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- 1. OBJECTIVES Inventory System Defined Types of Inventory Supply Chain Independent vs. Dependent Demand Inventory System Models Management Multi-Period Inventory Models: Basic Fixed-Order Quantity Models Inventory Costs Chapter 5 Multi-Period Inventory Models: Basic Inventory Control Fixed-Time Period Model Single-Period Inventory Model Miscellaneous Systems and IssuesInventory System Inventory Inventory is the stock of any item or resource used in an organization and One of the most expensive assets can include: raw materials, finished of many companies representing as products, component parts, supplies, much as 50% of total invested and work-in-process capital An inventory system is the set of policies and controls that monitor levels Inventory managers must balance of inventory and determines what levels inventory investment and customer should be maintained, when stock service should be replenished, and how large orders should be
- 2. Purposes of Inventory Types of Inventory 1. To maintain independence of operations Raw material 2. To meet variation in product demand Purchased but not processed Work-in-process Work- in- 3. To allow flexibility in production Undergone some change but not completed scheduling A function of cycle time for a product 4. To provide a safeguard for variation in Maintenance/repair/operating (MRO) raw material delivery time Necessary to keep machinery and processes 5. To take advantage of economic purchase- productive order size Finished goods Completed product awaiting shipment Cycle InventoryTypes of Inventory-2 Inventory that varies directly with lot size. Lot size varies with elapsed time between Cycle Inventory orders. Safety Stock Inventory The quantity ordered must meet the demand Anticipatory Inventory during the ordering period. Long gaps in the ordering period will require Pipeline Inventory larger cycle inventory. The inventory may vary between order size Q to zero just before the new lot is delivered. Average inventory size is therefore Q/2
- 3. Safety Stock Inventory Anticipation InventorySafety stock inventory protects against Inventory used to absorb uneven rate ofuncertainties in demand, lead time, and demand or supplysupply. Predictable seasonal demand pattern may justify anticipation inventory.It ensures that operations are not Uneven demand often makes the firm todisrupted when problems occur. stockpile during low production demand toTo build safety stock an order is placed make better use of production facilities andearlier than the item is needed or the avoid varying output rates and labor force.ordered quantity is larger than the Uncertainties such as threatened strikes,quantity required till the next delivery problem at suppliers facilities etc also justifyschedule. anticipation inventory. Pipeline Inventory Independent vs. Dependent Demand Inventory moving from point to point in the Independent Demand (Demand for the final end- material flow system is called pipeline product or demand not related to other items) inventory - from suppliers to plant, from one Finished operation to the next in processing, from product plant to distribution center and from Dependent distribution center to retailer Demand Pipeline Inventory between two points, can be (Derived demand expressed in terms of lead time and average items for demand (d) during the lead time (L). E(1) component parts, Pipeline Inventory = dL subassemblies, Component parts raw materials, etc)
- 4. Inventory Systems Models Inventory Models for •• Multi-Period Inventory Models Multi-Period Inventory Models -- Fixed-Order Quantity Models Independent Demand Fixed-Order Quantity Models Event triggered (Example: running out of Event triggered (Example: running out of stock) stock) Need to determine when and how -- Fixed-Time Period Models Fixed-Time Period Models much to order Time triggered (Example: Monthly sales Time triggered (Example: Monthly sales call by sales representative) call by sales representative) Basic economic order quantity•• Single-Period Inventory Models Single-Period Inventory Models -- One time purchasing decision (Example: One time purchasing decision (Example: Production order quantity vendor selling t-shirts at a football game) vendor selling t-shirts at a football game) -- Seeks to balance the costs of inventory Quantity discount model Seeks to balance the costs of inventory overstock and under stock overstock and under stockHolding, Ordering, and Holding CostsSetup Costs •Housing costs (including rent or depreciation, Holding costs - the costs of holding operating costs, taxes, insurance) or “carrying” inventory over time •Material handling costs (equipment lease or Ordering costs - the costs of depreciation, power, operating cost) placing an order and receiving goods •Labor cost Setup costs - cost to prepare a •Investment costs (borrowing costs, taxes, and machine or process for insurance on inventory) manufacturing an order •Pilferage, space, and obsolescence
- 5. Multi-Period Models: Multi-Period Models: Fixed-Order Quantity Model Fixed-Order Quantity Model Assumptions Model Assumptions (Contd.) Demand for the product is constant Inventory holding cost is based on and uniform throughout the period average inventory Lead time (time from ordering to Ordering or setup costs are constant receipt) is constant All demands for the product will be Price per unit of product is constant satisfied (No backorders are allowed) Basic Fixed-Order Quantity Model and Reorder Cost Minimization Goal Point Behavior By adding the item, holding, and ordering costs By adding the item, holding, and ordering costs together, we determine the total cost curve, which in together, we determine the total cost curve, which in 1. You receive an order quantity Q. 4. The cycle then repeats. turn is used to find the Qopt inventory order point that turn is used to find the Qopt inventory order point that minimizes total costs minimizes total costsNumber Total Costof units Con hand Q Q Q O S Holding T R Costs L Annual Cost of 2. You start using L Items (DC) them up over time. 3. When you reach down to Time a level of inventory of R, Ordering Costs R = Reorder point Q = [Economic] order quantity you place your next Q QOPT L = Lead time sized order. Order Quantity (Q)
- 6. D The EOQ Model Annual setup cost = D Q S The EOQ Model Annual setup cost = Q S Q Annual holding cost = H 2 Q = Number of pieces per orderQ* = Optimal number of pieces per order (EOQ) Q = Number of pieces per order D = Annual demand in units for the Inventory item Q* = Optimal number of pieces per order (EOQ) S = Setup or ordering cost for each order D = Annual demand in units for the Inventory item H = Holding or carrying cost per unit per year S = Setup or ordering cost for each order H = Holding or carrying cost per unit per year Annual setup cost = (Number of orders placed per year) year) x (Setup or order cost per order) (Setup order) Annual holding cost = (Average inventory level) level) x (Holding cost per unit per year) (Holding year) Annual demand Setup or order = Order quantity Number of units in each order cost per order = (Holding cost per unit per year) (Holding year) 2 = D (S) (S Q = Q (H) (H 2 The EOQ Model Basic Fixed-Order Quantity (EOQ) TC=Total annual TC=Total annual cost Model Formula cost D =Demand D =Demand Q = Number of pieces per order Total Annual Annual Annual C =Cost per unit C =Cost per unit Q* = Optimal number of pieces per order (EOQ) Q =Order quantity Annual = Purchase + Ordering + Holding Q =Order quantity D = Annual demand in units for the Inventory item S =Cost of placing S = Setup or ordering cost for each order Cost Cost Cost Cost S =Cost of placing an order or setup an order or setup H = Holding or carrying cost per unit per year cost cost R =Reorder point R =Reorder point Optimal order quantity is found when annual setup cost equals annual holding cost L =Lead time L =Lead time H=Annual holding H=Annual holding D D S = Q 2 H Annual setup cost = Q S D Q and storage cost and storage cost Solving for Q* Q Annual holding cost = Q H TC = DC + S+ H per unit of inventory per unit of inventory 2DS = Q2H 2 Q 2 Q2 = 2DS/H Q* = 2DS/H
- 7. Deriving the EOQ The Economic Ordering Quantity (EOQ) 2DS 2DS = 2(Annual Demand)(Order or Setup Cost) 2(Annual Demand)(Order or Setup Cost) QOPT = QOPT = H = H Annual Holding Cost Annual Holding Cost __ We also need a We also need a R eorder point, R = d L R eorder point, R = d L reorder point to reorder point to _ tell us when to tell us when to d = average daily demand (constant) place an order place an order L = Lead time (constant)EOQ Example-1 EOQ Example-1a Determine expected number of orders if: Determine optimal number of units to order D = 1,000 units D = 1,000 units Q* = 200 units S = $10 per order S = $10 per order H = $.50 per unit per year H = $.50 per unit per year 2DS Expected Q* = Demand D H number of = N = Order quantity = Q* * orders 2(1,000)(10) N= 1,000 = 5 orders per year Q* = = 40,000 = 200 units 200 0.50
- 8. EOQ Example- 1b EOQ Example- 1c Determine time between orders if: Determine carrying cost if: D = 1,000 units Q*= 200 units Q*= D = 1,000 units Q* = 200 units S = $10 per order N= 5 orders per year N= S = $10 per order N = 5 orders per year H = $.50 per unit/yr working days= 250 days/yr H = $.50 per unit per year T = 50 days Number of working Total carrying cost = Setup cost + Holding cost Expected days per year D Qtime between = T = TCC = S + H orders N Q 2 250 1,000 200 TCC = ($10) + ($.50) T= = 50 days between orders 200 2 5 TCC = (5)($10) + (100)($.50) = $50 + $50 = $100 Holding cost is often given as a fraction of unit cost Holding cost as a fraction of unit costQuantity Discount Model or 2DS 2(Annual Demand)(Or der or Setup Cost) Q OPT = =Price-Break Model iC Annual Holding Cost i = percentage of unit cost attributed to carrying inventory C = cost per unit
- 9. Price-Break Example- 2 Price-Break Model Formula Problem Data (Part 1) or Quantity Discount Model A company has a chance to reduce their inventory A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using ordering costs by placing larger quantity orders using Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula: the price-break order quantity schedule below. What the price-break order quantity schedule below. What should their optimal order quantity be if this company should their optimal order quantity be if this company purchases this single inventory item with an e-mail purchases this single inventory item with an e-mail 2DS 2(Annual Demand)(Or der or Setup Cost) ordering cost of $4, a carrying cost rate of 2% of the ordering cost of $4, a carrying cost rate of 2% of the Q OPT = = iC Annual Holding Cost inventory cost of the item, and an annual demand of inventory cost of the item, and an annual demand of 10,000 units? 10,000 units? i = percentage of unit cost attributed to carrying inventory C = cost per unit Order Quantity units) Price/unit($) 0 to 2,499 $1.20 Since “C” changes for each price-break, the formula above 2,500 to 3,999 $1.00 will have to be used with each price-break cost value 4,000 or more $0.98Price-Break Example-2 Solution (Part 2) Price-Break Example -3 Solution (Part 3) First, plug data into formula for each price-break value of “C” Since the feasible solution occurred in the first price- Since the feasible solution occurred in the first price- break, it means that all the other true Qopt values occur break, it means that all the other true Qopt values occur Annual Demand (D)= 10,000 units Carrying cost % of total cost (i)= 2% Cost to place an order (S)= $4 Cost per unit (C) = $1.20, $1.00, $0.98 at the beginnings of each price-break interval. Why? at the beginnings of each price-break interval. Why? Next, determine if the computed Qopt values are feasible or not Because the total annual cost function is Because the total annual cost function is Total a “u” shaped function annual a “u” shaped functionInterval from 0 to 2499, the 2DS 2(10,000)( 4)Qopt value is feasible Q OPT = = = 1,826 units costs iC 0.02(1.20) So the candidates So the candidates for the price- for the price-Interval from 2500-3999, the 2DS 2(10,000)( 4)Qopt value is not feasible Q OPT = = = 2,000 units breaks are 1826, breaks are 1826, iC 0.02(1.00) 2500, and 4000 2500, and 4000Interval from 4000 & more, 2DS 2(10,000)( 4) units unitsthe Qopt value is not feasible Q OPT = = = 2,020 units iC 0.02(0.98) 0 1826 2500 4000 Order Quantity
- 10. Price-Break Example -3 Solution (Part 4)Next, we plug the true Qopt values into the total costNext, we plug the true Qopt values into the total costannual cost function to determine the total cost under Price-Break Example -3 Solution (Part 5)annual cost function to determine the total cost undereach price-breakeach price-break D Q TC = DC + S + iC Q 2TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 = $12,043.82TC(2500-3999)= $10,041TC(2500-3999)= $10,041TC(4000&more)= $9,949.20 $9849,20TC(4000&more)= $9,949.20Finally, we select the least costly Qopt,,which is thisFinally, we select the least costly Qopt which is thisproblem occurs in the 4000 & more interval. Inproblem occurs in the 4000 & more interval. Insummary, our optimal order quantity is 4000 unitssummary, our optimal order quantity is 4000 units Production Order Quantity Model In EOQ Model, We assumed that the entire order was received at one Production Order Quantity Model time. However, some business firms may receive their orders over a period of time.
