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EXTRA PROBLEMS WITH SOLUTIONSExample 10: The ABC Co. is planning to stock a new product. The Co. Has developed thefollowing information:Annual usage = 5400 unitsCost of the product = 365 MU/unitOrdering cost = 55 MU/orderCarrying cost = 28% /year of inventory value held. a Determine the optimal number of units per order b. Find the optimal number of orders/year c. Find the annual total inventory costSolution: a. X0 = √2CB/zp = √2*5400*55/365*0.28 = ~76 units/order b. N0 = C/X0 =5400 / 76 = ~71 orders/year c. Ke= √2CBzp = √2*5400*55*365*0.28 = 7791.46 MU/year.Example 11: Holding costs are 35 MU/unit/year. The ordering cost is 120 MU/order and salesare relatively constant at 400 month. a. What is the optimal order quantity? b. What is the annual total inventory cost?Solution: a. X0 = √2CB/E= √2*(400*12)*120/35 = 181.42 units/order b. Ke= √2CBE= √2*(400*12)*120*35 = 6349.80 MU/year
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Example 12: Azim furniture company handles several lines of furniture, one of which is thepopular Layback Model TT chair. The manager, Mr. Farmerson, has decided todetermine by use of the EOQ model the best quantity to obtain in each order. Mr.Farmerson has determined from past invoices that he has sold about 200 chair duringeach of the past five years at a fairly uniform rate and he expects to continue at that rate.He has estimated that preparation of an order and other variable costs associated witheach order are about 10 MU, and it costs him about 1.5 % per month (or 18% per year) tohold items in stock. His cost for the chair is 87 MU. a. How many layback chairs should be ordered each time? b. How many orders would there be? c. Determine the approximate lenght of a supply order in days? d. Calculate the minimum total inventory cost e. Show and verify that the annual holding cost is equal to the annual ordering cost (due to rounding, show these costs are approximately equal)Solution: a. X0 = √2CB/Zp = √2*200*10 /0.18*87 = 15.98 =~16 chairs/order b. N0 = C/X0 =200 /16= 12.5 orders/year c. t0 = X0/C * 365 = 16 /200 *365 = 29.2 days d. Ke= √2CBZp = √2*200*10*0.18*87 = 250.28 MU/year e. N0 *B = 12.5 * 10 = 125 MU X0/2*Zp = 15.98/2 * 0.18*87 = 125.1 MUExample 13: A. Leyla Tas has determined that the annual demand for #6 screws is 100000screws. Leyla, estimates that it costs 10 MU every time when an order is placed. Thiscost includes wages, the cost of the forms used in placing the order and so on.Furthermore, she estimates that the cost of carrying are screw in inventory for a year isone-half of 0.01 MU. Assume that the demand is constant throughout the year.
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a. How many #6 screws should Leyla order at a time to minimize total inventory cost? b. How many orders per year would be placed? What would the annual ordering cost be? c. What would the average inventory be? What would the annual holding cost be?Solution: a. E= 0.01/2 = 0.005 X0 = √2CB/E X0 = √2*100000*10 / 0.005 = 20000 screws/order b. N0 = C/X0 = 100000/20000= 5 orders/year Total ordering cost = N*B = 5*10 = 50 MU/year c. Average inventory = x/2 = 20000/2 = 10000 units Total holding cost = x/2*E= 10000 * 0.005 = 50 MU/year B. It takes approximately 8 working days for an order of #6 screws to arrive oncethe order has been placed. The demand is fairly constant, and on the average the storesells 500 screws each day. What is the ROP?Solution: ROP = Ro= use rate * lead time = c*tlt = 500 screws/day * 8 days = 4 000 screws. C. The manager believes that Leyla places too many orders for screws /year. Hebelieves that an order should be placed only twice/year. If Leyla follows her manager`spolicy, how much more would this cost every year over the ordering policy that shedeveloped, if only two orders were placed each year, what effect would this have on theROP?Solution: Twice a year = Ke= NB + X/2*E = 2*10 + 50000/2*0.005 = 20 + 125 = 145 5 times a year = Ke= √2CBE = √2*100000*10*0.005 = 100 MU Extra cost for manager’s offer = 45 MU No effect on ROP.
