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Chapter 5

Chapter 5. Order Quantities When Demand Is Approximately Level. Order Quantities When Demand Is Approximately Level. How large a replenishment quantity to use when conditions are stable. A simplified situation but a reasonable approximation of reality on certain occasions. Assumptions.

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Chapter 5

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  1. Chapter 5 Order Quantities When Demand Is Approximately Level

  2. Order Quantities When Demand Is Approximately Level How large a replenishment quantity to use when conditions are stable. A simplified situation but a reasonable approximation of reality on certain occasions.

  3. Assumptions • Demand rate is constant and deterministic. • Order quantity need not be an integral number and no min or max restrictions on its size. • Unit variable cost, v, does not depend on order quantity, i.e. there are no discounts on unit purchase cost or unit transportation cost. • Cost parameters do not change considerably over time, inflation is negligible. • Item investigated is treated independently of other items, i.e. benefits from joint review or joint replenishment do not exist.

  4. Assumptions • Replenishment lead time is zero. • No shortages are allowed. • Entire order quantity is delivered at the same time • Planning horizon is very long. Paramaters continue at the same value for a long time. Most of these assumptions will be relaxed later.

  5. Derivation of the Economic Order Quantity (EOQ) Minimize the total relevant costs in determining the order quantity Notation: Q: order quantity, in units A: fixed cost incurred in each replenishment, in dollars v: unit variable cost, in $/unit r: carrying charge, cost of having one dollar of item tied up in inventory for one period, in $/$/unit time D: demand rate, in unit/unit time TRC(Q): cost per unit time, sum of those costs which can be influenced by Q, in $/unit time

  6. Derivation of the Economic Order Quantity (EOQ) • Parameters do not change  Q does not change • Demand is deterministic, L=0 and no shortages are planned  each replenishment will be made when inventory level is zero (Slope = – D) Time between replenishments = Q/D

  7. Derivation of the Economic Order Quantity (EOQ) Total cost = Replenishment cost + Inventory holding cost Replenishment cost = Fixed cost + Purchasing cost = A+Qv Number of replenishments = Replenishment cost per unit time = Cr = Inventory cost per unit time = Cc = Does not affect Q*  neglected

  8. Derivation of the Economic Order Quantity (EOQ) As Q increases, replenishment cost decreases but holding cost increases  tradeoff

  9. Derivation of the Economic Order Quantity (EOQ) Necessary condition that the slope is zero at minimum: Wilson lot size This is one of the earliest and most well-known results in inventory theory.

  10. Derivation of the Economic Order Quantity (EOQ) Substitute EOQ in the equation, two cost components are equal. We must also add the purchasing cost to the total cost.

  11. Derivation of the Economic Order Quantity (EOQ) EOQ as a time supply: number of months it will satisfy

  12. Example Jaydeep Company buys 6,000 units of an item every year with a unit cost of $30. It costs $125 to process an order and arrange delivery, while interest and storage costs amount to $6 a year for each unit held. What is the best ordering policy for the item? D = 6,000 units/year v = $30/unit A = $125/order vr = $6/year

  13. Example The optimal time between orders: Total cost:

  14. Sensitivity Analysis The total cost curve is shallow around EOQ, meaning that small deviations from EOQ costs slight changes in the total cost.

  15. Sensitivity Analysis Suppose we use Q' rather than EOQ, Q'= (1+p)EOQ 100p is the percent deviation of Q’ from EOQ.

  16. Sensitivity Analysis

  17. Sensitivity Analysis Example: Assume 10% deviation from EOQ. p=0.1 and Q'=1.1EOQ

  18. Sensitivity Analysis • Incorrect value of Q may be due to inaccurate estimates of one or more parameters (D, A, v, r). • Some Q values may have additional appeal over EOQ due to physical constraints like pallet size, not included in basic EOQ. Such values can be used provided that they are reasonably close to EOQ.

  19. Quantity Discounts In EOQ, v does not depend on Q. Discount structures: • All-units discount • Incremental discount Suppose: Qb= break point

  20. Incremental Discount If Q>Qb, Q-Qb units are discounted. Total cost v0(1-d) v0 Quantity Qb

  21. All-units Discount If Q≥Qb, all Qunits are discounted. Total cost v0(1-d) v0 Quantity Qb

  22. All-units Discount

  23. All-units Discount and (2) are valid for non-overlapping regions of Q. Term by term comparison gives (1)>(2) for the same Q. If minimum of (2) is feasible, then it is optimal.

  24. All-units Discount

  25. All-units Discount

  26. All-units Discount

  27. All-units Discount-The Algorithm • Compute • Compare EOQ(discount) with Qb. If EOQ(discount) ≤ Qb, EOQ(discount) is the best order quantity (case c). If EOQ(discount) < Qb, go to Step 3. • Compare TRC(EOQ) and TRC(Qb). If TRC(EOQ) < TRC(Qb), Qopt=EOQ (case b). If TRC(EOQ) > TRC(Qb), Qopt=Qb (case a).

  28. Example Consider three components used in the assembly of an X-ray film processor. The supplier offers the same discount structure for each of the items, and discounts are based on replenishment sizes of the individual items. The relevant characteristics of the items are given below: The supplier offers a 2% discount on any replenishment of 100 units or higher of a single item.

  29. Example

  30. Limits on Order Sizes • Maximum Time Supply or Capacity Restriction • Shelf Life (SL): perishability of the product warranty period Time supply (time during which Q will be depleted) If TEOQ>SL  Q=DSL • Long time supply may be unrealistic due to uncertainty in demand or obsolescence. • Storage capacity limitation

  31. Limits on Order Sizes • Minimum Order Quantity Supplier may set a minimum allowable order quantity. If EOQ is smaller than this quantity, best order size is supplier minimum. • Discrete Units • If EOQ is noninteger, round the value to nearest integer. • Items may be supplied in packs of more than one unit. • Replenishment may have to cover an integral number of periods of demand.

  32. Finite Replenishment Rate EOQ assumes the whole replenishment quantity arrives at the same time. If it becomes available at a rate of m per unit time (production rate) Imax Slope=m-D Slope=-D Q/D Q/m Imax/D

  33. Finite Replenishment Rate

  34. Finite Replenishment Rate

  35. Finite Replenishment Rate

  36. Finite Replenishment Rate

  37. Example Demand for an item is constant at 1,800 units a year. The item can be made at a constant rate of 3,500 units a year. Unit cost is $50, batch set-up cost is $650 and holding cost is 30% of value a year. What is the optimal batch size for the item? What are the optimal production time and cycle length? D=1,800 units/year m=3,500 units/year v=$50/unit A=$650/order r=30%

  38. Example

  39. Nonzero Lead Time (Slope = – D) Reorder point Lead time (L) Place order

  40. Nonzero Lead Time When inventory level hits DL, an order is placed.

  41. A Special Opportunity to Procure One time opportunity to procure an item at a reduced unit cost (ex: last opportunity to replenish before a price increase). Since the unit cost of the current replenishment (v1) is different from future replenishments (v2), order quantity is different. Two choices: a large Q to procure now, or the future EOQ with the higher unit cost. Exact approach is too complex. Therefore, we use an approximate approach.

  42. A Special Opportunity to Procure

  43. A Special Opportunity to Procure

  44. A Special Opportunity to Procure

  45. Example The manufacturer announces a price increase for a product from $28/box to $30/box. The dealer uses approximately 80 boxes a year and estimates the fixed cost per order to be $1.50 and the carrying charge as 0.20$/$/year. v1= $28/box v2= $30/box D=80 boxes/year A=$1.50/order r= 0.20$/$/year

  46. Example

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