14 11 8 6 9 9 12 6 11 5 4 9 22 4 9 18 10 7 8 11 1 14 12 4 17 18 14 15 7 12 15 15 19 9 10 8 9 16 8 11 11 18 15 17 19 14 14 17 13 12
Murat Kaya , Sabancı Üniversitesi
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Newsboy Example
• Normal approximation to demand: mean: 11.73, standard deviation: 4.74
• Perhaps, the newsboy should buy 12 magazines to satisfy
the average demand?
• The cost for each issue unsold (
cost of overage
): • The lost profit due to a lost customer (
cost of underage
): • Hence, intuition tells us that the newsboy should order more than the average demand. But, how much more?
Murat Kaya , Sabancı Üniversitesi
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Single Period Problem : Model
Inventory level
1) Order quantity Q is determined
Q leftover inventory
2) Random demand during the period
0
?
or, lost sales
3) Relevant costs are realized at the end of the period
Murat Kaya , Sabancı Üniversitesi
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Model Environment
• Relatively short selling season (weeks, 2 months,…) with a well-defined beginning and end • At the beginning of the period, a decision is made on how much to order or produce ( Q ) • The demand ( D ) is uncertain. Although we don’t know exactly what value D is going to take, we have a forecast on its distribution: F(a)= P(D Murat Kaya , Sabancı Üniversitesi 5 • When the total demand in the period exceeds the stock available, there is an associated underage cost, c u – cost per unit of unsatisfied demand • When the total demand is less than the stock available, overage cost is incurred, c o – cost per unit of positive inventory at the end of the period • Objective: Minimize the total underage and overage cost Murat Kaya , Sabancı Üniversitesi 6 • Define G(Q,D) as the total overage and underage cost incurred at the end of the period when Q units are ordered and the demand is realized as D G ( Q G ( Q ) , D ) = = c E [ G o max{ ( Q , D )] 0 , Q = c o max{ 0 , Q 0 - x } f ( - D x ) dx } + + c c u u 0 max{ 0 max{ 0 , , = c o Q ( Q 0 - x ) f ( x ) dx + c u Q ( Q ) f ( x ) dx D x - Q } Q } f ( x ) dx Murat Kaya , Sabancı Üniversitesi 7 • The optimal value of the order quantity Q? – if necessary conditions are satisfied, find Q that satisfies G ( Q ) = 0 • How to find the derivative of an integral? d dQ b a ( ( Q Q ) g ) ( Q , x ) dx = d dQ b ( Q ) g ( Q , b ( Q )) - d dQ a ( Q ) g ( Q , a ( Q )) + b a ( ( Q Q ) ) dg ( x dQ , Q ) dx Murat Kaya , Sabancı Üniversitesi 8 dG ( Q ) dQ = c o Q 0 1 f ( x ) dx + c u Q ( 1 ) f ( x ) dx = c o F ( Q ) - c u ( 1 - F ( Q )) d 2 G ( Q ) dQ 2 = ( c o + c u ) f X ( Q ) 0 The function is convex (the second order condition is satisfied) G ' ( Q * ) = ( c o + c u ) F ( Q * ) - c u = 0 F ( Q * ) = c o c u + c u = The critical ratio Murat Kaya , Sabancı Üniversitesi 9 Cumulative probability F(Q) 1 c u c u + c o Q * Murat Kaya , Sabancı Üniversitesi Q 10 Suppose X ~ N 2 = , Var(X) = 2 the standard normal random variable N(0,1) Murat Kaya , Sabancı Üniversitesi 11 Then, the optimality equation can be written as : Q = c u c u + c o we get Q * z normal random variable. c u c u + c o th percentile of the standard Murat Kaya , Sabancı Üniversitesi 12 Area = c u c u + c o ( z ) z 0 = 0 = 1 Murat Kaya , Sabancı Üniversitesi 13 Q * z • Q * increases when – mean demand increases • What happens when variability increases? – Q * increases if z > 0 (if c u /(c u + c o ) > 0.5) – Q * decreases if z < 0 (if c u /(c u + c o ) < 0.5) Murat Kaya , Sabancı Üniversitesi 14 F(Q*) = c u c u + c o = 1 1 + 0 . 5 = 0 . 67 • • Find Q* such that z = NORMSINV(0.67) = 0.439913 Q* = μ + zσ 11.73+ (0.44*4.74) = 13.81 ~ order 14 items Area=0.67 Murat Kaya , Sabancı Üniversitesi 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Sales 15 • Assume now that the newsboy cannot return unsold • copies to the publisher. In this case, Q* = μ + zσ Murat Kaya , Sabancı Üniversitesi Area=0.25 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Sales 16 • Expected lost sales = • where L(z) is the standard normal loss function (tabulated) • Expected sales = • where μ is the expected demand • Exp. leftover inventory = • Expected profit = • Fill rate = • the expected ratio of satisfied demand • In-stock probability = • the probability that all demand is satisfied (i.e., no stockout) Murat Kaya , Sabancı Üniversitesi 17 • Consider a case where demand is uniformly distributed over [50,150]. c u =$10, c o =$7. For uniform demand, Murat Kaya , Sabancı Üniversitesi Q * = Q 50 100 = c u c u + c o = 10 17 = 10 17 109 18 • Turkish Airlines has decided to offer a one-month advance purchase discount ticket on its Istanbul-Munich route for $225 instead of the regular price of $325 • The flight has a capacity of 150 passengers • Demand for regular tickets follow a normal distribution with mean 60 and standard deviation of 15 – independent of the number of discount ticket customers • Suppose that all discount tickets would be sold as soon as they are released • Determine the optimum number of seats that should be reserved for the regular passengers Murat Kaya , Sabancı Üniversitesi 19 • Suppose we reserved X seats for the full-fare passengers, and the number of full fare passengers that show up is D • We will lose $100*(D-X) if D>X and we will lose $225*(X-D) if X>D • This problem can be modeled as a newsboy problem with • The critical ratio is calculated as Murat Kaya , Sabancı Üniversitesi 20 • Suppose we have an item for which c u / (c u + c o ) = 0.68 Number of units, Q P(D=Q) P(D<=Q) 1 0.2 0.2 2 0.2 0.4 3 0.3 0.7 4 0.2 0.9 5 0.1 1.0 Total 1 - • Q* is the Murat Kaya , Sabancı Üniversitesi 21 • Semicon uses nitric acid to produce semiconductors • Nitric acid has a shelf-life of only three months • Semicon’s need for nitric acid: Uniform[1000, 3000] gallons for the coming three months • Cost of acid: $150/gallon • Acid storage cost: $35/gallon – assume that this cost is incurred at the end of the 3-month period • Leftover acid needs to be disposed of, costs $75/gal • If the company runs out of acid during the 3-month period, can place emergency order for $600/gal Murat Kaya , Sabancı Üniversitesi 22 Murat Kaya , Sabancı Üniversitesi 23Model Environment
Development of the Cost Function
Leibniz’s Rule
Optimal Policy
The CDF Plot
Normally Distributed Demand
Normally Distributed Demand
The Standard Normal Distribution
Normally Distributed Demand: Insights
Back to the Newsboy Example
Another Newsboy Example
Performance Measures
Example with Uniform Demand
Revenue Management Example
Revenue Management Example
The Case of Discrete Demand
In Class Exercise
In Class Exercise: