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Unformatted text preview: Ch3: Pr 15 DQ: 15 3.1 D/yr = UC = RC = HC/yr HC/yr EOQ = pp. 9798 1,000 50 100 25% 12.5 126.49 If I define r=Q*/Q, then I can try to solve: 0.5(r + 1/r) = 1.1 multiply by 2 r + 1/r = 2.2 multiply by r r2 + 1 = 2.2r r2 2.2r + 1 = 0 so a=1, b=2.2, c=1 r = [ 2.2 +/ sqrt(4.84 4*1*1) ]/2 r = [2.2 +/ 0.917 ]/2 r =1.558, r=0.642 page 79 0.64Q, or 1.56Q a x + b x+ c = 0
2 x= −b ± b 2 − 4ac 2a
126.49 790.57 790.57 1581.14 1739.25 80.95 505.96 1235.26 1741.23 Q Holding Ordering Total Cost 197.33 1233.29 506.78 1740.06 200 1250.00 500.00 1750.00 200 is about exactly at the point that costs 10% more than EOQ 3.2 Demand RC HC UC HC Smallest Largest 3.3 D/wk D/yr UC = RC = LTwks HC/yr = HC/yr = 3.4 D/wk = EOQ = LTwks = ROL = ROL2 = 100 50 20% 200 40 14.1 21.3 16 832 5,000 1,000 4 16% 800 40 100 4 160 60 EOQ = ROL = ROL2 = 45.6 64 18.4 130 70 25% 50 Raise it for safety by ROL2 = If LTwks = ROL = 2 80 10 units 70 If LTwks = ROL = ROL2 = 3.5 part a. D/wk UC = RC = LTwks = HC/yr = EOQ = HC = RC = sum = This is a savings of 8,000 4 900 8 25% 27,364 13,682 13,682 27,364 8,000 4,000 46,800 50,800 23,436 per year 6 240 40 part b. To maximize profits, we need to minimize costs, since we assume that sales are not affected by the inventory decisions we are making, since demand is stable. SP = Revenues = Profits = 8 3,328,000 1,636,636 DISCUSSION QUESTIONS 3.1 The main concern of an organization is customer service, and this often needs high stocks. Why, then, are we concentrating on minimizing the cost of stock, when we should be more concerned with raising stock levels? We want to keep the least amount necessary to meet a given service level target. We would need an infinite amount of inventory to get 100% service. There is no point in trying to carry as much as we possibly can. We need to try to carry a smart amount, to get the best service we can out of a given amount of inventory. Carrying a lot would cost so much, it would more than wipe out the sales gains. 3.2 What costs are incurred by holding stock? How would you set about finding these? Why are shortage costs so difficult to find? There are holding costs from capital tied up, facility costs, labor costs, spoilage, theft, damage, insurance, taxes, and many other costs. These all show up on an expense line somewhere that we could look up Shortage costs are hard to measure, because we will never know what might have happened, or what was going to happen saleswise, but didn't, on account of one stockout. 3.3 What are the assumptions of the EOQ? How valid are these? What factors in real inventory control are not included in the EOQ model? p. 67 1. Demand is known, is continuous, and constant over time 2. All costs are known exactly, and do not vary. 3. no shortages are allowed 4. lead time is zero, so a delivery is made as soon as an order is placed. 5. Consider a single item at a time 6. No quantity discounts 7. A single delivery is made for each item 3.3 What are the assumptions of the EOQ? How valid are these? What factors in real inventory control are not included in the EOQ model? p. 67 1. Demand is known, is continuous, and constant over time 2. All costs are known exactly, and do not vary. 3. no shortages are allowed 4. lead time is zero, so a delivery is made as soon as an order is placed. 5. Consider a single item at a time 6. No quantity discounts 7. A single delivery is made for each item 8. All of an order arrives at once. These are not great assumptions, but as we will see, if we relax them, the EOQ still will turn out to have given us a pretty good place to start. 