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Inventory Model (group 4)


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amanda arevalo
rachel evora
randy mendoza
marille faltado
louie lorenzo

Published in: Data & Analytics
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Inventory Model (group 4)

  1. 1. Presented by : Amanda R. Arevalo Rachel Evora Mariela Faltado Randy Mendoza
  2. 2. A stock or store of goods The raw materials, component parts, work-in-process, or finished products that are held at a location in the supply chain. The goods and materials that a business holds for the ultimate purpose of resale (or repair). Represents one of the most important assets that most businesses possess, because the turnover of inventory represents one of the primary sources of revenue generation and subsequent earnings for the company's shareholders/owners.
  3. 3. Inventory can be the most expensive and the most important asset for an organization
  4. 4. Raw materials inventory – purchased but not processed WIP (work in progress ) inventory - partially completed goods or goods that undergone some change but not completed
  5. 5. MRO (Maintenance, Repairs, and operating Supplies) inventory – Replacement parts, tools and supplies neccessary to keep machinery and processes productive Finished-goods inventory (manufacturing firm) or merchandise inventory(retail store) – completed products for delivery or shipment Pipeline inventory - Goods-in-transit to warehouses or customers
  6. 6. Suppliers Customers Inventory Storage Finished Goods Raw Materials Work in Process Fabrication and Assembly Inventory Processing
  7. 7. The supervision of supply, storage and accessibility of items in order to ensure an adequate supply without excessive oversupply. Maintaining the inventory at a desired level. The desired level keeps on fluctuating as per the demand and supply of goods.
  8. 8. To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds – Level of customer service – Costs of ordering and carrying inventory
  9. 9. to meet anticipated customer demand (to meet the anticipation stocks, average demand) to smooth production requirements (create seasonal inventories to meet seasonal demand) to decouple operations (eliminate sources of disruptions) to protect against stock-outs (hold safety stocks to prevent the risk of shortages) to take advantage of order cycles (buys more quantities than immediate requirements - cycle stock, periodic orders, or order cycles)
  10. 10. to hedge against price increases (purchase large order to hedge future price increase or implement volume discount) to permit operations (Little's Law: the average amount of inventory in a system is equal to the product of the average demand rate and the average time a unit is in the system) to take advantage of quantity discounts (supplies may give discount on large orders)
  11. 11. There are two basic decisions that must be made for every item that is maintained in inventory. These decisions have to do with the timing of orders for the item and the size of orders for the item.
  12. 12. • How much to order • When to order
  13. 13. Basic Inventory Decisions How much ? Lot sizing decision Determination of the Lot sizing decision Determination of the quantity to be quantity to be ordered ordered When ? Lot timing decision Determination of the timing for the orders Lot timing decision Determination of the timing for the orders
  14. 14. Some inventory items can be classified as independent demand items, and some can be classified as dependent demand items. The manner in which we make inventory decisions will differ depending upon whether the item has independent demand or dependent demand.
  15. 15. A Independent Demand: Finished Goods Independent Demand: Finished Goods Dependent Demand: Raw Materials, Component parts, Sub-assemblies, etc. Dependent Demand: Raw Materials, Component parts, Sub-assemblies, etc. CCUUSSTTOOMMEERR DDEEMMAANNDD
  16. 16. A technique for determining the best answers to the how much and when questions. Based on the premise that there is an optimal order size that will yield the lowest possible value of the total inventory cost.
  17. 17. 1. Demand for the item is known and constant. 2. Lead time is known and constant. (Lead time is the amount of time that elapses between when the order is placed and when it is received.) 3. When an order is received, all the items ordered arrive at once (instantaneous replenishment). 4. The cost of all units ordered is the same, regardless of the quantity ordered (no quantity discounts). 5. Ordering costs are known and constant (the cost to place an order is always the same, regardless of the quantity ordered). 6. Since there is certainty with respect to the demand rate and the lead time, orders can be timed to arrive just when we would have run out. Consequently the model assumes that there will be no shortages.
  18. 18. There are only two costs that will vary with changes in the order quantity: 1. the total annual ordering cost 2. the total annual holding cost. Shortage cost can be ignored because of assumption No. 6. Since the cost per unit of all items ordered is the same, the total annual item cost will be a constant and will not be affected by the order quantity.
