Problem $2$: Simulation

GLAP is one of the largest retailers in the DFW area, and it is always innovating to satisfy their customers. Their last innovation relates to a new refund policy. Specifically, if you buy a product at full$-$price and it is subsequently put on sale, GLAP allows to turn up with the receipt and get the difference refunded. This policy seems counterproductive: how could it possibly increase GLAP$$s overall income?

In this question, we will construct a very simple simulation model to explore the benefits of this policy. Assume that GLAP only sells two products, call them Product A and Product B$.$ GLAP knows that the demand for each product is normally distributed, but does not know what the actual demand will eventually be $($or$,$ indeed, whether one product will be more popular than the other$).$ The following table summarizes the mean and standard deviation of demand at a particular store for each product.

In addition, GLAP divides the season in two phases: $($i$)$ Phase $1,$ whereby both products are priced at their full$-$price of $$50$; and $($ii$)$ Phase $2,$ whereby each product is put on sale and, thus, their price is reduced to $$45,$ if their demand in Phase $1$ is less than $500$ units. Conversely, if the demand in Phase $1$ exceeds $500$ units, the price remains at $$50$ in Phase $2.$ For instance, if the demand for Products A and B in Phase $1$ are $600$ and $400,$ respectively, then the price of Products A and B in Phase $2$ are $$50$ and $$45,$ respectively.

The fraction of demand that arrives in each phase depends on the return policy as follows:

$$ If GLAP implements the return policy described above, then $90\%$ of customers buy the products in Phase $1($knowing that they will get a refund for the difference if the product

in on sale in Phase $2)$ and $10\%$ wait and buy products in Phase $2.$ For example, if the realized demand for Product A is $1,000$ units, $900$ would be bought in Phase $1$ and $100$ in Phase $2.$

$$ If GLAP does not implement the return policy, i$.$e$.,$ customers who buy in Phase $1$ do not get a refund if their product is on sale in Phase $2,$ then $60\%$ of customers buy in Phase $1$ and $40\%$ buy in Phase $2.$ For instance, if the realized demand for Product B is $1,200$ units, then $720$ customers buy in Phase $1$ and $480$ buy in Phase $2.$ The rationale for this behavior is that, if the refund policy is not in place, a larger fraction of customers choose to wait for a potential markdown.

Assume that the supply of each product is basically unlimited and that fractional number of customers is possible.

$[$Hint: model the demand for each product as an assumption, and then in a different cell make sure that your demand is always greater than or equal to zero, i$.$e$.,$ actual demand $=$ MAX$($Assumption cell, $0)]$

$1.$ Develop a model in Crystal Ball to determine GLAP$$s expected profit in both scenarios $($using the refund policy and not using it$),$ and estimate the expected difference in profit. What is the probability that the profit under the refund policy is higher than the profit without the refund policy?

$2.$ GLAP is also interested in the probability it will end up having to discount Product A$.$ Find this probability in each scenario, i$.$e$.,$ when GLAP is using the refund policy and when it is not.

$3.$ GLAP realized that it is very inefficient to have unlimited supply for each product and decides to implement some changes starting with Product A $($this part only focuses on Product A$).$ Before the season starts $($and before observing demand$),$ GLAP will decide the quantity of Product A to order for $$10$ per unit ordered. If GLAP runs out of the product in Phase $1,$ then the sale in Phase $2$ does not take place and none of the Phase $1$ customers gets the difference refunded. Otherwise, if GLAP has inventory left after Phase $1,$ then Phase $2$ takes place and follows the same procedure as in part $1,$ i$.$e$.,($i$)$ the price is reduced to $$45$ if demand in Phase $1$ is below $500,$ and $($ii$)$ customers get refunded if the price is reduced and the refund policy is in place, while they do not get refunded otherwise. GLAP estimates that each unit of unmet demand $($that is$,$ every customer that showed up and was unable to buy because there was no inventory left$)$ represents a $$2$ loss in future sales due to goodwill loss. Also, GLAP gives away every unit of unsold inventory after Phase $2$ to charity. Calculate the optimal quantity of Product A to order in both scenarios $($using the refund policy and not using the refund policy$).$

$[$Hint: you can use a step size of $100$ and a range from $300$ to $1500]$ in excel