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Example 14: Ahmet Uslu experiences an annual demand of 220 000 MU for pro quality tennisballs at the İzmir Tennis Supply Company. It cost Ahmet 30 MU to place an order andhis carrying cost is 18%. How many orders per year should Ahmet place for the balls?Solution: N0 = CpZ = ( 220 000 )(0.18) = 25.69orders/year 2B 2(30 )Example 15: Ayşe Çalışkan,owner of Computer Village, needs to determine an optimalordering policy for Porto-Pro computers,annual demand for the computers is 28 000 MUand carrying cost is 23 percent. Ayşe has estimated order costs to be 48 MU per order.What are the optimal MU per order?Solution: 2CpB 2( 28 000 )( 48) X0 = = =3418.62 mu/order Z 0.23Example 16: EMU uses 96 000 MU annually of a particular reagent in the chemistrydepartment of the EMU estimates the ordering cost at 45 MU and thinks that theuniversity can hold this type of inventory at an annual storage cost of 22% of thepurchase price. How many months’ supply should the purchasing department order at ontime to minimize total annual cost of inventory?Solution: 2B 2( 45) t 0 =12 =12 =0.784 month ssupply CpZ (96 000 )(0.22)
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Example 17: The ABC co.is planning to stock new product. The ABC co.has developed thefollowing information:Annual usage = 5400 unitsCost of the product = 365 MU/unitOrdering cost = 55 MU/orderCarrying cost = 28%/year of inventory value held a) Determine the optimal number of units per order? b) Find the optimal number of orders/year? c) Find the annual total inventory cost?Solution: 2CB 2(5400)(55) a) X 0 = = ≅ 76 units/order Zp 365(0.28) C 5400 b) N o = = ≅ 71 orders/year X0 76 c) K e = 2CBZp = 2(5 400)(55)(365)( 0.28) = 60 706 800 = 7 791.46mu/yearExample 18: A local artisan uses supplies purchased from an overseas supplier. The ownerbelieves the assumptions of the EOQ model are met reasonably well. Minimization ofinventory cost is her objective. Relevant data, from the files of the credit firm, are annualdemand (C) = 240 units, ordering cost (B) = 42 MU/order, and holding cost = 4MU/unit/year. a) How many should she order at one time? b) How many times per year will she replenish its inventory of this material? c) What will be the total inventory costs associated with this material?
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d) If she discovered that the carrying cost has been overstated, and was in reality only 1 MU/unit-year, what is the corrected value of EOQ?Solution: C = 240 units/year B =42 MU/order E = 4MU/unit/year 2CB 2( 240 )( 42 ) a) Xo = E = 4 = 70.99 = 71 Units/order C 240 b) N0 = = =3.38 Times/year X0 71 c) Ke 2CBE = 2(240 ) 42 ) 4 ) = ( ( 283.97 MU/year 2CB 2( 240 )( 42 ) d) X0 = E = 1 =141.99 ≅142 Units K e = 2CBE = 2(240 ) 42 ) 1) = ( ( 142 MU/yr.Example 19: Ground Coffee shop uses 3 kgs of a specialty tea weekly; each kg. costs 16 MU.Carrying costs are 2 MU/kg/week because space is very scarce. It costs the firm 7 MU toprepare an order. Assume the basic EOQ model with no shortages applies. Assume 52weeks/year. a) How many kgs should Ground to order at a time? b) What is total annual inventory cost? c) How many orders should ground place annually? d) How many days will there be between orders(assume 310 operating days)?Solution: 2CB 2(3)( 7 ) a) X0 = E = 2 = 4.58 kgs/order b) K e = 2CBE = 2( ) 7 ) 2 ) = .165 MU/week 3 ( ( 9 Ke = 165 × = 9. 52 476.59 MU/year C 3 ×52 c) N0 = = =34.06 Times/year X0 4.58 1 1 d) t = N ×310 = 34.06 ×310 =9.1 ≅9 days
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Example 20: Holding costs are 35 MU/unit/year. The ordering cost is 120 MU/order, andsales are reletively constant at 400/month. a) What is the optimal order quantity? b) What is the annual total inventory cost?Solution: 2CB 2( 400 × )(120 ) 12 X0 = = =181.42 Units/order E 35 K e = 2CBE = 2(400 × ) 120 ) 35) = 349.80 MU/year 12 ( ( 6Example 21: ABC, a company that sells pump housings to other manufacturers, would like toreduce its inventory cost by determining the optimal number of pump housings to obtainper order. The annual demand is 1 000 units, the ordering cost is 10 MU/order, and theaverage carrying cost per unit per year is 0.50 MU. Calculate EOQ .(i.e.the optimalnumber of units per order).Solution: 2CB 2(1 000 )(10 ) X0 = = = 40 000 = 200units E 0.50 C X 1 000 200 Ke = ⋅B + E= ⋅10 + ⋅ 0.5 = 100 MU X 2 200 2 C 1000 N0 = = = 5times / yr Q 200 Q 200 T = = = 0.20 yr = 73days C 1000Example 22: Lemar Supermarket sells about 200 000 kilos of milk annually. The milk ispurchased for 4 MU/kg. Holding costs are 1.40 MU/kg/yr. Each order costs 35 MU. If theshelf life of the milk is only 5 days, how many kilos should be ordered at a time.Solution:
