Slides prepared by JOHN LOUCKS St. Edward’s University

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Slidesprepared by

JOHN LOUCKS

St. Edward’sUniversity

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Chapter 13, Part AInventory Models: Deterministic Demand Economic Order Quantity (EOQ) Model Economic Production Lot Size Model Inventory Model with Planned Shortages

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Inventory Models

The study of inventory models is concerned with two basic questions:• How much should be ordered each time• When should the reordering occur

The objective is to minimize total variable cost over a specified time period (assumed to be annual in the following review).

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Inventory Costs Ordering cost -- salaries and expenses of

processing an order, regardless of the order quantity

Holding cost -- usually a percentage of the value of the item assessed for keeping an item in inventory (including finance costs, insurance, security costs, taxes, warehouse overhead, and other related variable expenses)

Backorder cost -- costs associated with being out of stock when an item is demanded (including lost goodwill)

Purchase cost -- the actual price of the items Other costs

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Deterministic Models

The simplest inventory models assume demand and the other parameters of the problem to be deterministic and constant.

The deterministic models covered in this chapter are:• Economic order quantity (EOQ)• Economic production lot size• EOQ with planned shortages• EOQ with quantity discounts

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Economic Order Quantity (EOQ)

The most basic of the deterministic inventory models is the economic order quantity (EOQ).

The variable costs in this model are annual holding cost and annual ordering cost.

For the EOQ, annual holding and ordering costs are equal.

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Economic Order Quantity

Assumptions• Demand is constant throughout the year at

D items per year.• Ordering cost is $Co per order.• Holding cost is $Ch per item in inventory per

year.• Purchase cost per unit is constant (no

quantity discount).• Delivery time (lead time) is constant.• Planned shortages are not permitted.

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Economic Order Quantity Formulas

• Optimal order quantity: Q * = 2DCo/Ch • Number of orders per year: D/Q * • Time between orders (cycle time): Q */D

years• Total annual cost: [(1/2)Q *Ch] + [DCo/Q *]

(holding + ordering)

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Example: Bart’s Barometer Business Economic Order Quantity Model

Bart's Barometer Business is a retail outlet thatdeals exclusively with weather equipment. Bart is trying to decide on an inventoryand reorder policy for home barometers.

Barometers cost Bart $50 each anddemand is about 500 per year distributedfairly evenly throughout the year.

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Example: Bart’s Barometer Business Economic Order Quantity Model

Reordering costs are $80 per order and holdingcosts are figured at 20% of the cost of the item. BBB isopen 300 days a year (6 days a week and closed twoweeks in August). Lead time is 60 working days.

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Example: Bart’s Barometer Business

Total Variable Cost Model Total Costs = (Holding Cost) + (Ordering

Cost) TC = [Ch(Q/2)] + [Co(D/Q)]

= [.2(50)(Q/2)] + [80(500/Q)] = 5Q + (40,000/Q)

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Optimal Reorder Quantity Q * = 2DCo /Ch = 2(500)(80)/10 =

89.44 90 Optimal Reorder Point Lead time is m = 60 days and daily

demand is d = 500/300 or 1.667. Thus the reorder point r = (1.667)(60) = 100. Bart should reorder 90 barometers when his inventory position reaches 100 (that is 10 on hand and one outstanding order).

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Number of Orders Per YearNumber of reorder times per year = (500/90) = 5.56 or once every (300/5.56) = 54 working days (about every 9 weeks).

Total Annual Variable CostTC = 5(90) + (40,000/90) = 450 + 444

= $894.

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We’ll now use a spreadsheet to implementthe Economic Order Quantity model. We’ll confirmour earlier calculations for Bart’s problem andperform some sensitivity analysis.

This spreadsheet can be modified to accommodateother inventory models presented in this chapter.

