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**Inventory Control Models**

Part II Applied Management Science for Decision Making, 1e © 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 OR USED WHENEVER THE FACTORY CANNOT PRODUCE THE ORDER ( Q* ) ALL IN ONE DAY

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**Variable Interpretations**

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

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**Variable Interpretations**

Service Sector Manufacturing Qp* Qp* 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

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**Production Order Quantity Model Cycle Chart**

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

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**Production Order Quantity Model Cycle Chart**

SAW TOOTH VERSION REPLENISHMENT RATE MAXIMUM INVENTORY LEVEL CONSUMPTION RATE ONLY UNITS P D P D CONSUMPTION OCCURS EVEN AS REPLENISHMENT IS TAKING PLACE AVERAGE INVENTORY LEVEL ELAPSED TIME

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**Production Order Quantity Model Cycle Chart**

SAW TOOTH VERSION REPLENISHMENT RATE MAXIMUM INVENTORY LEVEL CONSUMPTION RATE ONLY INVENTORY LEVEL PEAKS AT THIS POINT UNITS P D P D An EOQ of 100 units is delivered at the rate of 20 units per week over five weeks AVERAGE INVENTORY LEVEL INVENTORY FALLS TO ZERO OR REORDER POINT ELAPSED TIME

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**CYCLE CHART DISCUSSION**

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

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**Production Order Quantity Formula**

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

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**Production Order Quantity Model EXAMPLE**

DA = 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 ] Qp* = √ 20,000 .50 [ .25 ] √ = = 160,000 = 400 units

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**The Basic EOQ Model SOLUTION**

√ 2(1000)(10.00) .50 Q* = √ √ 20,000 .50 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**

ORDER RECEIPT TIME PERIOD ( t ) Qp* P t = = = 50 days

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**Production Order Quantity Model EXAMPLE**

TOTAL VARIABLE COSTS ( TVC ) Q*p D D 1 - TVC = X H + X S X 2 P Q*p 400 6 1,000 X 1 - X .50 + X = 8 2 400 = [ 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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**DAILY REPLENISHMENT RATE = 8 UNITS DAILY CONSUMPTION RATE = 6 UNITS**

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**

Backorder cost – B Optimal number of backorders – S* Optimal order quantity under a backordering scenario – Qb* Number of units going into stock after all backorders have been filled – b or [ Qb* - 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* POSITIVE INVENTORY INITIAL CYCLE CYCLE 2 CYCLE 3 CYCLE 4 NEGATIVE INVENTORY S*

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**Backorder Model Cycle Chart PICKET FENCE VERSION**

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

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**Backorder Model Formula**

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

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**Backorder Model Formula**

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

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**The Backorder Model EXAMPLE**

DA = 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) ( ) Qb* = X √ = x 1.05 = ≈ 92 units √

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**The Backorder Example SOLUTION**

.50 S* = x = ( x ) ≈ 4 units

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**Relationship Between ROP and S***

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

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**Relationship Between ROP and S***

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

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**Backorder Reorder Point EXAMPLE**

Given: L = 0 days Qb* = 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, Qb* = 92 units, S* = 4 units Order 92 units when there are 8 units still left in the account balance. THE INITIAL ROP = d x L = [ 2 units x 6 days ] = 12 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 Qb* Qb* POSITIVE INVENTORY REGION FINAL ROP FINAL ROP 12 – 4 = +8 12 – 4 = +8 INITIAL ROP = 12 UNITS INITIAL ROP = 12 UNITS NEGATIVE INVENTORY REGION S* = 4 units ( - 4 )

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**MAY OR MAY NOT BE AVAILABLE ON YOUR PARTICULAR SOFTWARE**

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**

RATIONALE 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 ).

