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Inventory Control Models Part II Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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The Production Order Quantity Model USED WHENEVER THE VENDOR CANNOT DELIVER THE ORDER ( Q* ) ALL IN ONE DAY USED WHENEVER THE FACTORY CANNOT PRODUCE THE ORDER ( Q* ) ALL IN ONE DAY OR

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Service Sector Variable Interpretations P or p The delivery rate of purchased items P or p The production rate of manufactured items Manufacturing

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Service Sector Variable Interpretations The optimal EOQ when purchased items are received in partial shipments The optimal EOQ when manufactured items cannot all be produced in a single day Manufacturing Qp*

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Production Order Quantity Model Cycle Chart Production Order Quantity Model Cycle Chart Maximum Inventory Level ( IMAX ) Replenishment Rate ( P ) Consumption Rate ( D ) Time Cycle Charts enhance understanding of basic inventory concepts Graphically depicts the relationship between

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SAW TOOTH VERSION Production Order Quantity Model Cycle Chart MAXIMUM INVENTORY LEVEL AVERAGE INVENTORY LEVEL P PDD CONSUMPTION RATE ONLY REPLENISHMENTRATE UNITS ELAPSED TIME CONSUMPTION OCCURS EVEN AS REPLENISHMENT IS TAKING PLACE 0

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Production Order Quantity Model Cycle Chart MAXIMUM INVENTORY LEVEL AVERAGE INVENTORY LEVEL P PDD CONSUMPTION RATE ONLY REPLENISHMENTRATE 0 ELAPSED TIME An EOQ of 100 units is delivered at the rate of 20 units per week over five weeks INVENTORY LEVEL PEAKS AT THIS POINT INVENTORY FALLS TO ZERO OR REORDER POINT UNITS SAW TOOTH VERSION

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CYCLE CHART DISCUSSION Pp The replenishment rate ( P or p ) is diminished Dd by the consumption rate ( D or d ). Average inventory will always be less than [ Q* / 2 ]. IMAX / 2 Average inventory is essentially [ IMAX / 2 ].

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Production Order Quantity Formula 2DS H [ 1 – d / p ] THE CONSUMPTION OR USAGE RATE ( D ) THE REPLENISHMENT OR PRODUCTION RATE ( P ) THE FINITE CORRECTION FACTOR PRODUCES A LARGER VALUE OF Q* IN ORDER TO COMPENSATE FOR PIECEMEAL REPLENISHMENT AND CONSUMER DEMAND Q p * = √

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Production Order Quantity Model EXAMPLE D A = 1000 units ( annual demand ) S = $10.00 ( cost per order ) H = $.50 ( annual unit carry cost ) P = 8 units ( daily supply rate ) D = 6 units ( daily usage rate ) How many units should be ordered at a time? How long to receive the entire order?

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Production Order Quantity Model SOLUTION 2(1000)(10.00).50 [ 1 – 6 / 8 ] 20, [.25 ] Q p * = √ = √ = 160,000 = 400 units √

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The Basic EOQ Model SOLUTION 2(1000)(10.00).50 20, Q* = √ = √ = 40,000 = 200 units √ Q* IS ONLY HALF OF WHAT IT WAS UNDER THE PRODUCTION ORDER QUANTITY MODEL

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Production Order Quantity Model EXAMPLE Q p * 400 P 8 ORDER RECEIPT TIME PERIOD ( t ) t === 50 days

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Production Order Quantity Model EXAMPLE TOTAL VARIABLE COSTS ( TVC ) TVC = Q*p 2 X 1 - D P X H+ D Q*p X S = X X.50+ 1, X = [ 200 x (.25)(.50) ] + [ 2.5 x ] = [ ] + [ ] = $50.00

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Post Solution Comments The Qp* must be larger than Q* since it is being drawn down even as it arrives on a piecemeal basis. The larger Qp* produces no increase in carry costs. Annual carry costs are actually less than they are under the basic EOQ model ( Q* ). “P” and “D” can be expressed as daily, weekly, monthly, and annual figures without changing the value of the finite correction factor [1-d/p]

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Production Order Quantity Model

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WE SELECT THE “INVENTORY” MODULE

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WE SELECT THE “PRODUCTION ORDER QUANTITY MODEL”

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THE DIALOGUE BOX ALLOWS US TO INSERT A TITLE FOR THIS PROBLEM

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THE DATA INPUT TABLE APPEARS