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A B1 BART'S ECONOMIC ORDER QUANTITY23 Annual Demand 5004 Ordering Cost $80.005 Annual Holding Rate % 206 Cost Per Unit $50.007 Working Days Per Year 3008 Lead Time (Days) 60

Partial Spreadsheet with Input Data

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Partial Spreadsheet Showing Formulas for Output A B C10 Econ. Order Qnty. =SQRT(2*B3*B4/(B5*B6/100))11 Request. Order Qnty12 % Change from EOQ =(C11/B10-1)*1001314 Annual Holding Cost =B5/100*B6*B10/2 =B5/100*B6*C11/215 Annual Order. Cost =B4*B3/B10 =B4*B3/C1116 Tot. Ann. Cost (TAC) =B14+B15 =C14+C1517 % Over Minimum TAC =(C16/B16-1)*1001819 Max. Inventory Level =B10 =C1120 Avg. Inventory Level =B10/2 =C11/221 Reorder Point =B3/B7*B8 =B3/B7*B82223 No. of Orders/Year =B3/B10 =B3/C1124 Cycle Time (Days) =B10/B3*B7 =C11/B3*B7

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Partial Spreadsheet Showing OutputA B C

10 Econ. Order Qnty. 89.4411 Request. Order Qnty. 75.0012 % Change from EOQ -16.151314 Annual Holding Cost $447.21 $375.0015 Annual Order. Cost $447.21 $533.3316 Tot. Ann. Cost (TAC) $894.43 $908.3317 % Over Minimum TAC 1.551819 Max. Inventory Level 89.44 7520 Avg. Inventory Level 44.72 37.521 Reorder Point 100 1002223 No. of Orders/Year 5.59 6.6724 Cycle Time (Days) 53.67 45.00

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Summary of Spreadsheet Results• A 16.15% negative deviation from the EOQ

resulted in only a 1.55% increase in the Total Annual Cost.

• Annual Holding Cost and Annual Ordering Cost are no longer equal.

• The Reorder Point is not affected, in this model, by a change in the Order Quantity.

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Economic Production Lot Size

The economic production lot size model is a variation of the basic EOQ model.

A replenishment order is not received in one lump sum as it is in the basic EOQ model.

Inventory is replenished gradually as the order is produced (which requires the production rate to be greater than the demand rate).

This model's variable costs are annual holding cost and annual set-up cost (equivalent to ordering cost).

For the optimal lot size, annual holding and set-up costs are equal.

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Assumptions• Demand occurs at a constant rate of D

items per year.• Production rate is P items per year (and P

> D ).• Set-up cost: $Co per run.• Holding cost: $Ch per item in inventory per

year.• Purchase cost per unit is constant (no

quantity discount).• Set-up time (lead time) is constant.• Planned shortages are not permitted.

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Formulas

• Optimal production lot-size: Q * = 2DCo /[(1-D/P )Ch]

• Number of production runs per year: D/Q *• Time between set-ups (cycle time): Q */D

years• Total annual cost: [(1/2)(1-D/P )Q *Ch] +

[DCo/Q *] (holding + ordering)

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Example: Non-Slip Tile Co.

Economic Production Lot Size Model

Non-Slip Tile Company (NST) has been usingproduction runs of 100,000 tiles, 10 times per yearto meet the demand of 1,000,000 tilesannually. The set-up cost is $5,000 perrun and holding cost is estimated at10% of the manufacturing cost of $1per tile. The production capacity of the machine is 500,000 tiles per month. The factoryis open 365 days per year.

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Total Annual Variable Cost ModelThis is an economic production lot size problem with

D = 1,000,000, P = 6,000,000, Ch = .10, Co = 5,000

TC = (Holding Costs) + (Set-Up Costs) = [Ch(Q/2)(1 - D/P )] + [DCo/Q]

= .04167Q + 5,000,000,000/Q

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Optimal Production Lot Size Q * = 2DCo/[(1 -D/P )Ch]

= 2(1,000,000)(5,000) /[(.1)(1 - 1/6)]

= 346,410

Number of Production Runs Per YearD/Q * = 2.89 times per year.

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End of Chapter 13, Part A

Slides prepared by JOHN LOUCKS St. Edward’s University

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Slides by JOHN LOUCKS St. Edward’s University

Slides by John Loucks St . Edward’s University