## Decision-making under conditions of risk, Mathematics

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Decision-making Under Conditions of Risk

With decision-making under conditions of risk all possible states of nature are known and the decision maker has sufficient knowledge to assign probabilities to their likelihoods of occurrence. These probabilities may range from subjective assignments based upon the decision maker's feelings and experience to objective assignments based on the collection and analysis of numerous data related to the states of nature.

The most popular method used to evaluate these types of decisions is the use of expected value.

### Example

A discounting clothing store chain purchases shirts from a manufacturer for Rs.100 each. The store will initially sell the shirts for Rs.150 each. It will then sell all unsold items for Rs.75 each to another discounter.

Historical data have been gathered which confirm that monthly demand for the item assumes four possible values. The table gives this demand information along with their respective probabilities of occurrence. The store is trying to decide how many units of the item to stock in a month. Its goal is to select the quantity which maximizes expected monthly profit.

 Estimate of demand Probability 3,000 0.20 5,000 0.25 8,000 0.45 10,000 0.10

To calculate the optimum stock level which maximizes the profit we have to construct the conditional profit table, as shown below. This table summarizes the monthly profit which would result given the selection of a particular stock level and the occurrence of a specific level of demand. The table also reflects the losses that occur when the remaining stock is sold to the discounter at the end of the month and it does not take into account the additional profit it lost when customers demand more than the store has stocked.

Conditional Profit Table

 Stock Decision (Probability) Possible demand 3,000 (0.2) 5,000 (0.25) 8,000 (0.45) 10,000 (0.1) 3,000 5,000 8,000 10,000 1,50,000 1,00,000   25,000  -25,000 1,50,000 2,50,000 1,75,000 1,25,000 1,50,000 2,50,000 4,00,000 3,50,000 1,50,000 2,50,000 4,00,000 5,00,000

The conditional profit values are determined by computing the total profit from units sold and subtracting from this any loss which would have to be absorbed because of overstocking. For example, if the chain store decides to stock 3,000 shirts it always results in a conditional profit of Rs.1,50,000 [3,000 (150 - 100)], because if the demand is for more than 3000 shirts, all the shirts stocked will be sold.

But, if the chain store decides to stock 5,000 shirts and the demand is 3,000 shirts, the conditional profit is equal to the total profit of selling 3,000 shirts and the loss incurred by overstocking 2,000 shirts, where loss incurred because of overstocking is the difference between the cost of the remaining shirts and the sale price of these shirts to another discounter for Rs.75. Thus, conditional profit = (3,000 x 50)          - (2000 x 25) = Rs.1,00,000.

The conditional profits for other stock decisions are also calculated in a similar manner.

The expected daily profits for each stock decision can be determined by weighing each conditional profit by its likelihood of occurrence (which is the probability of the corresponding level of demand). The table below gives the conditional profit for each level of stock decision.

Expected Monthly Profit Computation

Stock 3,000 shirts

 Conditional Profit (Rs.) Probability of Occurrence Expected Monthly Profit (Rs.) 1,50,000 0.20 30,000 1,50,000 0.25 37,500 1,50,000 0.45 67,500 1,50,000 0.10 15,000 1,50,000

Stock 5,000 units

 Conditional Profit (Rs.) Probability of Occurrence Expected Monthly Profit (Rs.) 1,00,000 0.20 20,000 2,50,000 0.25 62,500 2,50,000 0.45 1,12,500 2,50,000 0.10 25,000 2,20,000

Stock 8,000 units

 Conditional Profit (Rs.) Probability of Occurrence Expected Monthly Profit (Rs.) 25,000 0.20 5,000 1,75,000 0.25 43,750 4,00,000 0.45 1,80,000 4,00,000 0.10 40,000 2,68,750

Stock 10,000 units

 Conditional profit (Rs.) Probability of Occurrence Expected Monthly Profit (Rs.) -25,000 0.20 - 5,000 1,25,000 0.25 31,250 3,50,000 0.45 1,57,500 5,00,000 0.10 50,000 2,33,750

On the basis of expected monthly profits, the best decision is to stock 8,000 shirts, resulting in an expected (average) profit of Rs.2,68,750.

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