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- 1. Inventory System
- 2. INVENTORY SYSTEM InventoryInventory is the stock of any item or resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process An inventory systeminventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be
- 3. INVENTORY COSTS Holding (or carrying) costs Costs for storage, handling, insurance, etc Setup (or production change) costs Costs for arranging specific equipment setups, etc Ordering costs Costs of someone placing an order, etc Shortage costs Costs of canceling an order, etc
- 4. E(1 ) Independent vs. Dependent Demand Independent Demand (Demand for the final end- product or demand not related to other items) Dependent Demand (Derived demand items for component parts, subassemblies, raw materials, etc) Finished product Component parts
- 5. Inventory Systems Single-Period Inventory Model One time purchasing decision (Example: vendor selling t-shirts at a football game) Seeks to balance the costs of inventory overstock and under stock Multi-Period Inventory Models Fixed-Order Quantity Models Event Triggered (Example: running out of stock) Fixed-Time Period Models Time triggered (Example: Monthly sales call by sales representative)
- 6. Fixed-order quantity model also called economic order quantity (EOQ) an inventory control model where the amount requisitioned is fixed and the actual ordering is triggered by inventory dropping to a specified level of inventory
- 7. Fixed-Order Quantity Model Assumptions Demand for the product is constant and uniform throughout the period Lead time (time from ordering to receipt) is constant Price per unit of product is constant
- 8. Fixed-Order Quantity Model Assumptions Inventory holding cost is based on average inventory Ordering or setup costs are constant All demands for the product will be satisfied (No back orders are allowed)
- 9. Basic Fixed-Order Quantity Model and Reorder Point Behavior R = Reorder point Q = Economic order quantity L = Lead time L L Q QQ R Time Number of units on hand 1. You receive an order quantity Q. 2. Your start using them up over time. 3. When you reach down to a level of inventory of R, you place your next Q sized order. 4. The cycle then repeats.
- 10. Cost Minimization Goal Ordering Costs Holding Costs Order Quantity (Q) C O S T Annual Cost of Items (DC) Total Cost QOPT By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs
- 11. Basic Fixed-Order Quantity (EOQ) Model Formula H 2 Q +S Q D +DC=TC Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost + + TC=Total annual cost D =Demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory TC=Total annual cost D =Demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory
- 12. Deriving the EOQ Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt Q = 2DS H = 2(Annual Demand)(Order or Setup Cost) Annual Holding CostOPT Reorder point, R = d L _ (constant)timeLead=L (constant)demanddailyaverage=d _ We also need a reorder point to tell us when to place an order We also need a reorder point to tell us when to place an order
- 13. EOQ Example (1) Problem Data Annual Demand = 1,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = $2.50 Lead time = 7 days Cost per unit = $15 Given the information below, what are the EOQ and reorder point? Given the information below, what are the EOQ and reorder point?
- 14. EOQ Example (1) Solution Q = 2DS H = 2(1,000 )(10) 2.50 = 89.443 units orOPT 90 units d = 1,000 units / year 365 days / year = 2.74 units / day Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _ 20 units In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units. In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.
- 15. EOQ Example (2) Problem Data Annual Demand = 10,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = 10% of cost per unit Lead time = 10 days Cost per unit = $15 Determine the economic order quantity and the reorder point given the following… Determine the economic order quantity and the reorder point given the following…
- 16. EOQ Example (2) Solution Q = 2DS H = 2(10,000 )(10) 1.50 = 365.148 units, orOPT 366 units d = 10,000 units / year 365 days / year = 27.397 units / day R = d L = 27.397 units / day (10 days) = 273.97 or _ 274 units Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units. Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.
- 17. Economic Production Quantity (EPQ) Production done in batches or lots Capacity to produce a part exceeds the part’s usage or demand rate Assumptions of EPQ are similar to EOQ except orders are received incrementally during production
- 18. Assumptions: Only one item involved Annual demand is known The usage rate is constant Usage occurs continually, but production occurs periodically The production rate is constant Lead time does not vary There are no quantity discounts
- 19. Economic Production Quantity Model
- 20. Formulas: No. of runs per year = D/Q Annual setup cost = (D/Q)S TCmin = Carrying Cost + Setup Cost = ( )H+(D/Qo)S Qo = ; where p = production or delivery rate u = usage rate Cycle Time = Run Time = Imax = (p-u) and Iaverage =
- 21. Example 1 A toy manufacturer uses 48,000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. The firm operates 240 days per year.
- 22. Determine the a.Optimal run size. b.Minimum total annual cost for carrying and setup. c.Cycle time for the optimal run size. d.Run time.
- 23. Given: D = 48,000 wheels per year S = $45 H = $1 per wheel per year p = 800 wheels per day u = 48,000 wheels per 240 days, or 200 wheels per day
- 24. a. Qo= Qo= = 2,400 wheels b. TCmin = ( )H+(D/Qo)S Thus, you must first compute Imax: Imax = (p-u) = (800-200) = 1,800 wheels TC = ( )($1) + ( )($45) = $1,800 Solution:
- 25. c. Cycle Time = = = 12 days Thus, a run of wheels will be made every 12 days. d. Run Time = = = 3 days Thus, each run will require 3 days to complete.
- 26. Example 2 The Dine Corporation is both a producer & a user of brass couplings. The firm operates 220 days a year & uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 200 per day. Annual storage cost is $2 per coupling, & machine setup cost is $70 per run. a.Determine the economic run quantity. b.Approximately how many runs per year will there be? c.Compute the maximum inventory level.
