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A dairy product is ordered daily at a particular supermarket. The product, which costs $1.19 per unit, sells for $1.65 per unit. If units are unsold at the end of the day, the supplier takes them back at a rebate of $1 per unit. Assume that daily demand is approximately normally distributed with a mean of 150 and a standard deviation of 30.

a) What is your recommended daily order quantity for the supermarket?
b) What is the probability that the supermarket will sell all the units it orders?
c) In problems such as these, why would the supplier offer a rebate as high as $1? For example, why not offer a nominal rebate of, say 25 cents per unit? What happens to the supermarket order quantity as the rebate is reduced?

Mean demand (μ) = 150
Standard deviation (σ) = 30
(a) Daily order quantity Q* = μ + z* σ
Assuming 95% confidence level, z value is 1.65.
Q* = 150 + 30*1.65 = 199.5 or 200 units.
Q* =200 units.

b) What is the probability that the supermarket will sell all the units it orders?

Cost of product per unit = $1.19
Selling price per unit = $ 1.65
Rebate on unsold product per unit = $1
Cost of overestimating Co= Loss on each extra unit = Cost of product - Rebate on each unit = $1.19 - $1 =$0.19
Cost of underestimating Cu = Loss of profit per unit = Selling price - cost per unit = $1.65 -1.19 = $ 0.46
Probability that the supermarket will sell all the units it orders = Cu/Cu + Co = 0.46 /(0.46+0.19) = 0.707

P(Supermarket will sell all units it orders) = 0.707
c) In problems such as these, why would the supplier offer a rebate as high as $1? For example, why not offer a nominal rebate of, say 25 cents per unit? What happens to the supermarket order quantity as the rebate is reduced
Assuming the rebate is $0.25. Then the Co = $1.19 - 0.25 = $0.94
Now using incremental analysis with Co = 0.94 and Cu = 0.46, Cu/Cu + Co = 0.46 / (0.46+0.94) =0.328
Find Q * such that P (D<Q*) = 0.328
The probability of 0.328 corresponds to z= -0.47.
Thus, Q* = 150 -0.47 *30 = 135.97 or 136 units.

From the above analysis it can be seen that the optimal order quantity depends on the Cost of overestimating, which in turn depends on the rebate offered by the supplier. Above analysis shows that the probability reduces to more than half if the rebate is reduced to 25 cents from $1. In this case the Ordering quantity has reduced to 136 units from 200 units. Thus, reduction of 64 units.
The Revenue Loss to the supplier = 64*1.19 = $76.16
On the other hand, with 0.70 probability of using all the units ordered, expected no of units returned to the supplier = 200 - 0.707*200 = 58.6 or 57 units.
Loss to the supplier = 57* 1= $57.

It can be said that as the Co is in denominator of the probability formula, increasing the value of Co (Reducing the rebate) would reduce the probability and in turn would reduce the Quantity to be ordered. Thus, it is a trade off between the amount of rebate offered by the supplier and the no of units ordered by the supermarket.