02c probabilistic inventory models (1)

Probabilistic Inventory Models

(Part – III)

1 Sasadhar Bera, IIM Ranchi

Outline

2 Sasadhar Bera, IIM Ranchi

Probabilistic Model

Single period Discrete demand

Single period Continuous demand

Introduction

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Sasadhar Bera, IIM Ranchi

The major influencing factors for the inventory decisions are price, demand, and lead time. Other factors such as ordering cost, carrying cost, and stock-out cost are also affecting the inventory decisions but their nature is not so much disturbing. Sometimes price fluctuations are too much in the market and hence it influences inventory decisions. Similarly, variability in demand or consumption of an item as well as the variability in lead time influence the overall inventory policy.


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Probabilistic inventory models: I. Single period probabilistic models: The single period

inventory model applies to inventory situations in which only one order is placed for goods in anticipation of a future selling season where demand is uncertain. At the end of the period the product has either sold out or there is a surplus of unsold items to sell at salvage value.

II. Multi-period probabilistic models: Multi-period inventory models are usually addressed in finding the minimum expected cost over N periods. In these models both demand and lead time are uncertain.

Single Period Probabilistic Models

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In single period inventory situation, the only question is how much of the product to order at the start of the period. Newspaper sales are a typical example of the single period model. Thus the single period inventory problem is sometimes referred to as the newsvendor problem. These models deal with inventory situation of the items such as perishable goods, spare parts, and seasonal goods requiring one time purchase only.

Single Period Probabilistic Models (Contd.)

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Applications of single period model:

i. Order size of Periodicals, such as newspapers and magazines.

ii. Order size seasonal greeting cards.

iii. Order size Christmas trees.

iv. Seasonal clothing, such as winter coats, where any goods remaining at the end of the season must be sold at highly discounted prices to clear space for the next season.

v. Vital spare parts that must be produced during the last production run of a certain model of a product (e.g., an airplane) for use as needed throughout the lengthy field life of that model.

vi. Reservations provided by an airline for a particular flight.

vii. Reservations of hotel room.


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The demand for such item may be discrete or continuous. Since for a given period, purchase of item is made only once, the lead time factor is least important in these models. In such cases, there are two types of costs involved: Over-stocking cost: Loss due to excess stocks Under-stocking cost: Opportunity losses due to high demand D = Demand of an item in units (a random variable) Q = the number of units stocked (or to be purchased) P = Purchase price per unit S = Selling price per unit Ch = Carrying cost or holding cost per unit for entire period Cs = Shortage cost per unit V = Salvage value per unit


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Co = Over-stocking cost per unit = Loss associated with each unit left unsold = P + Ch – V Cu = Under-stocking cost per unit = Loss due to not meeting demand = S – P + Cs The single period can be solved using a technique called marginal economic analysis, which compares the cost or loss of ordering one additional item with the cost or loss of not ordering one additional item.

Single Period Discrete Demand Model

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Single Period Discrete Probabilistic Models

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Single Period Discrete Probabilistic Models (Contd.)

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Example: Discrete Demand Probability

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Example: Demand follows discrete distribution A newspaper boy buys papers for Rs. 2 each and sells them for Rs. 2.50 each. He cannot return unsold newspapers. Daily demand has the following distribution If each day’s demand is independent of the previous day’s demand, how many papers should he order each day?

Nos of customers 230 240 250 260 270 280 290 300 310 320 Probability 0.01 0.03 0.06 0.10 0.2 0.25 0.15 0.1 0.05 0.05

Example: Discrete Demand Probability (Contd.)

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Single Period Continuous Demand Model

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Single Period Continuous Probabilistic Models

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Single Period Continuous Probabilistic Models (Contd.)

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Example: Continuous Demand Probability

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Example: Demand follows uniform distribution An item sells for Rs. 20 per unit and cost Rs. 10. Unsold items can be sold for Rs. 4 each. It is assumed that there is no shortage penalty cost besides the lost revenue. The demand is known to be any value between 500 and 1200 items. Determine the optimal number of units to the item to be stocked.

Example: Continuous Demand Probability (Contd.)

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Example: Continuous Demand Probability (Contd.)

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02c probabilistic inventory models (1)

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