- 27. Solution a. Qp = = ≈ 1,013 units
- 28. b. No. of runs = D = 11,000 = 10.86 ≈ 11 per year Q0 1,013 c. Imax = Qp (p - u) = 1,013 (200 - 50) p Imax = 759.75 or 760 units
- 29. Quantity Discounts • Quantity Discounts, also called Price-Break Models, are price reductions for large orders offered to costumers to induce them to buy in large quantities. • The buyer must weigh the potential benefits of reduced purchase price and fewer orders that will result from buying in large quantities against the increase in carry costs caused by higher average inventories.
- 30. Quantity Discounts • The buyer’s goal with quantity discounts is to select the order quantity that will minimize total cost. • Total cost is the sum of carrying cost, ordering cost and purchasing cost.
- 31. Price-Break Model Formula CostHoldingAnnual Cost)SetuporderDemand)(Or2(Annual = iC 2DS =QOPT Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula: i = percentage of unit cost attributed to carrying inventory C = cost per unit Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value
- 32. Total Cost = Carrying Cost + Ordering Cost + Purchasing Cost TC = (Q/2)H + (D/Q)S + PD Where: Q = Quantity P = Unit Price D = Demand S = Ordering Cost
- 33. Example1 A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units? A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units? Order Quantity(units) Price/unit($) 0 to 2,499 $1.20 2,500 to 3,999 1.00 4,000 or more .98
- 34. Solution units1,826= 0.02(1.20) 4)2(10,000)( = iC 2DS =QOPT Annual Demand (D)= 10,000 units Cost to place an order (S)= $4 First, plug data into formula for each price-break value of “C” units2,000= 0.02(1.00) 4)2(10,000)( = iC 2DS =QOPT units2,020= 0.02(0.98) 4)2(10,000)( = iC 2DS =QOPT Carrying cost % of total cost (i)= 2% Cost per unit (C) = $1.20, $1.00, $0.98 Interval from 0 to 2499, the Qopt value is feasible Interval from 2500-3999, the Qopt value is not feasible Interval from 4000 & more, the Qopt value is not feasible Next, determine if the computed Qopt values are feasible or not
- 35. Solution Since the feasible solution occurred in the first price- break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why? Since the feasible solution occurred in the first price- break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why? 0 1826 2500 4000 Order Quantity Total annual costs So the candidates for the price- breaks are 1826, 2500, and 4000 units So the candidates for the price- breaks are 1826, 2500, and 4000 units Because the total annual cost function is a “u” shaped function Because the total annual cost function is a “u” shaped function
- 36. Solution iC 2 Q +S Q D +DC=TC Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units
- 37. Example2 Consider the following case, where D = 10,000 units (annual demand) i = 20% of cost (annual carrying cost, storage, interest, obsolescence, etc.) S = $20 to place an order C = Cost per unit (according to order size; 0-499 units, $5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9 per unit) Consider the following case, where D = 10,000 units (annual demand) i = 20% of cost (annual carrying cost, storage, interest, obsolescence, etc.) S = $20 to place an order C = Cost per unit (according to order size; 0-499 units, $5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9 per unit) What quantity should be ordered?What quantity should be ordered?
- 38. Solution units632.46= 0.2(5) 20)2(10,000)( = iC 2DS =QOPT Annual Demand (D)= 10,000 units Cost to place an order (S)= $20 First, plug data into formula for each price-break value of “C” units666.67= 0.2(4.50) 20)2(10,000)( = iC 2DS =QOPT units716.11= 0.2(3.90) 20)2(10,000)( = iC 2DS =QOPT Carrying cost % of total cost (i)= 20% Cost per unit (C) = $5, $4.5, $3.9 Interval from 0 to 499, the Qopt value is feasible Interval from 500-999, the Qopt value is not feasible Interval from 1000 & more, the Qopt value is not feasible Next, determine if the computed Qopt values are feasible or not
- 39. Solution iC 2 Q +S Q D +DC=TC Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5) = $50,632.46 TC(500-999)= $45,600 TC(1000&more)= $39,558.57 TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5) = $50,632.46 TC(500-999)= $45,600 TC(1000&more)= $39,558.57 Finally, we select the least costly Qopt, which in this problem occurs in the 1000 & more interval. In summary, our optimal order quantity is 1000 units Finally, we select the least costly Qopt, which in this problem occurs in the 1000 & more interval. In summary, our optimal order quantity is 1000 units
- 40. A-B-C Classification Classifies Inventory items according to some measure of importance, Usually annual dollar value, and then allocates control efforts accordingly
- 41. Three Classes of Items A (Very Important Items) Account for 10%-20% of the number of items in inventory 60%-70% of the dollar inventory of an inventory B (Moderately Important) Account for 20%-30% of the number of items in inventory 20%-50% of the dollar inventory of an inventory C (Least Important) Account for 50%-60% of the number of items in inventory 10%-15% of the dollar inventory of an inventory
- 42. A typical A-B-C breakdown in relative annual dollar value of items and number of items by category.
- 43. Example The annual dollar value of 12 items has been calculated according to annual demand and unit cost. The annual dollar values were then arrayed from highest to lowest to simplify classification of items.
- 44. Item Number Annual Demand Unit Cost Annual Dollar Value Classification 8 $1,000 $4,000 $4,000,000 A 5 $3,900 $700 $2,730,000 A 3 $1,900 $500 $950,000 B 6 $1,000 $915 $915,000 B 1 $2,500 $330 $825,000 B 4 $1,500 $100 $150,000 C 12 $400 $300 $120,000 C 11 $500 $200 $100,000 C 9 $8,000 $10 $80,000 C 2 $1,000 $70 $70,000 C 7 $200 $210 $42,000 C 10 $9,000 $2 $18,000 C

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