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**Class “A” Inventory Items**

Comprise only 15% of the total items in stock yet represent 70% - 80% of the total dollar volume. LARGE APPLIANCES AUTOMOBILES FURNITURE DIAMONDS

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**Class “B” Inventory Items**

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

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**Class “C” Inventory Items**

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

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

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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

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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

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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

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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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**ITEMS THAT NEED TO BE CLASSIFIED**

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**

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

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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 Variable Demand / Variable 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 ( 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**

KNOWN STOCKOUT COST MODEL REORDER POINT TOTAL COSTS ANNUAL SAFETY ( BUFFER ) STOCK CARRY COSTS Cost THE SELECTED SAFETY ( BUFFER ) STOCK LEVEL ( SS or B ) ANNUAL EXPECTED STOCKOUT COSTS ∞ Safety ( Buffer ) Stock in Units

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**Known Stockout Cost Model**

ASSUMPTIONS 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 )

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**Demand During Lead Time ( DL )**

EXAMPLE Probability 15% % % % % DEMAND IN UNITS LEAD TIME DEMAND IS APPROXIMATELY NORMALLY DISTRIBUTED AND RANGES BETWEEN THIRTY AND SEVENTY UNITS

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**Demand During Lead Time ( DL )**

THE HIGHEST PROBABILITY Probability 15% % % % % DEMAND IN UNITS 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.

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**Demand During Lead Time ( DL )**

Probability 15% % % % % SAFETY STOCK IS ZERO HERE DEMAND IN UNITS THE SAFETY OR BUFFER STOCK LEVEL AT THE MINIMUM REORDER POINT IS ZERO BY DEFINITION

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**Reorder Point Carry Cost**

ANNUAL UNIT CARRY COST SAFETY OR BUFFER STOCK in units X ANNUAL SAFETY STOCK CARRY COSTS =

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**Reorder Point of 50 Units ‘0’ Safety Stock x $5.00 per unit = $0.00**

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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**CREATE A STOCKOUT OF 10 UNITS CREATE A STOCKOUT OF 20 UNITS**

Reorder Point of 50 Units EXPECTED STOCKOUTS PER ORDER PERIOD EXPECTED STOCKOUTS Probability 15% % % % % 10 Stockouts 20 Stockouts Reorder Point Actual Demand 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

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**Reorder Point Stockout Cost**

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

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**NUMBER OF ORDER PERIODS**

Reorder Point of 50 Units ANNUAL EXPECTED STOCKOUT COSTS STOCKOUT COST ( per unit ) NUMBER OF ORDER PERIODS ( per year ) NUMBER OF STOCKOUTS EXPECTED STOCKOUTS COST Lead Time Demand of 60 10 2 $40.00 6 $480.00 Lead Time Demand of 70 20 2 $40.00 6 $480.00 Σ = $960.00

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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**

ANNUAL CARRY COSTS ANNUAL STOCKOUT COSTS TOTAL COSTS 50 $0.00 $960.00

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**RAISING THE REORDER POINT TO 60 UNITS AUTOMATICALLY**

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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**CREATE A STOCKOUT OF 10 UNITS**

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

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**NUMBER OF ORDER PERIODS**

Reorder Point of 60 Units ANNUAL EXPECTED STOCKOUT COSTS STOCKOUT COST ( per unit ) NUMBER OF ORDER PERIODS ( per year ) NUMBER OF STOCKOUTS EXPECTED STOCKOUTS COST 10 1 $40.00 6 $240.00 LEAD TIME DEMAND of 70 Σ = $240.00

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**Reorder Point Score Board**

ANNUAL CARRY COSTS STOCKOUT COSTS TOTAL COSTS 50 Units $0.00 $960.00 60 Units $50.00 $240.00 $290.00

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**RAISING THE REORDER POINT TO 70 UNITS AUTOMATICALLY**

Reorder Point of 70 Units ANNUAL CARRY COSTS ‘20’ Safety Stock x $5.00 per unit = $100.00 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% 30 40 50 60 70 EXPECTED**

EXPECTED STOCKOUTS PER ORDER PERIOD EXPECTED STOCKOUTS ZERO Probability 15% % % % % Stockouts Reorder Point BASED ON HISTORICAL DEMAND DATA, NO DEMAND SHOULD OCCUR THAT CREATES A STOCKOUT

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**NUMBER OF ORDER PERIODS**

Reorder Point of 70 Units ANNUAL EXPECTED STOCKOUT COSTS STOCKOUT COST ( per unit ) NUMBER OF STOCKOUTS NUMBER OF ORDER PERIODS ( per year ) EXPECTED STOCKOUTS COST $40.00 6 $0.00 LEAD TIME DEMAND of 70 Σ = $0.00