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ANNUAL DEMAND = 1,000 UNITS ORDER COST = $10.00 UNIT CARRY COST = $.50 DAILY REPLENISHMENT RATE = 8 UNITS DAILY CONSUMPTION RATE = 6 UNITS

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Optimal Order Quantity Total Variable Costs

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Total Variable Cost Annual Carry Cost Annual Order Cost

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Production Order Quantity Model

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Template and Sample Data

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Sensitivity Analysis also shows the optimal solution

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The Backorder Inventory Model USED WHENEVER THE FIRM IS APPEALING TO ITS CUSTOMERS TO BUY AND THEN WAIT FOR THEIR PURCHASES UNTIL A NEW SHIPMENT ( Q* ) IS ORDERED AND RECEIVED

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Backorder Model Variables B Backorder cost – B S* Optimal number of backorders – S* Optimal order quantity under a backordering Q b * scenario – Q b * Number of units going into stock after all b Q b * - S* backorders have been filled – b or [ Q b * - S* ]

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Backorder Model Cycle Chart Optimal Order Quantity, Optimal Number of Backorders, Remaining Units Going Into Stock after Backorders Have Been Filled Graphically depicts the relationship between Cycle charts enhance understanding of basic inventory concepts

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Backorder Model Cycle Chart PICKET FENCE VERSION Q* or EOQ b = Q*- S* S* 0 INITIAL CYCLE 2 CYCLE 3 CYCLE 4 POSITIVE INVENTORY NEGATIVEINVENTORY

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Backorder Model Cycle Chart PICKET FENCE VERSION Q* or EOQ = 92 units b = Q*- S* = 88 units S* = 4 UNITS 0 INITIAL CYCLE 2 CYCLE 3 CYCLE 4 POSITIVE INVENTORY NEGATIVEINVENTORY ASSUME AN EOQ OF 92 UNITS 88 UNITS GO INTO INVENTORY AFTER THE FOUR STOCKOUTS HAVE BEEN FILLED THE FIRM ALLOWS BACKORDERS TO REACH 4 UNITS THIS MODEL ASSUMES THAT THE NEW EOQ IS ORDERED WHEN BACKORDERS EQUAL FOUR UNITS

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Backorder Model Formula OPTIMAL ORDER QUANTITY IN BACKORDER-TOLERATED SITUATIONS UNIT BACKORDER COST H ANNUAL CARRY COST PER UNIT

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Backorder Model Formula OPTIMAL ORDER QUANTITY UNDER BACKORDERING UNIT BACKORDER COST THE OPTIMAL NUMBER OF BACKORDERS

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The Backorder Model EXAMPLE D A = 500 units ( annual demand ) S = $4.00 ( order cost ) H = $.50 ( annual unit carry cost ) B = $10.00 ( unit backorder cost ) How many units should be ordered? What are the number of backorders?

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The Backorder Model SOLUTION 2(500)(4.00) ( ) = 8000 x 1.05 = 8400 ≈ 92 units Q b * = √ √ √ X

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The Backorder Example SOLUTION S* = x = ( x.0476 ) ≈ 4 units

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Relationship Between ROP and S* IF LEADTIME IS ZERO, THE REORDER POINT OCCURS WHEN THE OPTIMAL NUMBER OF BACKORDERS IS REACHED If leadtime ( L ) = 0, then ROP = S*

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Relationship Between ROP and S* IF LEADTIME IS NOT ZERO, THE REORDER POINT OCCURS BEFORE THE OPTIMAL NUMBER OF BACKORDERS IS REACHED If leadtime ( L ) > 0, then ROP > S*

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Backorder Reorder Point EXAMPLE Given: L = 0 days Q b * = 92 units S * = 4 units Order 92 units when the number of backorders accumulate to 4

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Backorder Reorder Point EXAMPLE Given: d = 2 units, L = 6 days, Q b * = 92 units, S* = 4 units Order 92 units when there are 8 units still left in the account balance. 12 units THE INITIAL ROP = d x L = [ 2 units x 6 days ] = 12 units 8 units THE FINAL ROP = Initial ROP – S* = [ 12 units – 4 units ] = 8 units

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ROP under Backordering EXAMPLE WHERE L IS POSITIVE S* = 4 units ( - 4 ) NEGATIVEINVENTORYREGION POSITIVE INVENTORY REGION Qb*Qb* Qb*Qb* 0 FINAL ROP 12 – 4 = +8 INITIAL ROP = 12 UNITS

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THE BACKORDER MODEL MAY OR MAY NOT BE AVAILABLE ON YOUR PARTICULAR SOFTWARE

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Inventory Control Part II Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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The ABC Classification System Its purpose is to assist inventory specialists in establishing policies that focus their limited resources on the relatively few critical materials, components, and products...and not the many trivial ones. Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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ABC Classification System Purchasing personnel are relatively few in number in the firm. There are thousands of components, materials, and finished good inventory accounts in medium and large firms. There needs to be a priority system for establishing and updating inventory control doctrines ( Q* / R ). RATIONALE

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Class “A” Inventory Items LARGE APPLIANCES LARGE APPLIANCES AUTOMOBILES FURNITURE DIAMONDS Comprise only 15% of the total items in stock yet represent 70% - 80% of the total dollar volume.

