Question

To ensure a full line of outdoor clothing and accessories, the marketing department at Teddy Bower insists that it also sell waterproof hunting boots. Unfortunately, Teddy Bower does not have expertise in manufacturing those kinds of boots. Hence, Teddy Bower contacted several Taiwanese suppliers to request quotes. Due to competition, Teddy Bower knows that it cannot sell these boots for more than $98. However, $60 per boot was the best quote from the suppliers. In addition, Teddy Bower anticipates excess inventory will need to be sold off at a 50 percent discount at the end of the season. Given the $98 price, Teddy Bower’s demand forecast is for 400 boots, with a standard deviation of 260.

A) If Teddy Bower decides to include these boots in its assortment, how many boots should Teddy Bower order from the supplier?

B) Suppose Teddy Bower orders 380 boots. What is Teddy Bower's expected profit?

C) The marketing department insists that its in-stock probability be at least 98 percent. Given this mandate, how many boots does it need to order?

D) John Briggs, a buyer in the procurement department, overheard at lunch a discussion of the "boot problem." He suggested that Teddy Bower ask for a quantity discount from the supplier. After following up on his suggestion, the supplier responded that Teddy Bower could get a 10 percent discount if it were willing to order at least 775 boots. Compared to the optimal quantity in Part a, how many boots should Teddy Bower order given this new offer if the objective is to maximize expected profit?

Answer 1

Part a:

	Values
Selling Price (p)	$98
Purchase cost (c)	$60
Disposal Cost (s) (50% discount)	= 0.5*98 = $49
Cu = under-stocking cost = cost of shortage (underestimate demand) = Sales price/unit – Cost/unit	Cu = $98 – $60 Cu = $38
Co = Over-stocking cost = Cost of overage (overestimate demand) = Cost/unit – Salvage value/unit	Co = c – s = $60 – $49 Co = $11
The service level or optimal probability of not stocking out, is set at, Service Level = critical ratio = Cu/( Cu + Co)	SL = 38/(38 + 11) SL = 0.7755 SL = 77.55%
Mean Demand (µ)	µ = 400
Standard Deviation (σ)	σ = 260
Optimal Order Quantity = [=Norminv(service level,µ, σ)] **from excel	Q* =Norminv(0.7755,400, 260) Q* = 596.85 units Or Q* = 597 units

Part b.

The order quantity = O = 380 units

Expected Profit = (p)(expected sales) + (s)(expected overstock) – (c)(order quantity)

Expected Overstock = [ (O - µ)*NORMDIST((O - µ)/σ,0,1,1) ] + [σ*NORMDIST((O - µ)/σ,0,1,0) ]

Expected Overstock = (380 - 400)NORMDIST((380 - 400)/260,0,1,1) + (260)NORMDIST((380 - 400)/260,0,1,0)

From excel: Expected Overstock = 94.0317

94 units are expected to be sold at discount rate

Expected sales = Optimal Order quantity – Expected Overstock = 380 – 94.037 = 285.9683 ~ 286 units

Expected Profit = (p)(expected sales) + (s)(overstock) – (c)(Order quantity)

= (98)(286) + (49)(94) – (60)(380) = $9834

Expected Profit for order quantity of 380 = $9,834

Part c.

The in-stock probability be 98%, the Service level should be 98%,

The service level or optimal probability of not stocking out, is set at, 98% Service Level = critical ratio = Cu/( Cu + Co)	SL = 98%
Mean Demand (µ)	µ = 400
Standard Deviation (σ)	σ = 260
Optimal Order Quantity = [=Norminv(service level,µ, σ)] **from excel	Q* =Norminv(0.98,400, 260) Q* = 933.97 units Or Q* = 934 units

Thus, for 98% in-stock probability, the order quantity is 934 units

Part d.

If the order quantity (as per part a.)= O = 597 units

Expected Profit = (p)(expected sales) + (s)(expected overstock) – (c)(order quantity)

Expected Overstock = [ (O - µ)*NORMDIST((O - µ)/σ,0,1,1) ] + [σ*NORMDIST((O - µ)/σ,0,1,0) ]

Expected Overstock = (597 - 400)NORMDIST((597 - 400)/260,0,1,1) + (260)NORMDIST((597 - 400)/260,0,1,0)

From excel: Expected Overstock = 230.54

230.54 units are expected to be sold at discount rate

Expected sales = Optimal Order quantity – Expected Overstock = 597 – 230.54 = 366.314 units

Expected Profit = (p)(expected sales) + (s)(overstock) – (c)(Order quantity)

= (98)(366.314) + (49)(230.54) – (60)(597) = $11,384.03

Expected Profit for order quantity of 597 units (as per part a) = $11,384.03

If order quantity, O = 775, the purchase cost discount is 10%

Cost per unit = $60 – 0.1*60 = $54

Expected Profit = (p)(expected sales) + (s)(expected overstock) – (c)(order quantity)

Expected Overstock = [ (O - µ)*NORMDIST((O - µ)/σ,0,1,1) ] + [σ*NORMDIST((O - µ)/σ,0,1,0) ]

Expected Overstock = (775 - 400)NORMDIST((775 - 400)/260,0,1,1) + (260)NORMDIST((775 - 400)/260,0,1,0)

From excel: Expected Overstock = 383.68 units

383.68 units are expected to be sold at discount rate

Expected sales = Optimal Order quantity – Expected Overstock = 775 – 383.68 = 391.32 units

Expected Profit = (p)(expected sales) + (s)(overstock) – (c)(Order quantity)

= (98)(391.32) + (49)(383.68) – (54)(775) = $10,649.7

Expected Profit for order quantity of 775 units = $10,649.7

Since the expected cost of order quantity is more than the order quantity of 775, if is not advisable to purchase the boots at price discount rate.