- 11. Production Order Quantity Model Production Order Quantity Model Such cases require a different inventory model. In these cases inventory is being This version of the EOQ model is known as used while new inventory is still being “Noninstantaneous Receipt Model” also received and the inventory does not build referred to as the “Gradual Usage Model ” and up immediately to its maximum point. “Production Order Quantity Model”. In this, noninstantaneous receipt model, the order Instead, it builds up gradually when quantity is received gradually over time, and the inventory is received faster than it is inventory level is depleted at the same time it is being used; then it declines to its lowest being replenished. level as incoming shipments stop and the use of inventory continues. Here, we take into account the daily production rate and daily demand/input rate.Production Order Quantity Model Production Order Quantity Model Inventory Maximum Inventory Production occurs at a Demand rate of p occurs at a rate of d time t t
- 12. Production Order Quantity Production Order QuantityModel Model Since this model is especially suitable for p: Daily Production rate (units / day) production environments, It is called d: Daily demand rate (units / day) Production Order Quantity Model. t: Length of the cycle in days. Here, we use the same approach as we H: Annual holding cost per unit used in EOQ model. Lets define the following:Production Order Quantity Model Production Order Quantity Model Average Holding Cost = (Average In the period of production (until the end of Inventory) . H each t period): Max. Inventory = (Total Produced) – (Total Used) = (Max. Inventory / 2) . H = p.t - d.t Here, Q is the total units that are produced. Therefore, Q = p.t t = Q/p
- 13. Production Order Quantity Production Order QuantityModel Model If we replace the values of t in the Max. Ann. Holding Cost =(Max. Inventory / 2) . H Inventory formula: Annual Holding Cost = Q/2 (1 – d/p) . H Max. Inventory = p (Q/p) - d (Q/p) = Q - dQ/p = Q (1 – d/p) Annual Setup Cost = (D/Q) . SProduction Order Quantity Production Order QuantityModel Model Now we will set Annual Holding Cost = Annual Setup Cost Q/2 (1 – d/p) . H = (D/Q) . S
- 14. Production Order Quantity Production Order Quantity-Model Example-5 D = 1,000 units p = 8 units per day This formula gives us the optimum S = $10 d = 4 units per day production quantity for the Production H = $0.50 per unit per year Order Quantity Model. 2DS Q* = H[1 - (d/p)] It is used when inventory is consumed as it is produced. Q* = 2(1,000)(10) = 80,000 0.50[1 - (4/8)] = 282.8 or 283 units How Important is the Item? Segmentation of Inventory - Not all inventory is created equally - Different classes of inventory Miscellaneous Systems and - Result in different levels of profitability /revenue - Have different demand patterns and magnitudes Issues - Require different control policies ABC Analysis Commonly used in practice Classify items by revenue or value Combination of usage, sales price, etc.