  19. 19. Inventory levels will fluctuate over time as in the following graph:
  20. 20. D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity Total Annual Ordering Cost = Annual Demand Order Quantity x Cost per order = D Q x S Total Annual Ordering Cost = (D/Q) S
  21. 21. D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity Total Annual Holding Cost = Order Quantity 2 x = Q 2 x H Total Annual Holding Cost = (Q/2) H Holding cost per unit per year
  22. 22. D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity D = annual demand S = cost per order H = holding cost per unit per year Q = order quantity Total Inventory Cost = Total Annual Ordering Cost + Total Annual = Q 2 x H Total Inventory Cost = (D/Q)S + (Q/2) H Holding Cost D Q x S +
  23. 23. EOQ occurs when : Total Annual Ordering Cost = Total Annual Holding Cost D/Q)S = (Q/2)H Simplifying the equation, we have : This can also be written as : Q* represents the optimal value for Q This is what we call the EOQ Q2 = (2DS)/H Q* = √2DS/H
  24. 24. Given the following data for an inventory scenario whose characteristics fit the assumptions of the basic EOQ model: D = 15,000 units per year S = $3 per order H = $1 per unit per year LT = Replenishment = lead time = 2 days Operating days per year is assumed to be 300 days
  25. 25. Find the following: 1. Average daily demand 2. EOQ 3. Number of orders placed per year 4. Total annual ordering cost 5. Total annual holding cost 6. Time between orders 7. Reorder point (in units) 8. Average inventory level
  26. 26. 1. Average daily demand 15,000 units/yr ÷ 300 days/yr = 50 units per day 2. EOQ EOQ = √2DS/H = √(2)(15,000)(3)/(1) = 300 units/order 3. Number of orders placed per year D/Q = (15,000 units/yr)/(300 units/order) = 50 orders/yr
  27. 27. 4. Total annual ordering cost (D/Q)(S) = [(15,000units/yr)/(300 units/order)]($3/order) = $150/yr 5. Total annual holding cost (Q/2)H = [(300 units/order/2)]($1/unit/yr) = $150/yr 6. Time between orders (Q/d) = (300 units/order)/(50 units/day) = 6 days/order or 300days/yr÷50 orders/yr = 6 days/order
  28. 28. 7. Reorder point (in units) ROP = (daily demand)(Lead time) = (50 units/day)(2 days) = 100 units 8. Average inventory level Q/2 = 300 units/2 = 150 units
  29. 29. At optimal order quantity (Q*): Carrying cost = Ordering cost
  30. 30. IINNPPUUTT V VAALLUUEESS OOUUTTPPUUTT V VAALLUUEESS AAnnnnuuaal lD Deemmaanndd OOrrddeerriningg C Coosstt CCaarrrryyiningg C Coosstt LLeeaadd T Timimee DDeemmaanndd P Peerr D Daayy Economic Order Quantity (EOQ) Economic Order Quantity (EOQ) RReeoorrddeerr P Pooinintt ( (RROOPP)) EOQ Models
  31. 31. After Q* is determined, the second decision is when to order Orders must usually be placed before inventory reaches 0 due to order lead time Lead time is the time from placing the order until it is received The reorder point (ROP) depends on the lead time (LT)
  32. 32. Given the following data : D = 1,000 units per year Lead Time (LT) = 3 business days Operating days per year is assumed to be250 days Find the Reorder Point (ROP)
  33. 33. Solution : Daily Demand (d) = 1,000 units per year 250 days per year Daily demand (d) = 4 units per day Reorder Point (ROP) = Daily Demand (d) x Lead Time (LT) Reorder Point (ROP) = 4 units per day x 3 business days Reorder Point (ROP) = 12 units
  34. 34. Economic Production Quantity : Determining How Much to Produce The EOQ model assumes inventory arrives instantaneously In many cases inventory arrives gradually The economic production quantity (EPQ) model assumes inventory is being produced at a rate of p units per day There is a setup cost each time production begins
  35. 35. Parameters Q* = Optimal production quantity (or EPQ) Cs = Setup cost D = annual demand d = daily demand rate p = daily production rate
  36. 36. We will need the average inventory level for finding carrying cost Average inventory level is ½ the maximum Max inventory = Q x (1- d/p) Ave inventory = ½ Q x (1- d/p)
  37. 37. Total Cost consists of : Setup cost = (D/Q) x Cs Carrying cost = [½ Q x (1- d/p)] x Ch Production cost = P x D As in the EOQ model: The production cost does not depend on Q The function is nonlinear
  38. 38. As in the EOQ model, at the optimal quantity Q* we should have: Setup cost = Carrying cost (D/Q*) x Cs = [½ Q* x (1- d/p)] x Ch Rearranging to solve for Q* : Q* =
  39. 39. Brown Manufacturing produces mini refrigerators and has 167 business days per year. Other relevant data are given below : D = 10,000 units annually d = 1000 / 167 = ~60 units per day p = 80 units per day (when producing) Ch = $0.50 per unit per year Cs = $100 per setup P = $5 to produce each unit
  40. 40. The production cycle will last until Q* units have been produced Producing at a rate of p units per day means that it will last (Q*/p) days For Brown this is: Q* = 4000 units p = 80 units per day 4000 / 80 = 50 days
  41. 41. 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)
  42. 42. A mathematical model in operations management and applied economics used to determine optimal inventory levels. It is typically characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is q, each unit of demand above q is a lost in potential sales.
