INVENTORY THEORY

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INVENTORY THEORY
Probabilistic Demand
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The Effect of Demand Uncertainty

• Most companies treat the world as if it were
  predictable
• Production and inventory planning are based on
  forecasts of demand made far in advance of the
  selling season
• Companies are aware of demand uncertainty when
  they create a forecast, but they design their
  planning process as if the forecast truly
  represents reality
• Recent technological advances have increased the
  level of demand uncertainty
• Short product life cycles
• Increasing product variety

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Demand Forecasts

• The three principles of all forecasting
  techniques
• Forecasting is always wrong
• The longer the forecast horizon the worst is the
  forecast
• Aggregate forecasts are more accurate

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SnowTime Sporting Goods

• Fashion items have short life cycles, high
  variety of competitors
• SnowTime Sporting Goods
• New designs are completed
• One production opportunity
• Based on past sales, knowledge of the industry,
  and economic conditions, the marketing department
  has a probabilistic forecast
• The forecast averages about 13,000, but there is
  a chance that demand will be greater or less than
  this.

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SnowTime Demand Scenarios
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SnowTime Costs

• Production cost per unit (c) 80
• Selling price per unit (r) 125
• Salvage value per unit (V) 20
• Fixed production cost (K) 100,000
• Q is production quantity, D demand
• Profit Revenue - Variable Cost - Fixed Cost
  Salvage

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SnowTime Scenarios

• Scenario One
• Suppose you make 12,000 jackets and demand ends
  up being 13,000 jackets.
• Profit 125(12,000) - 80(12,000) - 100,000
• 440,000
• Scenario Two
• Suppose you make 12,000 jackets and demand ends
  up being 11,000 jackets.
• Profit 125(11,000) - 80(12,000) - 100,000
  20(1000) 335,000

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SnowTime Best Solution

• Find order quantity that maximizes expected
  profit.
• Question Will this quantity be less than, equal
  to, or greater than average demand?

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What to Make?

• ANSWER
• Average demand is 13,100
• Look at marginal cost Vs. marginal profit
• if extra jacket sold, profit is 125-80 45
• if not sold, cost is 80-20 60
• So we will make less than average

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SnowTime Expected Profit
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SnowTime Important Observations

• Several quantities have the same average profit
• Average profit does not tell the whole story
• Question 9000 and 16000 units lead to about the
  same average profit, so which do we prefer?

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Probability of Outcomes
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Key Points from this Model

• The optimal order quantity is not necessarily
  equal to average forecast demand
• The optimal quantity depends on the relationship
  between marginal profit and marginal cost
• As order quantity increases, average profit first
  increases and then decreases
• As production quantity increases, risk increases.
  In other words, the probability of large gains
  and of large losses increases

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Initial Inventory

• Suppose that one of the jacket designs is a model
  produced last year.
• Some inventory is left from last year
• Assume the same demand pattern as before
• If only old inventory is sold, no setup cost
• Question If there are 7000 units remaining, what
  should SnowTime do? What should they do if there
  are 10,000 remaining?

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Initial Inventory and Profit
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(s, S) Policies

• For some starting inventory levels, it is better
  to not start production
• If we start, we always produce to the same level
• Thus, we use an (s,S) policy.
• The difference between the two levels is driven
  by the fixed costs associated with ordering,
  transportation, or manufacturing

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Probabilistic Demand Single Period

• In General When there is no set up cost
• h unit holding cost/time, p unit shortage
  cost/time,
• Let y Inventory Level that maximize expected
  profit,
• then

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Probabilistic Demand Single Period
In General When there is set up cost (s,S)
policy S can be determined by
s can be determined by
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Probabilistic Demand Multiple Periods

• Sources of uncertainty
• Uncertain lead time
• Uncertain demand
• Use (s,Q) Policy

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Probabilistic Demand Multiple Periods with
Backorder
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Probabilistic Demand Multiple Periods with
Demand Loss
and