Inventory Management-Single Period

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Inventory management in supply chain

Single period models Impact of demand uncertainty

Only one ordering opportunity

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Source: MIT Sloan School of Management Arshinder,DoMS, IIT MAdras

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Decision

How much to Produce or Buy

One Time Shot

Perishable Goods

Excess Demand is Lost

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The Principal Trade-Off

Ordering Too Much

Inventory left over at Period End

Inventory sold at Loss

Ordering Too Little

Not all Demand is served

Loosing out on Revenue

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What should be the optimal order quantity?

Using historical data identify a variety of demand scenarios

determine probability each of these scenarios will occur

Given a specific inventory policy determine the profit associated with a particular scenario

given a specific order quantity weight each scenarios profit by the likelihood that it will occur

determine the average, or expected, profit for a particular ordering quantity.

Order the quantity that maximizes the average profit.

Example: Summer fashion items swimsuits

A company designs, produces, and sells summer fashion items such as swimsuits.

About six months before summer, the company must commit itself to specific production quantities for all its products.

Since there is no clear indication of how the market will respond to the new designs, the company needs to use various tools to predict demand for each design, and plan production and supply accordingly.

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Construct probabilistic forecast of demand

Probabilistic forecast based on five years historical data,

current economic conditions and other relevant factors

Average demand =13000

Additional Information

Fixed production cost: Rs.100,000

Variable production cost per unit: Rs. 80.

During the summer season, selling price: Rs. 125 per unit.

Salvage value: Any swimsuit not sold during the summer season is sold to a discount store for Rs. 20.

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Important decisions

Though average demand is 13000 but there is a probability that demand will be either larger than average or smaller than average.

What should be the production quantity?

Relationship between production quantity, customer demand and profits

Two Scenarios Manufacturer produces 10,000 units while demand

ends at 12,000 swimsuits Profit = 125(10,000) - 80(10,000) - 100,000 = Rs. 350,000 Manufacturer produces 10,000 units while demand

ends at 8,000 swimsuits Profit = 125(8,000) + 20(2,000) - 80(10,000) - 100,000 = Rs. 140,000

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Probability of Profitability Scenarios with Production = 10,000 Units

Probability of demand being 8000 units is 11%

Probability of profit of Rs. 140,000 = 11%

Probability of demand being 12000 units = 27%

Probability of profit of Rs. 140,000 = 27%

Total profit = Weighted average of profit scenarios

Order Quantity that Maximizes Expected Profit

FIGURE 2-6: Average profit as a function of production quantity

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Relationship Between Optimal Quantity and Average Demand

Compare marginal profit of selling an additional unit and marginal cost of not selling an additional unit

Marginal profit/unit = Selling Price - Variable Ordering (or, Production) Cost Marginal cost/unit = Variable Ordering (or, Production) Cost - Salvage Value

If Marginal Profit > Marginal Cost => Optimal Quantity > Average Demand If Marginal Profit < Marginal Cost => Optimal Quantity < Average Demand

For the Swimsuit Example

Average demand = 13,000 units.

Optimal production quantity = 12,000 units.

Marginal profit = Rs. 45

Marginal cost = Rs. 60.

Thus, Marginal Cost > Marginal Profit

=> optimal production quantity < average demand.

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Can you make out risk reward trade-off in the previous example?

Marginal cost analysis approach to single period uncertain demand

inventory model

News vendor case

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Newsvendor strategy

Consider the qth unit purchased by a newsvendor. The newsvendor will be able to sell qth unit if and only if

The total demand, d >=q

By buying and then making the qth unit available to the customer, newsvendor have avoided underage cost

Otherwise, an overage cost will be incurred for the qth unit

Newsvendor strategy The last unit q (or optimal quantity Q* ) we

would want to acquire is one where the expected overage cost incurred exactly equaled the expected underage cost saved.

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Notations

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Newsvendor model

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Optimal order quantity P (d < q)

Probability of meeting customer demand or cycle service level (CSL) at corresponding q is optimal order size q

Change the notation to Q* for optimal order size

P (d < Q* )=Optimal Cycle Service level (CSL*)

= ku / (ku + ko)

Optimal order size, Q* = F-1(CSL*, , )

Where, is the mean of demand distribution and is the standard deviation of demand

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