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# The EOQ Model

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An INVENTORY is an accumulation of a commodity that will be used to satisfy some future demand.
EOQ Is such an Inventory

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### The EOQ Model

1. 1. THE EOQ MODEL Syed Mohammed Sajl Semester 6 B Tech – PS & E Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 1
2. 2. To a pessimist, the glass is half empty. To an optimist, it is half full. - Anonymous Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 2
3. 3. SOME BASIC DEFINITIONS • An INVENTORY is an accumulation of a commodity that will be used to satisfy some future demand. • Inventories may be of the following form: - Raw material - Components (subassemblies) - Work-in-process - Finished goods - Spare parts - Pipeline Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 3
4. 4. EOQ History • Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once” • Interest on capital tied up in wages, material and overhead sets a maximum limit to the quantity of parts which can be profitably manufactured at one time; “set-up” costs on the job fix the minimum. Experience has shown one manager a way to determine the economical size of lots. • Early application of mathematical modeling to Scientific Management Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 4
5. 5. EOQ MODELING ASSUMPTIONS 1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously. 2. Delivery is immediate – there is no time lag between production and availability to satisfy demand. 3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand. 4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant. 6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products. Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 5
6. 6. TOTAL INVESTMENT IN INVENTORIES IN US Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 6
7. 7. CHARACTERISTICS OF INVENTORY SYSTEMS 1. Demand • constant versus variable • known versus uncertain 2. Replenishment Lead Time 3. Review • periodic or continuous 4. Replenishment • periodic or continuous 5. Excess demand • backordered or lost 5. Changing Inventory • perishability or obsolescence Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 7
8. 8. COST STRUCTURE • Order costs – fixed (Set-up cost, K) – variable, per-unit (c) • Holding costs (h) • Penalty cost (p) – if backordering allowed: loss of goodwill – if demand lost: loss of goodwill + loss of profit Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 8
9. 9. THE ECONOMIC ORDER QUANTITY (EOQ) MODEL Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 9
10. 10. ASSUMPTIONS LEADING TO EOQ • Demand rate, λ, is constant and deterministic over time (units/day, units/year, etc.) • Shortages not permitted • No replenishment lead time • The cost structure includes – fixed ordering cost, K, per order placed – unit variable cost, c, independent of order size (no discounts) – holding cost, h • Infinite planning horizon Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 10
11. 11. WHY EOQ? • Easy to compute • Does not require data that is hard to obtain • Policies are surprisingly robust • Assumptions can be relaxed • Gives a good overall idea • Can be starting point for more complicated models Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 11
12. 12. EOQ - NOTATION Q = order quantity (we want to find the optimal Q) K = fixed order cost c = unit variable cost, \$ / unit h = holding cost, \$ / (unit * (unit time)) h = I c λ = demand rate, units / (unit time) Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 12
13. 13. EOQ – THE OBJECTIVE • Our aim is to determine the best replenishment strategy (when and how much to order) under the criterion that the relevant costs (order and holding costs) will be minimized over time. Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 13
14. 14. EOQ – THE INTUITION • “When” should we place a new order ? – assume zero starting inventory – demand is deterministic and at a constant rate – lead time is negligible – no backorders are allowed Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 14
15. 15. EOQ – THE INTUITION • “How much” we have to order each time ? – parameters do not change over time – there is no reason for ordering different quantities – so, order the same quantity Q in each order Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 15
16. 16. EOQ – THE INTUITION Decision Variable: Q, the order size T Time Q Inventory Slope= -λ Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 16
17. 17. DERIVING THE EOQ: THE “CYCLE” • The same picture occurs over and over again – Why? – any one of the triangles will be called a “cycle” • Why not minimize the total cost in a cycle? • Minimize the average cost per unit time (annual cost) • Convert the order and holding costs to “average annual cost”, using the total cost in a cycle Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 17 lengthCycle cycleaincostTotal T timeofunitsTperCosts lim T
18. 18. DERIVING THE EOQ: THE ORDER COST • Cycle length T =Q / λ (slope, - λ, = -Q/T) • Since Q / λ is the time between two orders (the cycle time), λ /Q is the number of orders per unit time • In each order, we pay K+Qc • Order cost per unit time Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 18
19. 19. DERIVING THE EOQ: THE HOLDING COST Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 19 • Holding cost per unit time = 2 2 lengthcycle cycleoneovercostholding 2 0 Q h Q Q h Q T dttQh Q T 2 levelinventoryAverage Q hh • This can also be calculated as
20. 20. DERIVING THE EOQ Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 20 2 )(' 2 h Q KQG 0 2 )('' 3 Q K QG h K Q 2* = I c
21. 21. THE AVERAGE ANNUAL COST CURVE Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 21 unit time cost Q 2 hQ G(Q) Q K Q* Annual fixed ordering and holding cost The minimum
22. 22. AN EXAMPLE • Suppose that you are working for a retail store and have to determine how many boxes of detergent to order. • Suppose every time you make an order you pay \$20 for transportation and \$10 to prepare the ordering request. • Suppose every dollar tied in the inventory would earn 10 cents annually, if you had invested it. • Suppose there is a regular weekly demand of 20 boxes and each box costs you \$10. • What is the order quantity? • What is the trade-off here? Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 22
23. 23. THE SOLUTION • λ=20*52 = 1040 boxes annually • K (fixed ordering cost) = \$30 each time • h (holding cost) Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 23
24. 24. INVENTORY TURNOVER RATIO WITH EOQ MODEL Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 24 • Turnover Ratio: A measure of effective inventory control – “How many times I sold my inventory?” demand rate 2 avg. inventory / 2opt h Q K
25. 25. EOQ IN A PRODUCTION ENVIRONMENT • So far we have discussed a case in which we “order” from an outside supplier • If we produce in-house, the relevant problem is called the “Economic Production Quantity” • In this case, K denotes the “setup cost” • Why do we have a setup cost? Examples include: – cost of production setup independent of the number produced: heating an oven, cost of dyes, etc. – time required for the setup: lost production! – first few units might need to be scrapped Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 25
26. 26. THANK YOU!  Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 26
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