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Class “B” Inventory Items MID-SIZED APPLIANCES MID-SIZED APPLIANCES LAWN MOWERS MOST SUITS & COATS Comprise 30% of the total items in stock and represent 15% - 25% of the total dollar volume

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Class “C” Inventory Items TOASTERS / BLENDERS TOASTERS / BLENDERS NUTS, BOLTS, SCREWS STATIONERY SUPPLIES MOST ACCESSORIES Comprise 55% of the total items in stock yet represent only 5% of the total dollar volume

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Policies Based On ABC items would be inventoried in a more secure area “A” items would be inventoried in a more secure area

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Policies Based On ABC items would be inventoried in a more secure area “A” items would be inventoried in a more secure area “A” items would warrant more care in forecasting

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Policies Based On ABC items would be inventoried in a more secure area “A” items would be inventoried in a more secure area “A” items would warrant more care in forecasting “A” item records would be verified more frequently

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Policies Based On ABC items would be inventoried in a more secure area “A” items would be inventoried in a more secure area “A” items would warrant more care in forecasting “A” item records would be verified more frequently “A” items would justify closer attention to customer service levels service levels

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Policies Based On ABC “A” items would be inventoried in a more secure area “A” items would warrant more care in forecasting “A” item records would be verified more frequently “A” items would justify closer attention to customer service levels “A” items would qualify for real-time inventory tracking systems and more sophisticated ordering rules

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Additional Criteria for ABC Anticipated engineering changes Delivery problems Quality problems High unit production costs WE FOCUS ON THE RELATIVELY FEW ITEMS WITH MAJOR PROBLEMS

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ABC Analysis

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WE FIRST SCROLL TO “INVENTORY”

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WE THEN SELECT THE SUB MENU “ABC Analysis”

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THE DIALOG BOX APPEARS

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WE HAVE FIVE ( 5 ) ITEMS THAT NEED TO BE CLASSIFIED

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THE DATA INPUT TABLE

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FIVE ITEMS, TOGETHER WITH THEIR ANNUAL DEMANDS, AND UNIT VALUES WE DESIRE 20% OF ALL ITEMS BE “A” CATEGORY WE DESIRE 30% OF ALL ITEMS BE “B” CATEGORY

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One “A” Item One “B” Item Three “C” Items

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ABC Analysis

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Template and Sample Data

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ABC Classification System Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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Reorder Point Models Known Stockout Cost Model Service Level Model Variable Demand / Constant Lead Time Model Constant Demand / Variable Lead Time Model Variable Demand / Variable Lead Time Model Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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Reorder Point Models The original reorder point formula ROP = d x L computes the mean demand during lead time, that is, the average demand during the waiting period for the item. Where d = daily demand L = lead time L = lead time ( in days ) ( in days )

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Reorder Point Models However, actual demand during lead time can be much higher than the mean (average) demand. For this reason, the reorder point should contain a built-in safety stock ( SS or B ) that will meet unexpectedly higher demands and consequently reduce stockout costs.

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Reorder Point Models The reorder point formula now becomes ROP = d x L + SS But larger safety stocks involve higher carry costs. We must find a safety stock that minimizes both carry costs and expected stockout costs annually

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Reorder Point Cost Tradeoff ANNUALSAFETY ( BUFFER ) STOCKCARRYCOSTS ANNUALEXPECTED STOCKOUT COSTS REORDER POINT TOTAL COSTS 0 Safety ( Buffer ) Stock in Units Cost KNOWN STOCKOUT COST MODEL ∞ THE SELECTED SAFETY ( BUFFER ) STOCK LEVEL ( SS or B )

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Known Stockout Cost Model Stockout cost per unit is known. Lead time is known and constant. Lead time demand is variable. IT IS ALSO ASSUMED THAT THE ANNUAL NUMBER OF ORDERING PERIODS IS KNOWN ( n ) OF ORDERING PERIODS IS KNOWN ( n ) ASSUMPTIONS