- 15. ABC Analysis ABC Analysis Identify the items that management should spend time on Prioritize items by their value to firm Create logical groupings Adjust as neededABC Analysis Miscellaneous Systems: What is different between the classes? Bin SystemsA Items Very few high impact items are included Two-Bin System Require the most managerial attention and review Expect many exceptions to be madeB Items Order One Bin of Many moderate impact items (sometimes most) Inventory Automated control w/ management by exception Rules can be used for A (but usually too many exceptions) Full EmptyC Items One-Bin System Many if not most of the items that make up minor impact Control systems should be as simple as possible Reduce wasted management time and attention Order Enough to Group into common regions, suppliers, end users Refill Bin But these are arbitrary classifications Periodic Check
- 16. Miscellaneous Systems: Inventory Accuracy and Cycle CountingOptionalInventory Level, M SystemMaximum Replenishment Inventory accuracy refers to how q=M-I well the inventory records agree Actual Inventory Level, I with physical count M Cycle Counting is a physical I inventory-taking technique in which inventory is counted on a frequentQ = minimum acceptable order quantity basis rather than once or twice a yearIf q > Q, order q, otherwise do not order any.Question On average, I sell 150,000 units a year, which I obtain from a wholesaler. I estimate that the cost to me of placing an order is $50 with the Supply Chain average inventory storage cost being 20% per year of the cost of a unit ($5). Management1. What would be the optimal order quantity?2. I currently order 5 times a year. How much Inventory Control Part 2 would I save by switching to the optimal order Safety Stock, Fixed Period Model quantity as compared with my current policy of ordering 5 times a year? and Single Period Model
- 17. Planned Shortages with Back-Orders Uncertain Demand Shortage: when customer demand cannot be met Planned shortages could be beneficial Cost of keeping item is more expensive than the profit from selling it e.g. carUncertain Demand- Safety Stock Service Level A target for the proportion of demand that Buffer added to on hand is met directly from stock inventory during lead time Extra reserved stock The maximum acceptable probability that To prevent stock-out a demand can be met from the stock under uncertain demand For example 90% service level Safety stock will not 90% chance of meeting demand during lead time or normally be used, but it is 10% chance of not meeting demand (having back- available under uncertain order or lost sales) demand How much safety stock should we hold? Judgment on service level
- 18. Probabilistic Models Probabilistic Models So far we assumed that demand is One method of reducing stock outs is to constant and uniform. hold extra inventory (called Safety Stock). However, In Probabilistic models, demand In this case, we change the ROP formula is specified as a probability distribution. to include that safety stock (ss). Uncertain demand raises the possibility of a stock out (or shortage). Reorder Level (ROL) = LT x D Safety Stock Example Reorder Level (ROL) = (LT x D) + Safety Stock ROP = 50 units Stock-out cost = $40 per unit Orders per year = 6 Carrying cost = $5 per unit per yearReorder Number of Units ProbabilityLevel 30 0.2 40 0.2 ROP 50 0.3 60 0.2 70 0.1 1.0 Safety Stock
- 19. Safety Stock Example ExampleROP = 50 units Stock-out cost = $40 per unitOrders per year = 6 Carrying cost = $5 per unit per yearA safety stock of 20 units gives the lowest total cost ROP = 50 + 20 = 70 units Probabilistic Demand
- 20. Reorder Point for a Service Level Using the Standard Normal Probability TableUsing the Standard Normal Probability Table =
- 21. Probabilistic Demand Example- 3: Safety Stock Demand is variable and lead time is constant Safety stock, SS: Daily usage at a drug store follows a = Z × standard deviation of lead time normal distribution with a mean of 500 gm = Z × σ × √LT and a standard deviation of 50 gm. If the = Z × σdlt lead time for procurement is 7 days and Reorder level: the drug store wants a risk of only 2% ROL = lead time demand + safety stock determine = LT × D + Z × σ × √LT where σ = standard deviation of demand per day and a) reorder point and b) safety stock σdlt = σ × √LT Standard deviation of demand during necessary lead timeExample-3: Safety Stock Example: Safety Stock using Z-Score Mean daily demand, D =500 gm/day Lead Time, LT = 7days Mean Demand in lead period, µL =3500 gm Standard deviation, σ = 50 gm/day Standard deviation, σ = 50 gm/day Service level required = 98% or 0.98 σL = σ √ Lt= 50 √7 gm From normal distribution level Z is determined as z =2.05 Z= 2.05 from TableROL = (LT × D)+ z σ √LT X − µL Z= = (500 x 7)+ 2.05 * 50 * √7 σL = 3771 gm where X is a normal random variable Safety Stock = z σ √LT X=3771gm = 2.05 * 50 * √7 = 271 gm Safety stock = 3771 gm- 3500 gm =271 gm