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**Reorder Point Score Board**

ANNUAL CARRY COSTS STOCKOUT COSTS TOTAL COSTS 50 Units $0.00 $960.00 60 Units $50.00 $240.00 $290.00 70 Units $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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**TO FIND THE KNOWN STOCKOUT COST**

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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**DISCRETE DEMANDS DURING LEAD TIME**

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 = 50 UNITS Reorder Point = 60 UNITS Reorder Point = 70 UNITS THE KNOWN STOCKOUT COST REORDER POINT MODEL Reorder Point = 70 UNITS Safety Stock = 20 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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**5 discrete demand possibilities during any **

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

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**Service Level Model When the stockout cost per unit ( Cs ) is**

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. 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.

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**Service Level Model Assuming that the demand during lead time**

REORDER POINT DETERMINATION Assuming that the demand during lead time ( DL ) 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. SALES DATA ARE USUALLY ADEQUATE FOR COMPUTING THE MEAN AND STANDARD DEVIATION

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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 Z AREAS UNDER THE STANDARD NORMAL CURVE .03 .04**

.05 .06 .07 1.4 .92364 .92507 .92647 .92785 .92922 1.5 .93699 .93822 .93943 .94062 .94179 1.6 .94845 .94950 .95053 .95154 .95254 1.7 .95818 .95907 .95994 .96080 .96164 Z 95% SERVICE LEVEL IS REPRESENTED BY z = 1.65 standard normal deviates

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**Service Level Model μ ROP Z = + 1.65 0 units**

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

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**Service Level Model Reorder Point ( R ) = μ + ( z )( σ )**

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 Reorder Point ( R ) = 400 + ( 1.65 )( 15 )**

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 Reorder Point ( R ) = 400 + ( 2.33 )( 15 )**

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 .03

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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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**“ Reorder Point / Safety Stock “ ( Normal Distribution )**

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

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THE DIALOG BOX APPEARS

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**IF THE DAILY DEMAND AND/OR THEN THEIR STANDARD DEVIATION(S) = 0**

THE DATA INPUT TABLE REQUIRES DAILY DEMAND DURING LEAD TIME THE STANDARD DEVIATION OF DAILY 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 ( we choose the first model )**

reorder point models. ( we choose the first model )

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The Reorder Point = = 425 !

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**Stochastic Reorder Point Models**

WHEN DEMAND AND / OR LEAD TIME ARE VARIABLE 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.

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**Stochastic Reorder Point Models**

MODELS CONSIDERED IN THIS PRESENTATION variable demand rate / constant lead time constant demand rate / variable lead time 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 σD = standard deviation of demand rate σL = standard deviation of lead time

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**Stochastic Reorder Point Models**

LEAD TIME IS CONSTANT DEMAND RATE IS VARIABLE 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 )

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**Reorder Point Solution**

PIZZA PARLOR _ ROP = ( d x L ) + ( z ) ( L ) ( σD ) = ( 40 x 4 ) ( 4 ) ( 3 ) = ( 6 ) = = 174 cans Given: d = 40 cans daily σD = 3 cans daily L = 4 days _

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**Safety Stock Solution SS or B = ( z ) ( L ) ( σD ) = 2.33 ( 4 ) ( 3 )**

PIZZA PARLOR SS or B = ( z ) ( L ) ( σD ) = 2.33 ( 4 ) ( 3 ) = 2.33 ( 6 ) ≈ 14 cans

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Reorder Point Where Lead Time Is Constant Demand Rate 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**

LEAD TIME IS VARIABLE DEMAND RATE IS CONSTANT 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 )

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**Reorder Point Solution**

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

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**Safety Stock Solution SS or B = ( z ) ( d ) ( σL )**

THE GENERATOR SS or B = ( z ) ( d ) ( σL ) = ( 2.1 ) ( 2 ) = ( 4.2 ) = 8.63 gallons

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Reorder Point Where Lead Time Is Variable Demand Rate Constant

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**Stochastic Reorder Point Models**