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Demand During Lead Time ( D L ) 15% 25% 30% 20% 10% DEMAND IN UNITS Probability LEAD TIME DEMAND IS APPROXIMATELY NORMALLY DISTRIBUTED AND RANGES BETWEEN THIRTY AND SEVENTY UNITS EXAMPLE

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Demand During Lead Time ( D L ) 30% 15% 25% 30% 20% 10% DEMAND IN UNITS Probability SINCE THERE IS A 30% CHANCE THAT LEAD TIME DEMAND WILL BE FIFTY ( 50 ) UNITS WE WOULD AT LEAST SET THE REORDER POINT AT 50 UNITS. OTHERWISE, WE ARE EXTREMELY VULNERABLE TO LARGE AND RECURRING STOCKOUTS. THE HIGHEST PROBABILITY

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Demand During Lead Time ( D L ) 15% 25% 30% 20% 10% DEMAND IN UNITS Probability THE SAFETY OR BUFFER STOCK LEVEL AT THE MINIMUM REORDER POINT IS ZERO BY DEFINITION SAFETY STOCK IS ZERO HERE

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Reorder Point Carry Cost X = ANNUAL UNIT CARRY COST SAFETY OR BUFFER STOCK in units ANNUAL SAFETY STOCK CARRY COSTS

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Reorder Point of 50 Units ANNUAL SAFETY STOCK CARRY COSTS ‘0’ Safety Stock x $5.00 per unit = $0.00 per year carry cost ( BUFFER STOCK )

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Reorder Point of 50 Units 30% 15% 25% 30% 20% 10% ReorderPoint Probability Actual Demand 10 Stockouts 20 Stockouts A DEMAND OF 60 UNITS WOULD CREATE A STOCKOUT OF 10 UNITS WITH A 20% PROBABILITY A DEMAND OF 70 UNITS WOULD CREATE A STOCKOUT OF 20 UNITS WITH A 10% PROBABILITY EXPECTED STOCKOUTS PER ORDER PERIOD EXPECTED STOCKOUTS 2 + 2

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Reorder Point Stockout Cost X X = Lead Time Expected Stockouts in units Annual Number of Lead Time Periods ( ordering periods ) Stockout Cost per unit Annual Expected Stockout Costs

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Reorder Point of 50 Units ANNUAL EXPECTED STOCKOUT COSTS NUMBER OF STOCKOUTS EXPECTED STOCKOUTS STOCKOUT COST ( per unit ) NUMBER OF ORDER PERIODS ( per year ) 10 2$ COST $ $40.006$ Σ = $ Lead Time Demand of 60 Lead Time Demand of 70

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Reorder Point Total Cost ANNUAL SAFETY STOCK CARRY COSTS + ANNUAL EXPECTED STOCKOUT COSTS THE REORDER POINT THAT HAS THE LOWEST TOTAL COST IS SELECTED

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Reorder Point Score Board REORDER REORDERPOINT ANNUAL CARRY COSTS ANNUAL STOCKOUT COSTS TOTAL COSTS 50$0.00$960.00$960.00

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Reorder Point of 60 Units ANNUAL SAFETY STOCK CARRY COSTS ‘10’ Safety Stock x $5.00 per unit = $50.00 per year carry cost ( BUFFER STOCK ) RAISING THE REORDER POINT TO 60 UNITS AUTOMATICALLY CREATES A SAFETY STOCK OF TEN UNITS

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Reorder Point of 60 Units 20% 15% 25% 30% 20% 10% ReorderPoint Probability Actual Demand 10 Stockouts A DEMAND OF 70 UNITS WOULD CREATE A STOCKOUT OF 10 UNITS WITH A 10% PROBABILITY EXPECTED STOCKOUT 1 EXPECTED STOCKOUTS PER ORDER PERIOD

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Reorder Point of 60 Units ANNUAL EXPECTED STOCKOUT COSTS NUMBER OF STOCKOUTS EXPECTED STOCKOUTS STOCKOUT COST ( per unit ) NUMBER OF ORDER PERIODS ( per year ) 10 1$ COST $ Σ = $ LEAD TIME DEMAND of 70

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Reorder Point Score Board REORDERPOINTANNUAL CARRY COSTS ANNUAL STOCKOUT COSTS TOTAL COSTS 50 Units$0.00$ Units$50.00$240.00$290.00