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C = 200 000kg/yr P = 4 MU/kg E = 1.40 MU/kg/yr B = 35 MU/order Shelf Life = 5 days 2CB 2( 200 000 ×35) EOQ = = = 3 162.28 kilos E 1.40 X 3 162.28 t 0 = 0 ×365 = ×365 = 5.77 days C 200 000 With an EOQ of 3 162.28 kilos an order must be placed every 5.77 days. But theshelf life is 5 days so some milk will start to go sour. Therefore 200 000 ⋅ (5) =2 739.73 kilos should be ordered every 5 days. 365Example 23: A building materials stockist obtains its cement from a single supplier. Demandfor cement is reasonably constant throughout the year. Last year the company sold 2 000tonnes of cement. It estimates the costs of placing an order at around 25 MU each timean order is placed and charges inventory holding at 20% of purchase cost. The companypurchases cement at 60 MU per tonne. a) How much cement should the company order at a time? b) Instead of ordering EOQ, why not a order convenient 100 tonnes?Solution: 2CB 2( 25 )( 2 000 ) a) EOQ for cement = Zp = 0.2( 60 ) = 91.287 tonnes b) Total cost of ordering plan for X 0 = .287 tonnes; 91 K e = 2CBZp = 2(25) 2000 ) 0.2 ) 60 ) = ( ( ( 1095.454 MUor we can calculate Ke by C X 2000 Ke = ⋅B + ⋅ Zp = (25) +91.287 (0.2 )(60 ) =1095.454 MU X 2 91.287 2Total cost of ordering plan for X 0 =100 tonnes;
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2 000 100 Ke = ⋅ 25 + (0.2 )(60 ) =1100 MU 100 2Result: The extra cost of ordering 100 tonnes at a time is 1 100 MU-1095.454 MU= 4.55MU. The production/operations manager therefore should feel confident in using themore convenient order quantity.Example 24: # 2 pencils at the EMU bookstore are sold at a fairly steady rate of 60 per week.The pencil cost the bookstore 0.02 MU each and sell for 0.15 MU each. It costs thebookstore 12 MU to initiate an order and holding costs are based on an annual interetrate of 25%. Determine the optimal number of pencils for the bookstore to purchase andthe time between placement of orders. What are the holding and setup costs for this item?Solution: C = 60 × 52 = 3 210 Units Zp = E E = 0.25( 0.02 ) = 0.005 MU/unit/yr 2CB 2( 3120 )(12 ) X0 = = = 3 780 Units E 0.005 X 3 870 T = = = 1.24 yr. C 3 120 X 3 870 Average Holding Cost = ⋅ E = ⋅ 0.005 = 9.675 MU 2 2 C 3 120 Average Ordering Cost = ⋅ B = ×12 = 9.675 MU X 3 870Example 25: A wholesaler has a steady demand for 50 items of a given product each month.The purchase cost of each item is 6 MU and the holding cost for this item is estimated tobe 20% of the stock value per annum. Every order placed by the wholesaler cost 10 MUin administration charges regardless of the number ordered?Solution:C = annual demand = 50x12 = 600 itemsp = price unit = 6 MU
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B = order cost = 10 MUE = holding cost of one item per year = Zp= 0.20x6 2CB 2 × ×600 10 X0 = = =100items Zp 0.2 ×6 C 600An order size of 100 is used at an order frequency of N0 = = = 6times / year X0 100Example 26:C = Demand/week = 400 unit.P =Purchase price = 3 MU/unitE = Holding cost = 2 MU/100 items/weekB = Order cost =12 MU/order X 0 =?Solution: 2CB 2( 400 )(12 ) X0 = = = 692.8 ≅ 693items E 2 / 100 An order size of 693 items is recommendedExample 27: TT Beverage Co. is a distributor of beer, wine and soft drinks product. From amain warehouse located in Magusa, TRNC, TT supplies nearly 1000 retail stores withbeverage products. The beer inventory, which constitutes about 40% of the company’stotal inventory, averages approximately 50 000 cases. With an average cost per case ofapproximately 8MU, TT estimates the value of its beer inventory to be 400 000 MU. The warehouse manager has decided to do a detailed study of inventory costsassociated with Sergio Beer, the # 1 selling TT beer. The purpose of the study is toestablish the “how-much to order,” and “when to order” decisions for Segio Beer thatwill result in the lowest possible cost. The manager found that the demand is constant and 2 000 cases/week.The cost ofholding for the TT beer inventory is 25% of the value of the inventory. (Note that definding the holding cost as a % of the value of the product isconvenient, because it is easily transferable to other products)
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TT is paying 32 MU/order regardless of the quantity requested in the order.Suppose TT is open 5 days each week, and the manufacturer of Sergio Beer guarantees2-day delivery on any order placed. a) Find Economic Order Quantity. b) Find reorder point. c) How frequently theorder will be placed? d) Find the cycle time? e) Calculate minimum total inventory cost.Solution: a) Find EOQ 2CB 2( 2000 ×52 )(32 ) X0 = = =1824cases Zp 0.25 ×8 The use of an order quantity of 1824 cases will yield the minimum-cost inventorypolicy for TT beer. b) Find the reorder point R0 = c ⋅ t Lt 2000 c= = 400 5 R0 = 400units × 2days = 800cases c) How frequently the order will be placed? C 104000 N = = =57order / year X 1824 d) Find the cycle time. 1 1 T = = ×260 = 4.6days N 57 The cycle time is 4.6 working days. e) Calculate minimum total inventory cost. Ke = 2CBEZp = 2(104000 )( 32 )( 0.25)(8) K e = 13312000 ≅ 3648MU 1824 Total order cost ( 57 ×32) 1824 1824 Total holding cost ( ⋅ 2 ≅1824) 2
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