- 22. Periodic Review System Maximum Inventory Level, M Supply Chain q=M-I Management M Actual Inventory Level, I I Inventory Control Periodic Review SystemP-System: Periodic Review System P-System: Periodic Review System-2 In this system, costs are not explicitly In this system, we are interested in actual considered and order quantity is not fixed. and average consumption over a period of time i.e. time between two reviews and lead Time is taken into account and given more time. Order quantity can be computed as emphasis follows: Inventory is periodically reviewed at fixed intervals and any difference between the If L< R then Q= M - I If L> R then Q= M – I - Q ord present and the last review is made up by replenishment order. Where The order quantity is thus equal to L= Lead Time R = Review Period replenishment level minus actual inventory M= Replenishment Level in Units I = Inventory on hand in Units on hand. Q =Quantity to be Ordered Qord= Quantity on order (in pipeline)
- 23. Example: Fixed Period Inventory ControlSystem (P-System) Example-Solution: P-System L<RThe average Replen. Lvl. = M Replenishment Level, M = 60monthly Safety Stock (B)+ consumption, D* (Review Time+ Lead Time)consumption of an M= 20+ 40(1+0.5) = 80 Unitsitem is 40 units, 40 Inventory on Hand, I = B + consumption/2Safety Stock is 20units, review time I = 20+ 40/2 = 40 units Safety Stock=B 20is 1 month and R The Order Quantity, Q = M – I LTlead time is 15 Q= 80- 40 = 40 Unitsdays, calculate 1 2 3replenishmentlevel MExample 2: (P-System) Fixed Order Vs. Periodic Review Consider a case where Lead time > review time Buffer/safety stock= 50 units D= 100 units/month Fixed-order quantity Fixed-time period Review Time= 1 month L= 2 months models–when holding models—when holding costs are high (usually costs are low (i.e., M= replinsh. Lvl. = B +D (1+2) = 50+ 100*3 expensive items or high associated with low-cost M= 350 Units deprecation rates), or items, low-cost storage), I = B+D/2 = 50+50 = 100 units when items are ordered or when several items are from different sources. ordered from the same Order Qty Q= M – I = 250 units source (saves on order If Qty already on order is 100 units (review after 1 mth) placement and delivery charges). Q= M-I- Qord= 150 units
- 24. Fixed Order Vs. Periodic Review A fixed-order quantity The main disadvantage of system can operate with a fixed-time period a perpetual count inventory system is that (keeping a running log of inventory levels must be every time a unit is withdrawn or replaced) or higher to offer the same protection against Single-Period Inventory through a simple two-bin or flag arrangement stockout as a fixed-order quantity system. Model wherein a reorder is It also requires a periodic placed when the safety count and closer stock is reached surveillance than a fixed- order quantity system. Decision under uncertainity & risk Single-Period Inventory Model In inventory control, sometimes management has to take In inventory control, sometimes management has to take This model states that we This model states that we IG is the profit per item risk under uncertainity, though wanting to keep the risk should stock up to the point IG is the profit per item risk under uncertainity, though wanting to keep the risk should stock up to the point times the probability of times the probability of factor to a minimum. factor to a minimum. where incremental gain (IG) where incremental gain (IG) selling ‘x’ items selling ‘x’ items is equal to incremental loss is equal to incremental loss •• How many World Cup shirts to produce, when the shirts (IL) (IL) IG= m. P(x) IG= m. P(x) How many World Cup shirts to produce, when the shirts will be of little or no value after the Cup. will be of little or no value after the Cup. IL is the cost per item times IL is the cost per item times m= margin of profit item m= margin of profit