A perishable diary product is ordered daily at a particular supermarket. The product, which cost $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 normal distributed with μ = 150 and σ = 30. 1. What is your recommended daily order quantity for the supermarket? 2. What is the probability that the supermarket will sell all the units it orders? 3. 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₡ per unit? What happens to the supermarket order quantity as the rebate is reduced?

To answer the given questions, we need to calculate the recommended daily order quantity, the probability of selling all ordered units, and understand the supplier's rebate strategy. Let's break down each question and explain how to get the answers step by step.

1. What is the recommended daily order quantity for the supermarket?
To determine the recommended daily order quantity, we need to find the "economic order quantity" (EOQ) which minimizes the total cost of ordering and holding inventory. The EOQ formula is given by:
EOQ = √[(2 * D * S) / H]
Where:
D = Average daily demand
S = Cost per order
H = Holding cost per unit per day

In this case, the average daily demand is given as μ = 150, and we are assuming a normal distribution with a standard deviation of σ = 30. The cost per order (S) can be calculated by multiplying the quantity of the order by the order cost, which is not provided. Similarly, the holding cost per unit per day (H) is not provided. Without these values, it is not possible to calculate the EOQ or the recommended daily order quantity accurately.

2. What is the probability that the supermarket will sell all the units it orders?
To find the probability of selling all ordered units, we need to calculate the area under the normal distribution curve. Since the demand is assumed to be normally distributed, we can use the Z-score formula and lookup tables to find the probability.

The Z-score formula is given by:
Z = (X - μ) / σ
Where:
X = Number of units to be sold
μ = Mean demand
σ = Standard deviation

In this case, we want to find the probability of selling all units, so X would be equal to or greater than the recommended daily order quantity (which we haven't calculated yet). We can use the Z-score to find the probability associated with that value.

3. Why would the supplier offer a rebate as high as $1? What happens to the supermarket order quantity as the rebate is reduced?
The supplier offers a rebate of $1 per unit to incentivize the supermarket to order a higher quantity. This is known as a quantity discount or volume-based rebate. The supplier benefits from economies of scale when the supermarket orders larger quantities, as they can produce or supply the product more efficiently and reduce their costs.

If the supplier were to offer a lower rebate, such as $0.25 per unit as suggested in the question, the incentive for the supermarket to order higher quantities would be lower. The order quantity would decrease because the smaller rebate does not compensate for the holding costs incurred by the supermarket for unsold units. The supermarket would prefer to minimize its investment in inventory and reduce the risk of having excess unsold units.

In summary, to accurately answer the questions, we would need additional information regarding the order cost and the holding cost per unit per day. These factors are essential for calculating the recommended daily order quantity and making further assessments about the supplier's rebate strategy.