Inventory Control with Stochastic Demand

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Inventory Control with Stochastic Demand
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Lecture Topics

• Week 1 Introduction to Production Planning and
  Inventory Control
• Week 2 Inventory Control Deterministic Demand
• Week 3 Inventory Control Stochastic Demand
• Week 4 Inventory Control Stochastic Demand
• Week 5 Inventory Control Stochastic Demand
• Week 6 Inventory Control Time Varying Demand
• Week 7 Inventory Control Multiple Echelons

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Lecture Topics (Continued)

• Week 8 Production Planning and Scheduling
• Week 9 Production Planning and Scheduling
• Week 12 Managing Manufacturing Operations
• Week 13 Managing Manufacturing Operations
• Week 14 Managing Manufacturing Operations
• Week 10 Demand Forecasting
• Week 11 Demand Forecasting
• Week 15 Project Presentations

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• Demand per unit time is a random variable X with
  mean E(X) and standard deviation s
• Possibility of overstocking (excess inventory) or
  understocking (shortages)
• There are overage costs for overstocking and
  shortage costs for understocking

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Types of Stochastic Models

• Single period models
• Fashion goods, perishable goods, goods with short
  lifecycles, seasonal goods
• One time decision (how much to order)
• Multiple period models
• Goods with recurring demand but whose demand
  varies from period to period
• Inventory systems with periodic review
• Periodic decisions (how much to order in each
  period)

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Types of Stochastic Models (continued)

• Continuous time models
• Goods with recurring demand but with variable
  inter-arrival times between customer orders
• Inventory system with continuous review
• Continuous decisions (continuously deciding on
  how much to order)

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Example

• If l is the order replenishment lead time, D is
  demand per unit time, and r is the reorder point
  (in a continuous review system), then
• Probability of stockout P(demand during lead
  time ? r)
• If demand during lead time is normally
  distributed with mean E(D)l, then choosing r
  E(D)l leads to
• Probability of stockout 0.5

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The Newsvendor Model
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Assumptions of the Basic Model

• A single period
• Random demand with known distribution
• Cost per unit of leftover inventory (overage
  cost)
• Cost per unit of unsatisfied demand (shortage
  cost)

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• Objective Minimize the sum of expected shortage
  and overage costs
• Tradeoff If we order too little, we incur a
  shortage cost if we order too much we incur a an
  overage cost

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Notation
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The Cost Function
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The Cost Function (Continued)
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Leibnitzs Rule
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The Optimal Order Quantity

• The optimal solution satisfies

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The Exponential Distribution

• The Exponential distribution with parameters l

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The Exponential Distribution (Continued)
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Example

• Scenario
• Demand for T-shirts has the exponential
  distribution with mean 1000 (i.e., G(x) P(X ?
  x) 1- e-x/1000)
• 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

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Example (Continued)

• Solution
• Sensitivity
• If co 10 (i.e., shirts must be
  discarded) then

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The Normal Distribution

• The Normal distribution with parameters m and s,
  N(m, s)
• If X has the normal distribution N(m, s), then
  (X-m)/s has the standard normal distribution
  N(0, 1).
• The cumulative distributive function of the
  Standard normal distribution is denoted by F.

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The Normal Distribution (Continued)

• G(Q)a
• ?
• Pr(X ?Q) a
• ?
• Pr(X - m)/s ? (Q - m)/s a
• ?
• Let Y (X - m)/s, then Y has the the standard
  Normal distribution
• Pr(Y ? (Q - m)/s F(Q - m)/s a

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The Normal Distribution (Continued)

• F((Q - m)/s) a
• ?
• Define za such that F(za)a
• ?
• Q m zas

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The Optimal Cost for Normally Distributed Demand
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The Optimal Cost for Normally Distributed Demand
Both the optimal order quantity and the optimal
cost depend only on the variance of demand. Both
increase linearly in the standard deviation of
demand.
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Example

• Demand has the Normal distribution with mean m
  10,000 and standard deviation s 1,000
• cs 1
• co 0.5 ? a 0.67

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Example

• Demand has the Normal distribution with mean m
  10,000 and standard deviation s 1,000
• cs 1
• co 0.5 ? a 0.67

Q m zas From a standard normal table, we
find that z0.67 0.44 Q m sza 10,000
0.44(1,000) 10,440
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Service Levels

• Probability of no stockout
• Fill rate

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Service Levels

• Probability of no stockout
• Fill rate
• Fill rate can be significantly higher than
• the probability of no stockout

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Discrete Demand
X is a discrete random variable
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Discrete Demand (Continued)
The optimal value of Q is the smallest integer
that satisfies This is equivalent to choosing
the smallest integer Q that satisfies or
equivalently
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The Geometric Distribution
The geometric distribution with parameter r , 0 ?
r ? 1
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The Geometric Distribution
The optimal order quantity Q is the smallest
integer that satisfies
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Extension to Multiple Periods

• The news-vendor model can be used to a solve a
  multi-period problem, when
• We face periodic demands that are independent
  and identically distributed (iid) with
  distribution G(x)
• All orders are either backordered (i.e., met
  eventually) or lost
• There is no setup cost associated with
  producing an order

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Extension to Multiple Periods (continued)

• In this case
• co is the cost to hold one unit of inventory in
  stock for one period
• cs is either the cost of backordering one unit
  for one period or the cost of a lost sale

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Handling Starting Inventory/backorders