3.4 The variable cost rises slowly around the EOQ, and the analysis is based on a series of assumptions and approximations. Why, then, do we bother with the calculations rather than allowing inventory managers to design policies based on their experience? Would they get good results without bothering with the formal analysis? When we relax the assumptions, the model will be much more realistic, and perform much better than we could do just making decisions "by feel." 3.5 What are the benefits of short lead times? How can these be achieved in practice? In practice, shorter lead times means less time to potentially run out, so it means less need for safety stock. We can achieve shorter lead times by good relationships with our suppliers, and having them located close to us. 4.840 0.840 0.917 1.558 0.642 stocks. Why, cerned with ed an infinite sibly can. We entory. Carrying Why are insurance, taxes, r what was going entory control entory control to have given us of assumptions inventory thout bothering r than we could afety stock. cated close to Ch4 pp. 1424 Problems 18, DQ 1, 3, 4, 5 4.1 We've got two price break systems here. Should we try to combine them into one, or solve them separately? D/mo = RC = HC / yr = Min 1,500 2,000 2,500 1,500 Max 400 1240 30% 1,499 1,999 2,499 1,000,000 1,499 1,000,000 Price 12.6 12.2 11.8 11.2 12 11.4 EOQ Valid? Holding Ordering 1,774.6 1,499.0 2,833 3,971 1,803.5 1,803.5 3,300 3,300 1,833.8 2,000.0 3,540 2,976 1,882.2 2,500.0 4,200 2,381 1,818.4 1,865.7 1,499.0 1,865.7 2,698 3,190 3,971 3,190 4.2 D/yr = RC = Holding Min 0 100 400 1000 Max 2500 50 40% 99 399 999 1,000,000 Price 20 19.4 18.8 18 EOQ 176.8 179.5 182.3 186.3 Valid? 99.0 179.5 400.0 1000.0 Holding 396 696.4 1504.0 3600.0 Ordering 1262.63 696.42 312.50 125.00 If the reordering cost becomes zero, buying 400 is still optimal, to get the price discount. 4.3 D/mo = prod/mo = UC = RC = HC/mo = 500 1,000 10 2,000 1 POQ = A = EOQ = 2000.0 1000 1414.2 CoGS = Setup = Holding = Total Cost = Vco = 4.4 D/yr = RC = UC = HC/yr = Prod/yr = Ltmos = If the production rate can be varied, how would costs change? If we can produce faster, then the production quantity would get smaller, closer to the EOQ. Holding costs go up considerably. Looking at the formula n p. 116, Vco = sqrt(2*RC*HC*D)*sqrt(PD/P), if P=D, then ther no holding or ordering costs. As P increases, order sizes get smaller, so ordering costs go up, and holding also go up, because we produce the batch faster and carry more inventory. EOQ = 645.5 2,500 500 30 20% 10,000 2 POQ = POQ = ROL = 745.36 A = 745.36 416.67 559.0 Order/yr = Setup costs Holding cost 3.35 1,677.1 1,677.1 Total Cost = 4.5 HC/yr = 25% backorder/yr 150% UC = 400 RC = 100 D/yr = 300 Backorders on all unfulfilled demand Q0 = 26.5 S0 = 3.78 % backordered p. 123 14.3% 100 600 3,354.1 CoGS Reorder Holding Backorder 120,000 1,134 972 162 122,268 Without Backordering, we would get; EOQ = 24.5 CoGS Holding Ordering Savings of 120,000 1,225 1,225 122,449 182 0.15% 20 200 200 5,000 5,400 4.6 D/yr = UC = RC = HC/yr = lost demands 100 50 40 40% 20 EOQ = holding Ordering CoGS Total Cost Minimum Revenue per unit = 54.00 4.7 Item 1 2 3 4 Demand 500 300 200 400 UC 200 100 300 200 HC 35 20 45 40 RC 80 100 120 80 space used 4 3 6 5 RC = 1 2 3 4 EOQ 50 47.81 54.77 32.66 40.00 space space cost = 20 95.6 26.4 82.2 27.4 98.0 17.1 100.0 21.4 375.8 New interest rate = We don’t know the ordering cost to compute are, so we don't know how they would affect 52.8 EOQ and raise costs. 41.1 Just for fun, I assumed an ordering cost of $ 51.2 53.5 we only had 200 cubic feet available, and pla 198.5 0.5 RC = 1 EOQ 50 47.81 value 4,781 EOQ 2 28.28 Value 2 2,828 2 3 4 54.77 32.66 40.00 2,739 4,899 4,000 16,419 34.64 17.89 25.30 1,732 2,683 2,530 9,774 4.