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Reorder Point of 70 Units ANNUAL CARRY COSTS ‘20’ Safety Stock x $5.00 per unit = $ per year carry cost ( BUFFER STOCK ) RAISING THE REORDER POINT TO 70 UNITS AUTOMATICALLY CREATES A SAFETY STOCK OF 20 UNITS

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Reorder Point of 70 Units 15% 25% 30% 20% 10% ReorderPoint Probability 0 Stockouts BASED ON HISTORICAL DEMAND DATA, NO DEMAND SHOULD OCCUR THAT CREATES A STOCKOUT EXPECTED STOCKOUTS ZERO EXPECTED STOCKOUTS PER ORDER PERIOD

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Reorder Point of 70 Units ANNUAL EXPECTED STOCKOUT COSTS NUMBER OF STOCKOUTS EXPECTED STOCKOUTS STOCKOUT COST ( per unit ) NUMBER OF ORDER PERIODS ( per year ) 0 0$ COST $0.00 Σ = $0.00 LEAD TIME DEMAND of 70

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Reorder Point Score Board REORDER REORDERPOINTANNUAL CARRY COSTS ANNUAL STOCKOUT COSTS TOTAL COSTS 50 Units$0.00$ Units$50.00$240.00$ Units$100.00$0.00$100.00

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The Conclusion The lowest cost option: ROP = 70 units SS = 20 units

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Known Stockout Cost Model

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WE SCROLL TO “INVENTORY” TO FIND THE KNOWN STOCKOUT COST REORDER POINT MODEL

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WE SELECT THE “Reorder Point / Safety Stock” ( Discrete Distribution ) SUB MENU

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THE DIALOG BOX APPEARS

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WE HAVE FIVE ( 5 ) DISCRETE DEMANDS DURING LEAD TIME

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THE DATA INPUT TABLE

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THE COMPLETED DATA INPUT TABLE

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THE KNOWN STOCKOUT COST REORDER POINT MODEL Reorder Point = 70 UNITS Safety Stock = 20 UNITS Reorder Point = 50 UNITS Reorder Point = 60 UNITS Reorder Point = 70 UNITS

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Known Stockout Cost Model

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Template and Sample Data Insert Selected Reorder Point, etc. This is the conditional payoff matrix for this problem. The strategies are the safety stocks, the events are the five possible demands, and the conditional payoffs are the cost consequences (expected stockouts + carry costs) of selecting a particular safety stock and a certain demand materializing.

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Selected Reorder Point 5 discrete demand possibilities during any given lead time ( with probabilities ) Safety stocks associated with all 5 discrete demands For ROP = 50, total cost = $960.00, H = $ 0.00 For ROP = 60, total cost = $290.00, H = $ For ROP = 70, total cost = $100.00, H = $ Conditional Payoffs EMV Criterion Maxi-Min Criterion

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Service Level Model FOR EXAMPLE, THE FIRM MAY DESIRE A SERVICE LEVEL THAT MEETS 95% OF THE DEMAND, OR CONVERSELY, RESULTS IN STOCKOUTS ONLY 5% OF THE TIME. REORDER POINT DETERMINATION When the stockout cost per unit ( Cs ) is difficult or impossible to determine, the firm may elect to establish a policy of keeping enough safety stock on hand to satisfy a prescribed level of customer service.

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Service Level Model REORDER POINT DETERMINATION SALES DATA ARE USUALLY ADEQUATE FOR COMPUTING THE MEAN AND STANDARD DEVIATION Assuming that the demand during lead time ( D L ) follows a normal distribution, only the mean ( µ ) and standard deviation ( σ ) are required to define the reorder point as well as the safety stock ( SS or B ) for any given service level.

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Service Level Model REORDER POINT DETERMINATION EXAMPLE A firm stocks an item that has a normally-distributed demand during the lead time period. The average or mean demand during the lead time period is 400 units and the standard deviation is 15 units. The firm wants a reordering policy that limits stockouts to only 5% of the time. Requirement: 1. How much safety stock should the firm maintain? 2. What should be the reorder point?