LEAD TIME IS VARIABLE DEMAND RATE IS VARIABLE 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 )

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**Reorder Point Solution**

TAVERN EXAMPLE _ Given: d = 150 bottles daily σD = 10 bottles daily L = 6 days σL = 1 day _

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**Reorder Point Solution**

TAVERN EXAMPLE _ _ _ _ 2 2 2 ROP = ( d x L ) + ( z ) ( L )( σD ) + ( d ) ( σL ) 2 2 2 = ( 150 x 6 ) + ( 1.28 ) (6)(10) + (150) (1) = ,500 = ( ) ≈ 1,095 bottles

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**Safety Stock Solution SS or B = ( z ) ( L )( σD ) + ( d ) ( σL )**

TAVERN EXAMPLE _ _ 2 2 2 SS or B = ( z ) ( L )( σD ) + ( d ) ( σL ) 2 2 2 = (1.28 ) (6)(10) + (150) (1) = ,500 = 1.28 ( ) ≈ 195 bottles

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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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**REQUIRES DAILY DEMAND AND STANDARD DEVIATION AND STANDARD DEVIATION**

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

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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

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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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**Quantity Discount Model**

Solved Problem Quantity Discount Model Ivonne Callen Computer-Based Manual

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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?

174
**Ivonne Callen Problem D = 1,000 units S = $20.00 I = 10%**

P = $10.00 if Q < 300 units P = $ if Q > 300 units

175
**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 ?

176
**√ √ √ EOQ at $10.00 Item Cost 2 (1,000 ) ( 20 ) 2 x D x S**

( .10 ) ( ) 2 x D x S I x P √ = Q*1 = √ 40,000 1.00 = 200 units =

177
**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] = $ $ $10,000.00 = $10,200.00

178
**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] = $ $ $9,800.00 = $10,013.67

179
**Ivonne Callen Problem Therefore, she should order 300 units.**

CONCLUSION Therefore, she should order 300 units.

180
**Serial Rate Production Model The Handy Manufacturing Company**

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

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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 = $200.00 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 H ( 1 - d / p ) Qp* = 2 (12,000)(200.00) =**

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

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

960,000 = 979.8 units

185
**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-Based Manual

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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?

190
**Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets 20 sets 30 sets**

10 sets 20 sets 30 sets Expected Stockouts [.20 x 10] + [.20 x 20] + [.10 x 30] = 9 sets

191
**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 x 20 sets x 30 sets] x 7 orders x $50.00 [ ] x 7 x $50.00 = $3,150.00 Carrying Cost: SS = 0 sets x $5.00 / set / year = $0.00 TOTAL COST = $3,150.00

192
**Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets 20 sets**

10 sets 20 sets Expected Stockouts [.20 x 10] + [.10 x 20] = 4 sets

193
**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 x 20 sets] x 7 orders x $50.00 [ ] x 7 x $50.00 = $1,400.00 Carrying Cost: SS = 10 sets x $5.00 / set / year = $50.00 TOTAL COST = $1,450.00

194
**Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets**

10 sets Expected Stockouts [.10 x 10] = 1 set

195
**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 = $350.00 Carrying Cost: SS = 20 sets x $5.00 / set / year = $100.00 TOTAL COST = $450.00

196
**Mr. Beautiful Problem ROP 40 50 60 70 80 90 .10 .20 .20 .20 .20 .10**

Expected Stockouts [ .00 x 0 ] = 0 sets

197
**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 = $150.00 TOTAL COST = $150.00

198
**Mr. Beautiful Problem Reorder Point Stockout Cost Carry Total 60 sets**

$3,150.00 $0.00 70 sets $1,400.00 $50.00 $1,450.00 80 sets $350.00 $100.00 $450.00 90 sets $150.00 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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**“Reorder Point / Safety Stock” ( Discrete Distribution )**

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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**PROBABILITY DISTRIBUTION OF**

THE DATA INPUT TABLE REQUIRES: STARTING REORDER POINT WHERE 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

204
**Expected Stockout Cost = $0.00 Carry ( Holding ) Costs = $150.00**

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

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DOM 511 Inventory Control 2.

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