item •• How many suits to stock for Eid or Xmas season, profit How many suits to stock for Eid or Xmas season, profit the probability that ‘x’ items P (x)= probability of selling the item margin is high but the leftover stock will probably be of the probability that ‘x’ items P (x)= probability of selling the item margin is high but the leftover stock will probably be of will not be sold will not be sold C= Cost of the item C= Cost of the item no value no value IL= C. [1-P(x)]. IL= C. [1-P(x)]. Equating IG& IL and Equating IG& IL and P(x) = P(x) = C CSingle-period inventory model Applies in these cases solving the equation we get: solving the equation we get: m+C m+C
- 25. Single-Period Inventory Model Single Period Model Example-4 This model states that we This model states that we should continue to increase should continue to increase Our college basketball team is playing in a the size of the inventory so the size of the inventory so tournament game this weekend. Based on our past long as the probability of long as the probability of Cu experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every selling the last unit added is selling the last unit added is P≤ shirt we sell at the game, but lose $5 on every shirt equal to or greater than the equal to or greater than the Co + Cu not sold. How many shirts should we make for the ratio of: Cu/Co+Cu ratio of: Cu/Co+Cu game? Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667 Where : Z.667 = .432 Co = Cost per unit of demand over estimated therefore we need 2,400 + .432(350) = 2,551 shirts Cu = Cost per unit of demand under estimated P = Probability that the unit will be sold Example-6 Example-5 (Solution) Ahmed Juices makes a variety of juices for on-the- counter sales. Ahmed uses ice, which he grates in Where : making these drinks. Ice is supplied to Ahmed in large Cu blocks, each costing Rs 10. Ice blocks not used during P≤ Co = Cost per unit of demand over estimated a day gets wasted as the ice melts and cannot be used Co + Cu Cu = Cost per unit of demand under estimated the next day. If Ahmed is short of ice blocks on any day, P = Probabilit y that the unit will be sold he buys them from elsewhere, but at a premium of Rs 5 per block. Each block of ice can be used for 20Co = Rs 1.5 [(Cost) Loss if demand is overestimated] glasses of juice. The probability distribution for the demand of ice blocks is as followsCu = Rs 2.5 [(Cost) Profit Loss if demand is underestimated] What is the least cost stocking policy for Ahmed Juices? Probability of meeting demand is P ≤ [2.5/(2.5+1.5)] 0.65 at 700 buns. The baker x ice blocks: 20 21 22 23 24 25 26 27 28 P ≤ 0.625 should make 700 buns. p Probability 0 0.05 0.10 0.20 0.25 0.20 0.15 0.05 0
- 26. Practice NumericalThe end Example (Contd.)Example The present cycle and pipeline inventories are: The present cycle and pipeline inventories are: A plant makes monthly shipments of electric Cycle Inventory = Q/2 = 280/2 = 140 drills Cycle Inventory = Q/2 = 280/2 = 140 drills A plant makes monthly shipments of electric drills to a wholesaler in average lot sizes of drills to a wholesaler in average lot sizes of Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills 280 drills. The wholesaler’s average demand 280 drills. The wholesaler’s average demand is 70 drills a week and the lead time from the is 70 drills a week and the lead time from the plant is 3 weeks. The wholesaler must pay for plant is 3 weeks. The wholesaler must pay for Under the new offer, cycle and pipeline inventories are: Under the new offer, cycle and pipeline inventories are: the order the moment it leaves the plant. the order the moment it leaves the plant. Cycle Inventory = Q/2 = 350/2 = 175 drills Cycle Inventory = Q/2 = 350/2 = 175 drills If the wholesaler is willing to increase its If the wholesaler is willing to increase its Pipeline Inventory, dL= (70 drills/week)* (2 weeks) Pipeline Inventory, dL= (70 drills/week)* (2 weeks) purchase quantity to 350 units, the plant will purchase quantity to 350 units, the plant will = 140 drills = 140 drills guarantee a lead time of two weeks. What is guarantee a lead time of two weeks. What is effect on cycle and pipeline inventories? Under the new offer, cycle inventory increases by 25% but Under the new offer, cycle inventory increases by 25% but effect on cycle and pipeline inventories? pipeline inventories reduce by 33% (Decision Point) pipeline inventories reduce by 33% (Decision Point)