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Service Level Model AREAS UNDER THE STANDARD NORMAL CURVE Z 95% SERVICE LEVEL IS REPRESENTED BY z = 1.65 standard normal deviates

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Service Level Model ROP 95% STOCKAGE PROBABILITY 5% STOCKOUT PROBABILITY SAFETY STOCK Z = μ LEAD TIME MEAN DEMAND = 400 UNITS 0 units REORDER POINT AND SAFETY STOCK FOR 95% SERVICE LEVEL

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Service Level Model FORMULAE Reorder Point ( R ) = μ + ( z )( σ ) Safety Stock ( SS ) = ( z )( σ ) μ = MEAN DEMAND DURING LEAD TIME z = NUMBER OF STANDARD NORMAL DEVIATES σ = STANDARD DEVIATION OF LEAD TIME DEMAND

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Service Level Model EXAMPLE – 95% SERVICE LEVEL Reorder Point ( R ) = ( 1.65 )( 15 ) = 425 units Safety Stock ( SS ) = ( 1.65 )( 15 ) = 25 units μ = 400 units z = 1.65 ( 95% service level ) σ = 15 units

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Service Level Model EXAMPLE – 99% SERVICE LEVEL Reorder Point ( R ) = ( 2.33 )( 15 ) = 435 units Safety Stock ( SS ) = ( 2.33 )( 15 ) = 35 units μ = 400 units z = 2.33 ( 99% service level ) σ = 15 units Z

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Service Level Reorder Point Model

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TO FIND THE SERVICE LEVEL REORDER POINT, SCROLL TO “ INVENTORY “

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SELECT THE SUB MENU ENTITLED “ Reorder Point / Safety Stock “ ( Normal Distribution )

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THE DIALOG BOX APPEARS

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THE DATA INPUT TABLE REQUIRES - DAILY DEMAND DURING LEAD TIME - THE STANDARD DEVIATION OF DAILY DEMAND DURING LEAD TIME DEMAND DURING LEAD TIME - THE SERVICE LEVEL DESIRED ( i.e. 95% ) - LEAD TIME IN DAYS - THE STANDARD DEVIATION OF LEAD TIME IF THE DAILY DEMAND AND/OR LEADTIME ARE CONSTANT, THEN THEIR STANDARD DEVIATION(S) = 0

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The mean demand during lead time was given as 400 units. Since this model requires both daily demand during the lead time, and the lead time ( in days ), we will assume here that daily demand = 50 units, and lead time = 8 days.

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THE 95% “SERVICE LEVEL” REORDER POINT = 423 UNITS THE 95% “SERVICE LEVEL” SAFETY STOCK = 23 UNITS

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Service Level Reorder Point Model

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Templates for three ( 3 ) different reorder point models. ( we choose the first model )

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The Reorder Point = = 425 !

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Stochastic Reorder Point Models These models generally assume that any variability in either the demand rate or lead time can be adequately described by a normal distribution. This, however, is not a strict requirement. These models will provide approximate reorder points even in cases where the actual probability distributions depart substantially from normal. In all models shown, stockout costs are assumed to be unknown. WHEN DEMAND AND / OR LEAD TIME ARE VARIABLE

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Stochastic Reorder Point Models MODELS CONSIDERED IN THIS PRESENTATION I. variable demand rate / constant lead time II. constant demand rate / variable lead time III. variable demand rate / variable lead time

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Stochastic Reorder Point Models THE VARIABLES d = constant demand rate d = average demand rate L = constant lead time L = average lead time σ D = standard deviation of demand rate σ L = standard deviation of lead time

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Stochastic Reorder Point Models Jack’s Pizza Parlor uses 1,000 cans of tomatoes per month at an average rate of 40 per day for each of 25 days per month. Usage can be approximated by a normal distribution with a standard de- viation of 3 cans per day. Lead time is constant at 4 days. Jack desires a service level of 99%, that is, a stockout risk of only 1% Requirement: 1. Determine the reorder point ( ROP ) 2. Determine the safety or buffer stock ( SS or B ) LEAD TIME IS CONSTANT DEMAND RATE IS VARIABLE

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Reorder Point Solution Given: d = 40 cans daily σ D = 3 cans daily L = 4 days _ ROP = ( d x L ) + ( z ) ( L ) ( σ D ) _ = ( 40 x 4 ) ( 4 ) ( 3 ) = ( 6 ) = = 174 cans PIZZA PARLOR

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Safety Stock Solution SS or B = ( z ) ( L ) ( σ D ) = 2.33 ( 4 ) ( 3 ) = 2.33 ( 6 ) ≈ 14 cans PIZZA PARLOR

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Reorder Point Where Lead Time Is Constant Demand Rate Is Variable

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Template for pizza parlor reorder point

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Reorder point = 174 Safety stock = 14

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Stochastic Reorder Point Models An oil-driven generator uses 2.1 gallons per day. Lead time is normally distributed with a mean of 6 days. The standard deviation of lead time is 2 days. The service level is 98%, that is, the stockout risk is 2% Requirement: 1. Determine the reorder point ( ROP ) 2. Determine the safety or buffer stock ( SS or B ) LEAD TIME IS VARIABLE DEMAND RATE IS CONSTANT

