Brian Kelly is the founder and CEO of Kelly$$s Barbecue and Grill $($KBG$).$ KBG is a local retailer of outdoor cooking equipment, ranging from small portable gas grills to competition grade barbecue grills and smokers. Kelly has operated his business profitably for nearly $15$ years. In recent years, sales revenues and profits have been stagnant. As an old$-$fashioned type, Kelly has always operated his business on a cash$-$only basis. After consulting with his accountant, Josh Adams CPA, Kelly believes that his sales policy has left little room for additional growth.

Kelly is considering offering $30-$day credit to customers in an effort to drive increased sales. Initially, he plans to only offer credit on his high$-$end propane grill model before extending this new policy to his full inventory. His luxury gas grill model, the Caliber Cross Flame Pro retails for $$12,590$ and costs Kelly $$10,470$ per unit from his supplier. There should be no change to his cost per unit, but Kelly believes that he might be able to mark$-$up his retail price to $$13,000$ since he will be offering more flexible payment terms. KBG currently sells an average of $15$ units of the Caliber model each month. Kelly thinks that he may be able to sell up to an average of $22$ units per month if he begins offering the proposed $30-$day credit terms. Kelly has further decided that a $3\%$ per month required return would be appropriate for evaluating the proposed credit policy change.

$1)$ Calculate the net present value $($NPV$)$ of switching from the cash$-$only policy to the $30-$day credit terms.

$2)$ Determine the minimum increase, or maximum decrease, in monthly sales volume for Kelly to deem the switch appropriate.

Kelly currently places a monthly order of $15$ units of the Caliber model from his supplier. It costs him $$50$ to place each order and he estimates that it costs him about $$23$ to store each grill in inventory for a year.

$3)$ Under his current ordering system, with monthly sales of $15$ units;

a$)$ What is the average inventory balance $($in units$)$ for the Caliber model, assuming the grills are sold evenly throughout the month?

b$)$ How much does Kelly pay per year in ordering costs?

c$)$ How much does Kelly pay per year in storage costs?

$4)$ How much money would Kelly save per year if he were to optimize his order size?

$5)$ Assuming his goal would be to minimize the total costs incurred for ordering$/$storing inventory, how many units of the Caliber model should Kelly order each time he places an order when his monthly sales volume is $22$ units?