SINGLE PERIOD INVENTORY MODEL

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SINGLE PERIOD INVENTORY MODEL

• Supplement to Chapter 13

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Motivating Example

• You started working as the new cafeteria manager
  in Samuel Bronfman Building
• Its Friday, Exam Day
• Students dont have much time
• They wont wait for you to prepare a sandwich
• If nothing is ready they will head to
  super-sandwich.
• You are asked to make decisions about the
  sandwich section.
• The sandwiches are prepared early in the morning
• Unfortunately you can not order again during the
  day, all orders must be placed early in the
  morning

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Motivating Example

• You need to think about
• Purchase cost
• Selling price
• Uncertainty in demand
• Is there any value to leftovers?
• You need to decide on
• How much to order so as to maximize profit?
• Note that When to order ? is not a valid
  question anymore you can only order once

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Marginal Analysis for Stock-and-Sell Decisions

• The Newsvendors Problem
• (A Single Period Inventory Model).
• A newsvendor is faced with the problem of
  deciding how many newspapers to order daily so as
  to maximize the daily profit
• The problem actually is very similar to the
  sandwich problem
• Daily demand (d) for newspapers is a random
  variable.
• No reordering is possible during a day,
• If the newsvendor orders fewer papers than
  customers demand he or she will lose the
  opportunity to sell some papers.
• If supply exceeds demand, the newsvendor will be
  stuck with papers which cannot be sold.
• A single period can be of any time unit
• Day, week, month, quartile, year

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Demand Data

• Based on observations over several weeks, the
  newsvendor has established the following
  probability distribution of daily demand
• The newsvendor purchases daily papers at 0.20
  and sells them at 0.50 apiece. Leftover papers
  are valueless and are discarded (i.e. no salvage
  value).

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Analysis of Costs in the Newsvendors Problem.

• The newsvendor identifies two penalty costs which
  he/she will incur, regardless of his/her
  decision
• Cost of Overage
• CO Purchase Price - Salvage Value c - s
• For each paper overstocked the newsvendor incurs
  a penalty cost of CO 0.20 - 0.00 0.20.
• Cost of Underage
• CU Selling Price - Purchase Price p - c
• For each paper understocked the newsvendor
  incurs a penalty (opportunity) cost of CU
  0.50 - 0.20 0.30.

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Marginal Analysis The Critical Fractile Method
(Discrete Demand Distribution)

• Assume that there is already a policy in place to
  order a certain number of papers daily, say 38.
• Consider the decisions
• D1 Continue the present policy Stock 38
  papers.
• D2 Order one more paper Stock 39 papers.
• The possible events are
• E1 The 39th paper sells (i.e. demand ? 39
  demand gt 38).
• E2 The 39th paper does not sell (i.e. demand ?
  39 demand ? 38).

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Marginal Analysis (Discrete Demand Distribution)

• Payoffs are incremental profits (i.e. the change
  in profit associated with the events). The
  following decision tree summarizes the
  newsvendors problem

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A note on the computation of probabilities
associated with the events sell item 39 and
fail to sell item 39

• Item 39 will not sell on a given day only if
  demand on that day is for 38 or fewer items
• P(D ? 38) F(38) 0.20.
• The probability that an item will not sell is the
  cumulative probability associated with the
  previous item.
• Item 39 will sell on a given day only if demand
  on that day is for 39 or more items
• P(D ? 39) 1 - P(D ? 38).
• 1 - F(38).
• 1 - 0.2.
• 0.80.
• The expected payoff for the branch Stock 39 is
  computed by
• 0.30(0.8) (-0.20)(0.2) 0.20.
• This implies an increase in profit of 0.20 as
  compared to the alternative decision which has a
  payoff of 0.00
• the optimal decision is to stock the 39th paper.

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Marginal Analysis (Discrete Demand Distribution)

• Applying the previous principles to the general
  case of deciding between stocking Q items or Q
  1 items, we obtain the following decision tree

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Marginal Analysis (Discrete Demand Distribution)

• The expected monetary value for the decision to
  stock Q1 items is
• CU 1 - F(Q) - CO F(Q)
• The decision maker will choose this option as
  long as
• CU 1 - F(Q) - CO F(Q) ? 0.
• ? CU - CU F(Q) - CO F(Q) ? 0.
• ? CU - CU CO F(Q) ? 0.
• ? CU ? CU CO F(Q).
• ? CU / CU CO ? F(Q).
• The quantity on the left-hand side is called the
  critical ratio or critical fractile or cycle
  service level.

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DECISION RULE
Marginal Analysis (Discrete Demand Distribution)

• Order item Q 1 if F(Q) ? CU / CU CO
• Do not order item Q 1 if F(Q) ? CU / CU
  CO
• Indifferent if F(Q) CU / CU CO
• The optimal order quantity Q is the first value
  of Q for which F(Q) is larger than the critical
  ratio.
• EXAMPLE
• CU p-c 0.50 - 0.20 0.30
• CO c-s 0.20 - 0.00 0.20
• ? CU / CU CO 0.6
• ? Q 41

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Cumulative Probability
0.95
0.85
0.70
critical ratio or cycle service level.
0.50
0.35
39
40
41
42
43
Stocking Level
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Computation of Expected Profit for Optimal
Decision Q 41.

• Profit Total Revenue Total Salvage Value
  Total Purchase Cost.
• For this example (salvage 0.00)

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Continuous Demand Distribution

• The decision rule is to select Q such that

DECISION RULE

• is also referred as Customer Service Level (CSL)

If the selling price (p) increases, Q
increases If the purchase cost (c) increases, Q
decreases If the salvage value (s) increases, Q
increases
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Example Normal Distribution

• Suppose that in SBB cafeteria the price for each
  sandwiches is 12. Cost of production is 6/
  unit. If there are leftovers at the end of the
  day you sell them for 2/unit. Demand is
  estimated to be normally distributed with a mean
  of 60 and a standard deviation of 15.
• What is the optimal order quantity?

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Example Normal Distribution

• c 6/unit, s 2/unit, p 12/unit
• What is the optimal order quantity?
• CU p c 12 - 6 6
• CO c s 6 - 2 4
• CU / CU CO 0.6

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areaF(z)0.6
f(z)
z
z

• Transform D N(m,s) to z N(0,1)
• z (D - m) / s.
• F(z) Prob( N(0,1) lt z)

• Transform back, knowing z
• Q m zs.

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F(z)0.6 F(0.255)0.5987 F(0.256)0.6026 z?0.25
33
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Example Uniform distribution

• Suppose that the demand in SBB cafeteria is
  estimated to be uniformly distributed between 5
  and 55. What is the order quantity?
• If X U(A,B)

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Example Uniform distribution

• c 6/unit, s 2/unit, p 12/unit
• What is the optimal order quantity?
• CU p-c 12-6 6
• CO c-s 6 -2 4
• CU / CU CO 0.6
• F(Q)0.6

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Numerous Other Applications

• Fashion items
• Seasonal hot items
• High-tech goods
• Holiday items
• Christmas trees, toys
• Flowers on Valentines day
• Perishables
• Meals in cafeteria
• Dairy foods