Question

Green Thumb, a manufacturer of lawn care equipment, has introduced a new product. The anticipated demand is normally distributed with a mean of  = 100 and a standard deviation of = 40. Each unit costs $150 to manufacture and the introductory price is to be $200 to achieve this level of sales. Any unsold units at the end of the season are unlikely to be very valuable and will be disposed off in a fire sale for $50 each. It costs $20 to hold a unit in inventory for the entire season. How many units should Green Thumb manufacture for sale? What is the expected profit from this policy?

Answer 1

Cost of understocking Cu= selling price-cost price = 200-150 = 50
Cost of overstocking Co= Cost price+holding cost-salvage value = 150+20-50 = 120

Critical fractile = Cu/(Cu+Co) = 50/(50+120) = 0.294117647

Corresponding z score = normsinv(0.294117647) = -0.541395085

Green Thumb should manufacture 100+(-0.541395085)*40 = 78.3441966 units for sale

Expected profit calculation

L(z) = =normdist(z,0,1,0)-z*(1-normdist(z,0,1,1)) = NORMDIST(-0.541395085,0,1,0)-(-0.541395085)*(1-NORMDIST(-0.541395085,0,1,1)) = 0.726719233

Expected lost sales = standard deviation*L(z) = 40*0.726719233 = 29.06876932

Expected sales = demand - Expected lost sales = 100-29.06876932 = 70.93123068

Expected leftover inventory = order quantity-expected sales = 78.3441966-70.93123068 = 7.41296592

Expected profit = Cu*Expected sales-Co*Expected leftover inventory = 50*70.93123068-120*7.41296592 = 2657.005624 (Ans)