  43. 43. Also known as the Newsvendor Problem or Newsboy Problem by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.
  44. 44. A one-time business decision that occurs in many different business contexts such as: Buying seasonal goods (sometimes called style goods) - A “season” can be a day, week, year, etc. For example, most swimsuits can only be purchased seasonally. If a buyer orders too few swimsuits, the retailer will have lost sales and dissatisfied customers. If the buyer orders too many swimsuits, the retailer will have to sell them at a clearance price or even throw some away.
  45. 45. Making the last buy or last production run decision for a product (or component) that is near the end of its life cycle. - If the order size is too small, the firm will have stock outs and disappointed customers. If the order size is too large, the firm will only be able to sell the items for their salvage value. Setting safety stock levels for an item. - If the safety stock is too low, stock outs will occur. If safety stock is too high, the firm has too much carrying cost. Nearly all safety stock models are newsvendor problems with the selling season being one order cycle or one review period. empty seats.
  46. 46. Setting target inventory levels – A salesperson carries inventory in the trunk of a vehicle. The inventory is controlled by a target inventory level. If the target is too low, stock outs will occur. If the target is too high, the salesperson will have too much carrying cost. Selecting the right capacity for a facility or machine – If the capacity of a factory or a machine over the planning horizon is set too low, stock outs will occur. If capacity is set too high, the capital costs will be too high.
  47. 47. Overbooking customers – If an airline overbooks too many passengers, it incurs the cost of giving away free tickets to inconvenienced passengers. If the airline does not overbook enough seats, it incurs an opportunity cost of lost revenue from flying with empty seats.
  48. 48. All of these newsvendor problem contexts share a common mathematical structure with the following four elements: A decision variable (Q) – The newsvendor problem is to find the optimal Q for a one-time decision, where Q is the decision quantity (order quantity, safety stock level, overbooking level, etc.). Q* denotes the optimal (best) value for Q.
  49. 49. Uncertain demand (D) – Demand is a random variable defined by the demand distribution and estimates of the distribution parameters. Demand may be either discrete (integer) or continuous. Unit overage cost (Co) – This is the cost of buying one unit more than the demand during the one-period selling season. In the standard retail context, the overage cost is the unit cost (c) less the unit salvage value (s), i.e., Co = c – s. The salvage value is the salvage revenue less the salvage cost required to dispose of the unsold product.
  50. 50. Unit underage cost (Cu) – This is the cost of buying one unit less than the demand during the one-period selling season. This is also known as the stock out (or shortage) cost. In the retail context, the underage cost is computed as the lost contribution to profit, which is the unit price (p) less the unit cost (c), i.e., Cu = p – c
  51. 51. The “too much/too little problem”: Order too much and inventory is left over at the end of the season Order too little and sales are lost. Given : Given : Each suit sells for p = $180 Seller charges c = $110 per suit Discounted suits sell for v = $90 Each suit sells for p = $180 Seller charges c = $110 per suit Discounted suits sell for v = $90
  52. 52. Co = overage cost (order “one too many” --- demand < order amount) The cost of ordering one more unit than what you would have ordered had you known demand – if you have left over inventory the increase in profit you would have enjoyed had you ordered one fewer unit. Co = Cost – Salvage value Co = c – v Co = 110 – 90 Co = 20
  53. 53. Cu = underage cost (order “one too few” – demand > order amount) The cost of ordering one fewer unit than what you would have ordered had you known demand If you had lost sales (i.e., you under ordered), Cu is the increase in profit you would have enjoyed had you ordered one more unit. Cu = Price – Cost Cu = p – c Cu = 180 – 110 Cu = 70
  54. 54. To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected gain on the Qth unit: Rearrange terms in the above equation -> The ratio Cu / (Co + Cu) is called the critical ratio (CR).
  55. 55. We shall assume demand is distributed as the normal distribution with mean μ and standard deviation s. Find the Q that satisfies the above equality use NORMSINV(CR) with the critical ratio as the probability argument. (Q - μ ) / s = z - score for the CR so Q = μ + z * s NNoottee:: w whheerree F F((QQ)) = = P Prroobbaabbiilliittyy D Deemmaanndd < <== Q Q
  56. 56. Inputs: Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20 Evaluate the critical ratio: NORMSINV(.7778) = 0.765 Find an order quantity Q such that there is a 77.78% probability that demand is Q or lower. Find an order quantity Q such that there is a 77.78% probability that demand is Q or lower.