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Reorder Point Solution Given: d = 2.1 gallons daily σ L = 2 days L = 6 days ROP = ( d x L ) + ( z ) ( d ) ( σ L ) _ = ( 2.1 x 6 ) ( 2.1 ) ( 2 ) = ( 4.2 ) = = gallons _ THE GENERATOR

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Safety Stock Solution SS or B = ( z ) ( d ) ( σ L ) = ( 2.1 ) ( 2 ) = ( 4.2 ) = 8.63 gallons THE GENERATOR

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Reorder Point Where Lead Time Is Variable Demand Rate Is Constant

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Stochastic Reorder Point Models Beer consumption at a local tavern is known to be normally distributed with a mean of 150 bottles daily and a standard deviation of 10 bottles daily. Delivery time is also normally distributed with a mean of 6 days and a standard deviation of 1 day. The service level is 90% Requirement: 1. Determine the reorder point ( ROP ) 2. Determine the safety or buffer stock ( SS or B ) LEAD TIME IS VARIABLE DEMAND RATE IS VARIABLE

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Reorder Point Solution Given: d = 150 bottles daily σ D = 10 bottles daily L = 6 days σ L = 1 day _ TAVERN EXAMPLE _

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Reorder Point Solution ROP = ( d x L ) + ( z ) ( L )( σ D ) + ( d ) ( σ L ) ____ 222 = ( 150 x 6 ) + ( 1.28 ) (6)(10) + (150) (1) 222 = ,500 = ( ) ≈ 1,095 bottles TAVERN EXAMPLE

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Safety Stock Solution SS or B = ( z ) ( L )( σ D ) + ( d ) ( σ L ) _ _ 222 = (1.28 ) (6)(10) + (150) (1) 222 = ,500 = 1.28 ( ) ≈ 195 bottles TAVERN EXAMPLE

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Stochastic Reorder Point Model Demand and Lead Time Variable

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TO SOLVE A STOCHASTIC REORDER POINT AND SAFETY STOCK PROBLEM

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THE DIALOG BOX APPEARS

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THE DATA INPUT TABLE REQUIRES DAILY DEMAND AND STANDARD DEVIATION ( if any ) LEAD TIME ( DAYS ) AND STANDARD DEVIATION ( if any )

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STOCHASTIC MODEL REORDER POINT = 1,095 UNITS SAFETY STOCK = 195 UNITS THE LOCAL TAVERN

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Stochastic Reorder Point Model Demand Variable Lead Time Variable

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Reorder Point Models Inventory Control Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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Solved Problem Quantity Discount Model Ivonne Callen Computer-BasedManual

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Ivonne Callen Problem Ivonne Callen sells beauty supplies. Her annual demand for a particular skin lotion is 1,000 units. The cost of placing an order is $20.00, while the holding cost per unit per year is 10 percent of the cost. The item currently costs $10.00 if the order quantity is less than 300. For orders of 300 units or more, the cost falls to $9.80. To minimize total cost, how many units should Ivonne order each time she places an order? What is the minimum total cost?

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Ivonne Callen Problem D = 1,000 units S = $20.00 I = 10% P = $10.00 if Q < 300 units P = $ 9.80 if Q > 300 units

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Ivonne Callen Problem To minimize total costs, how many units should Ivonne order each time she places an order ? What is the minimum total cost ?

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EOQ at $10.00 Item Cost Q* 1 = 2 x D x S I x P √ 2 (1,000 ) ( 20 ) (.10 ) ( ) √ = = √ 40, = 200 units

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Total Cost at $10.00 Item Cost TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ] = [ 200 / 2 ] (.10)(10.00) + [ 1,000 / 200 ] (20.00) + [1,000 x 10.00] = $ $ $10, = $10,200.00

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Total Cost at $9.80 Item Cost TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ] = [ 300 / 2 ] (.10)(9.80) + [ 1,000 / 300 ] (20.00) + [1,000 x 9.80] = $ $ $9, = $10,013.67

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Ivonne Callen Problem Therefore, she should order 300 units. CONCLUSION

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Solved Problem Serial Rate Production Model The Handy Manufacturing Company Computer-BasedManual

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The Handy Mfg. Company The Handy Manufacturing Company manufactures small air conditioner compressors. The estimated demand for the year is 12,000 units. The setup cost for the production process is $ per run, and the carrying cost is $10.00 per unit per year. The daily production rate is 100 units per day, and demand has been 50 units per day. Determine the number of units to produce in each batch, the number of batches that should be run each year, and the time interval, in days, between each batch. ( Assume 240 operating days. )