- 27. Benefit of Better Inventory Control Example -1 A firms inventory turnover (IT) is 4 times on a Fleming sells distributor rebuild kits used on cost of goods sold (COGS) of $800,000. Ford V-8 engines. Fleming purchases these kits Through better inventory control, inventory for $20 and sells about 250 kits a year. Each time Fleming places an order, it costs him $25 to turnover is improved to 8 times while the cover paperwork. He estimated that the cost of COGS remains the same, a substantial holding a rebuild kit in inventory is about $3.5 amount of funds is released from inventory. per kit per year. What is the amount released? a) What is the economic order quantity b) How many times per year will Fleming place $ 100,000 is released an order?Example -1 (Contd.) EOQ Example (2) Problem Data S = Cost of placing order = $ 25 D= Annual demand = 250 units/year Given the information below, what are the EOQ and Given the information below, what are the EOQ and H= Annual per-unit carrying cost =$3.5 per reorder point? reorder point? kit/year Annual Demand = 1,000 units Q = order quantity Days per year considered in average Qopt= √ [2 S D/H] daily demand = 365 Cost to place an order = $10 Qopt= √ [(2*25*250)/3.5] Holding cost per unit per year = $2.50 = 59.75 round to 60 kits Lead time = 7 days Orders per year =D/Qopt = 250/59.75 Cost per unit = $15 = 4.18
- 28. EOQ Example (2) Solution EOQ Example (3) Problem Data 2DS 2 (1,00 0 )(1 0) Determine the economic order quantityQ O PT = = = 8 9.443 u nits o r 90 un its Determine the economic order quantity H 2.50 and the reorder point given the following… and the reorder point given the following… 1,000 units / year d = = 2.74 units / day 365 days / year Annual Demand = 10,000 units Days per year considered in average daily _R e order po int, R = d L = 2.7 4units / d ay (7d ays) = 1 9.18 or 20 u n its demand = 365 Cost to place an order = $10 Holding cost per unit per year = 10% of cost In summary, you place an optimal order of 90 units. In In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when per unit the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 you only have 20 units left, place the next order of 90 Lead time = 10 days units. units. Cost per unit = $15EOQ Example (3) Solution Example- 8 2D S 2(10,000 )(10) Demand for Deskpro computer at Best Buy is 1000 units Q OPT = = = 365.148 units, or 366 units per month. Best Buy incurs a fixed order placement, H 1.50 transportation and receiving cost of $4000 each time an order is placed. Each computer costs Best Buy $500 and 10,000 units / year the retailer has a holding cost of 20%. Evaluate the d= = 27.397 units / day 365 days / year number of computers that the store manager should order in each replenishment lot. _ R = d L = 2 7 .3 9 7 u n its / d ay (10 d ays) = 2 7 3 .9 7 o r 2 7 4 u n its Annual Demand, D = 1000 x12 = 12000 units Order cost per lot, S = $4000 Place an order for 366 units. When in the course of Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, Unit Cost per computer, C =$500 using the inventory you are left with only 274 units, place the next order of 366 units. Holding cost per year as a fraction of the inv. Value, h = 0.2 place the next order of 366 units.
- 29. Example-8 Solved Example-9 Q opt = √ 2 * 12000 * 4000 In the above example, the manager at Best Buy 0.2 * 500 would like to reduce the lot size from 980 to 200. For this lot size to be optimal, the store manager = 980 units wants to evaluate how much the order cost perOther Info lot should be reduced.Cycle Inventory = Qopt/2 = 980/2 = 490 Desired Qopt = 200 unitsNo. of orders/year = D/Q = 12000/980 = 12.24 Annual Demand, D = 1000 x12 = 12000 unitsAnnual ordering & holding costs =(D/Q)*S + (Q/2)hC New Order cost per lot, S = ? = $97,980 Unit Cost per computer, C =$500Average Flow time= Q/2D = 490/12000 = 0.041year Holding cost per year as a fraction of the inv. Value, h = 0.2 = 0.49 monthsExample-9 (Contd.) Problem-10 S = H [Qopt]2/2D The Acer Co. sells 10,000 units per year. H =hC= 0.2*500 The cost of placing one order is $50 and it S = [0.2*500* 2002]/ [2*12000] costs $4 per year to carry one unit of inventory. What is Acer’s EOQ? S = $166.7THUS THE STORE MANAGER AT BEST BUY WOULD HAVE TOREDUCE THE ORDER COST PER LOT FROM $4000 TO $166.7 FORA LOT SIZE OF 200 TO BE OPTIMAL

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