8/12/2019 Quantity Discounts

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Question (quantity discounts):A company faces an annual demand for 10,000 footballs. The holdingcost are 10 per football & year, & the cost of placing one order are $80.

(a) Determine the optimal order quantity. How many orders should be placed at optimum & what is thecycle time (given 360 working days per year)? What are the total inventory cost? What is the reorderpoint, given a lead time of 160 days?

(b) The present company policy is to place orders on a quarterly basis. What are the costs of this policyand what are the percentage savings if the optimal policy as determined above would be adopted?

(c) Suppose now that we are now allowed to include planned shortages. It has been estimated that the costof being one unit short for one year is 50. Compute the optimal order quantity, the optimal plannedshortage, & the total cost with this policy. Compare the result to the costs computed under (a) &explain the difference in one sentence.

(d) Suppose now that the companys supplier offers discounts, provided that the company orders largerquantities. Without the discount (the situation under (a)), the price of one unit of the product is $2,and the holding costs are 5% of the price. The supplier offers a % discount in case the company

orders at least 6,000 units. As an alternative, the supplier also offers a 1% discount, if the companyorders at least 15,000 units. Consider all alternatives, compute the total costs in each case & makeyour recommendation.

Solution:D= 10,000, Ch=0.1, Co= 80.

(a) ,000,410.

)80)(000,10)(2(* ==Q N* = 10,000/4,000 = 2.5, tc= 360/2.5 = 144 [days].

TC(Q*) = )10)(.80)(000,10)(2( = $400.

R= (10,000/360) (160) )000,4(144160 = 4,444.44 4,000 = 444.44.

(b) Present policy: N = 4, so that Q = 2,500 & TC(2,500) = 320 + 125 = 445; savings are 45/445 =10.11%.

(c) Q* = 4,381.78 & S* = 730.30, so that TC* = 182.57 + 152.14 + 30.43 = 365.14. It is cheaper than thesolution under (a) as we have the option to be short a few units; however, shortage costs (as compared toholding costs) are high, so that shortages are comparatively low.

(d) Without discount, we have the same situation as under (a), this time withp= 2 & Ch= 5% ofp= .1.Then Q* = 4,000 & TC* = 20,400.

With the % discount, we havep= 1.99 &pCh= .0995, so that we obtain Q*= 4,010.038 Q= 6,000.Then TC(6,000) = 133.33 + 298.50 + 19,900 = 20,331.83.

With the 1% discount, we have p= 1.98 &pCh= .099, so that we obtain Q*= 4,020.15 Q= 15,000.Then TC(15,000) = 53.33 + 742.50 + 19,800 = 20,595.83.

Comparing the three options, we should order 6,000 footballs & get a % discount for a total cost of$20,331.83.