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The Data D = 12,000 units S = $ H = $10.00 / unit / year p = 100 units / day d = 50 units / day 240 working days / year Use Production Order Quantity Model

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Production Run Model Qp* = √ 2 (D)(S) H ( 1 - d / p ) = √ 2 (12,000)(200.00) (10.00)( / 100 )

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Production Run Model Qp* = √ 4,800,000 (10.00)(.5) = √ 4,800,000 5 = √ 960,000 = units

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Batches Run Annually n = D / Q*p n = 12,000 / 980 n ≈ 12 production runs

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Time Between Runs t = 240 days / n = 240 days / 12 = every 20 days

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Solved Problem Reorder Point Model Mr. Beautiful Computer-BasedManual

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Mr. Beautiful Problem Mr. Beautiful, an organization that sells weight training sets, has an ordering cost of $40.00 for the BB-1 set. The carry cost for the BB-1 set is $5.00 per set per year. To meet demand, Mr. Beautiful orders large quantities of BB-1 (7) seven times a year. The stockout cost for BB-1 is estimated to be $50.00 per set. Over the past several years, Mr. Beautiful has ob- served the following demand during lead time for BB-1: Lead Time Demand Probability Starting with a ROP of 60 units, what level of safety stock should be maintained for BB-1?

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Mr. Beautiful Problem ROP Expected Stockouts [.20 x 10] + [.20 x 20] + [.10 x 30] = 9 sets 10 sets 20 sets 30 sets

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Mr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7 At ROP = 60 sets, SS = 0 sets Stockout Cost: [.20 x 10 sets +.20 x 20 sets +.10 x 30 sets] x 7 orders x $50.00 [ ] x 7 x $50.00 = $3, Carrying Cost: SS = 0 sets x $5.00 / set / year = $0.00 TOTAL COST = $3,150.00

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Mr. Beautiful Problem ROP Expected Stockouts [.20 x 10] + [.10 x 20] = 4 sets 10 sets 20 sets

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Mr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7 At ROP = 70 sets, SS = 10 sets Stockout Cost: [.20 x 10 sets +.10 x 20 sets] x 7 orders x $50.00 [ ] x 7 x $50.00 = $1, Carrying Cost: SS = 10 sets x $5.00 / set / year = $50.00 TOTAL COST = $1,450.00

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Mr. Beautiful Problem ROP Expected Stockouts [.10 x 10] = 1 set 10 sets

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Mr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7 At ROP = 80 sets, SS = 20 sets Stockout Cost: [.10 x 10 sets] x 7 orders x $50.00 [ 1 ] x 7 x $50.00 = $ Carrying Cost: SS = 20 sets x $5.00 / set / year = $ TOTAL COST = $450.00

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Mr. Beautiful Problem ROP Expected Stockouts [.00 x 0 ] = 0 sets

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Mr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7 At ROP = 90 sets, SS = 30 sets Stockout Cost: [.0 x 0 sets] x 7 orders x $50.00 [ 0 ] x 7 x $50.00 = $0.00 Carrying Cost: SS = 30 sets x $5.00 / set / year = $ TOTAL COST = $150.00

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Mr. Beautiful Problem Reorder Point Stockout Cost Carry Cost Total Cost 60 sets$3,150.00$0.00$3, sets$1,400.00$50.00$1, sets$350.00$100.00$ sets$0.00$ Mr. Beautiful should select a ROP = 90 sets with a SS = 30 sets

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We Scroll Down To INVENTORY

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WE SELECT THE SUB MENU “Reorder Point / Safety Stock” ( Discrete Distribution )

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THE DIALOG BOX APPEARS

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THERE ARE SIX ( 6 ) DEMANDS THAT CAN OCCUR DURING LEADTIME

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THE DATA INPUT TABLE REQUIRES: - STARTING REORDER POINT WHERE THE SAFETY STOCK IS ZERO ( 0 ) THE SAFETY STOCK IS ZERO ( 0 ) - CARRY COST PER UNIT PER YEAR - STOCKOUT COST PER UNIT PER YR - NUMBER OF ORDERS PER YEAR ( 7 ) THE DISCRETE PROBABILITY DISTRIBUTION OF LEAD TIME DEMAND

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THE ROP = 90 SETS THE SS = 30 SETS Expected Stockout Cost = $0.00 Carry ( Holding ) Costs = $150.00

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