  57. 57. Other Inputs: mean = μ = 3192; standard deviation = s = 1181 Convert into an order quantity Q = μ + z * s Q = 3192 + 0.765 * 1181 Q = 4095
  58. 58. Useful when demand is uncertain and is a continuous variable while the product is perishable. The only real Example :al The demand is approximately normally distributed with mean 11.731 and standard deviation 4.74. Each copy is purchased for 25 cents and sold for 75 cents, and he is paid 10 cents for each unsold copy by his supplier.
  59. 59. One obvious solution is approximately 12 copies. Suppose the vendor purchases a copy that he doesn't sell. His out-of-pocket expense is 25 cents - 10 cents = 15 cents. Suppose on the other hand, he is unable to meet the demand of a customer. In that case, he loses 75 cents - 25 cents = 50 cents profit.
  60. 60. Notation :al Co = Cost per unit of positive inventory remaining at the end of the period (known as the overage cost). Cu = Cost per unit of unsatisfied demand. This can be thought of as a cost per unit of negative ending inventory (known as the underage cost).
  61. 61. The demand D is a continuous nonnegative random variable with density function f (x) and cumulative distribution function F(x). The decision variable Q is the number of units to be purchased at the beginning of the period.
  62. 62. The cost function G(Q) is convex. The only real The optimal solution equation
  63. 63. Determining the optimal policy for function : The only real
  64. 64. Example (continuation) : Normally distributed with mean μ = 11.73 and standard deviation s = 4.74. Co = 25 - 10 Cu = 75 - 25 = 15 cents = 50 cents The critical ratio is = 0.50/0.65 = 0.77. Purchase enough copies to satisfy all of the weekly demand with probability 0.77. The optimal Q* is the 77th percentile of the demand distribution.
  65. 65. Area = 0.77 11.73 Q* x f(x)
  66. 66. Example (continuation) : Using the data of the normal distribution we obtain a standardized value of z = 0.74. The optimal Q is Hence, he should purchase 15 copies every week.
  67. 67. Recall the two general types of multi-period inventory models : 1. Fixed-order quantity models Also called the economic order quantity, EOQ, and Q-model Event triggered 2. Fixed-time period models Also called the periodic system, periodic review system, fixed-order interval system, and P-model Time triggered
  68. 68. Key Differences : To use the fixed–order quantity model, the inventory remaining must be continually monitored In a fixed–time period model, counting takes place only at the review period The fixed–time period model Has a larger average inventory Favors more expensive items Is more appropriate for important items Requires more time to maintain
  69. 69. Fixed-Order Quantity Model 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 Inventory holding cost is based on average inventory Ordering or setup costs are constant All demands for the product will be satisfied
  70. 70. Recall the Basic Fixed-Order Quantity (EOQ) Model
  71. 71. Recall the Basic Fixed-Order Quantity (EOQ) Formula Annual Ordering Cost S + Q Q H 2 Total Annual = Cost Annual Purchase Cost TC = DC + D Annual Holding Cost + + TC=Total annual 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 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 Q* = √2DS/H unit of inventory
  72. 72. Establishing Safety Stocks Safety stock is the amount of inventory carried in addition to the expected demand Safety stock can be determined based on many different criteria A common approach is to simply keep a certain number of weeks of supply
  73. 73. Establishing Safety Stocks (con’t) A better approach is to use probability Assume demand is normally distributed Assume we know mean and standard deviation To determine probability, we plot a normal distribution for expected demand and note where the amount we have lies on the curve
  74. 74. Fixed Order Quantity Model
  75. 75. Fixed Order Quantity Model with Safety Stock
  76. 76. Fixed Time Period Model
  77. 77. Fixed Time Period Model with Safety Stock Formula q = Average demand + Safety stock – Inventory currently on hand q = Average demand + Safety stock – Inventory currently on hand Where : q = quantity to be ordered T = the number of days between reviews L = lead time in days d = forecast average daily demand z = the number of standard deviations for a specified service probability I = current inventory level, including items on order sT+L = standard deviation of demand over the review & lead time
  78. 78. Determining the value of sT+L The standard deviation of a sequence of random events equals the square root of the sum of the variances
  79. 79. Fixed Time Period Model Sample Problem : Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units. Given the above information, how many units should be ordered ?
  80. 80. Solution : s s ( )( ) T+L d 2 2 = (T+ L) = 30 +10 4 = 25.298 s q = d(T + L) + Z - I T+L q = 20(30 +10) + (1.75)(25.298) - 200 + 645 units q = 800 44.272 - 200 = 644.272, or