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 6 10 21 18 7 4 3 2 4 15 27 38 51 40 22 12 8 6 7 2 3 Ending Inv 268.38 262.38 252.38 231.38 213.38 206.38 202.38 199.38 197.38 193.38 178.38 151.38 113.38 62.38 22.38 269.75 Avg = 257.75 stdev = 249.75 243.75 UC = 236.75 RC = 234.75 HC/yr = 231.75 HC/yr = EOQ = 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 13.95 13.97 80 800 20% 16 269.38 HC = RC = Clearly, demand is not stable. There is a huge amount of seasonality, which is either growing, or it's an annual pattern. In either case, our EOQ is probably not a good choice here. We would need to use some kind of trial and error or lot sizing, which is a subject for SCM 461. DISCUSSION QUESTIONS 4.1 We have described some simple extensions to the EOQ, and could have looked at many more complicated models. When do you think that it is worth using a more sophisticated model, rather than using the guidelines given by a simple model? If the cost savings of the more complicated model would outweigh the time and data collection costs needed to be able to use the model, then it would make sense to use the more complicated model. 4.3 With a finite replenishment rate the order quantity is the EOQ mulitplied by sqrt(P/(PD)). As the replenishment rate gets closer to the demand rate, this value gets larger, and the order size and average stock level both increase. But if the repenishment rate is equal to the demand we should not need any stock. What is happening? When they are equal, the prodution quantity is infinite, because we never stop producing. Since demand is stable, we sell products exactly as we make them. using the guidelines given by a simple model? If the cost savings of the more complicated model would outweigh the time and data collection costs needed to be able to use the model, then it would make sense to use the more complicated model. 4.3 With a finite replenishment rate the order quantity is the EOQ mulitplied by sqrt(P/(PD)). As the replenishment rate gets closer to the demand rate, this value gets larger, and the order size and average stock level both increase. But if the repenishment rate is equal to the demand we should not need any stock. What is happening? When they are equal, the prodution quantity is infinite, because we never stop producing. Since demand is stable, we sell products exactly as we make them. 4.4 Nobody likes waiting for a product they have decided to buy, so why would an organization deliberately work with shortages? Many companies work on a maketoorder basis, because customers are accustomed to it. This is more likely to be true when there are many, many possible order configurations, and it would be impossible to keep all possible products, or even a bestselling subset of them. 4.5 It is often difficult to find reliable costs for stocks. With shortages this seems almost impossible how do you find a cost for loss of goodwill or reduced future sales? Does this mean that any analysis of shortages is inevitably based on flimsy evidence? It is very hard to find stockout costs. But if we can make a ballpark estimate, it can at least point us into the right general direction. And we might be able to make a ballpark estimate by thinking of it in terms of how much we would be willing to spend to AVOID a stockout. solve them separately? CoGS Total 60,480 67,284 58,560 65,161 56,640 63,156 53,760 60,341 57,600 54,720 Difference 64,269 61,101 760 CoGS Total 50,000 51,659 48,500 49,893 47,000 48,817 45,000 48,725 unt. Annual Monthly 60,000 5,000 6,000 500 6,000 500 72,000 6,000 the EOQ. Holding costs would t(PD/P), if P=D, then there are costs go up, and holding costs ordering cost to compute EOQs, and we don't know what the space or capital constraints ow how they would affect anything. But those constraints could force us away from the ts. med an ordering cost of $50, which lead to average space usage of 276. Then I assumed ubic feet available, and played with the cost to get the average usage down below 200 10 11 12 13 14 15 16 17 18 19 20 21 22 2,155 2,155 4,310 r growing, or it's would need to any more del, rather than costs needed to D)). As the size and average ld not need any e demand is costs needed to D)). As the size and average ld not need any e demand is ization is is more likely le to keep all t impossible at any analysis int us into the right of how much we ...
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 Spring '09
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