Inventory systems

Inventory SystemInventory System

INVENTORY SYSTEMINVENTORY SYSTEMInventoryInventory is the stock of any item or

resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process

An inventory systeminventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be

InventoryInventory is the stock of any item or resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process

An inventory systeminventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be

INVENTORY COSTSINVENTORY COSTS

Holding (or carrying) costsCosts for storage, handling,

insurance, etcSetup (or production change) costs

Costs for arranging specific equipment setups, etc

Ordering costsCosts of someone placing an order,

etcShortage costs

Costs of canceling an order, etc

Holding (or carrying) costsCosts for storage, handling,

insurance, etcSetup (or production change) costs

Costs for arranging specific equipment setups, etc

Ordering costsCosts of someone placing an order,

etcShortage costs

Costs of canceling an order, etc

E(1)

Independent vs. Dependent Demand

Independent vs. Dependent Demand

Independent Demand (Demand for the final end-product or demand not related to other items)

Dependent Demand

(Derived demand items for

component parts,

subassemblies, raw materials,

etc)

Finishedproduct

Component parts

Inventory SystemsInventory SystemsSingle-Period Inventory Model

One time purchasing decision (Example: vendor selling t-shirts at a football game)

Seeks to balance the costs of inventory overstock and under stock

Multi-Period Inventory ModelsFixed-Order Quantity Models

Event Triggered (Example: running out of stock)

Fixed-Time Period Models Time triggered (Example: Monthly sales call by sales representative)

Single-Period Inventory ModelOne time purchasing decision

(Example: vendor selling t-shirts at a football game)

Seeks to balance the costs of inventory overstock and under stock

Multi-Period Inventory ModelsFixed-Order Quantity Models

Event Triggered (Example: running out of stock)

Fixed-Time Period Models Time triggered (Example: Monthly sales call by sales representative)

Fixed-order quantity modelFixed-order quantity model

also called economic order quantity (EOQ)

an inventory control model where the amount requisitioned is fixed and the actual ordering is triggered by inventory dropping to a specified level of inventory

also called economic order quantity (EOQ)

an inventory control model where the amount requisitioned is fixed and the actual ordering is triggered by inventory dropping to a specified level of inventory

Fixed-Order Quantity Model Assumptions


Demand for the product is constant and uniform throughout the period

Lead time (time from ordering to receipt) is constant

Price per unit of product is constant

Demand for the product is constant and uniform throughout the period

Lead time (time from ordering to receipt) is constant

Price per unit of product is constant



Inventory holding cost is based on average inventory

Ordering or setup costs are constant

All demands for the product will be satisfied (No back orders are allowed)

Inventory holding cost is based on average inventory

Ordering or setup costs are constant

All demands for the product will be satisfied (No back orders are allowed)

Basic Fixed-Order Quantity Model and Reorder Point

Behavior

Basic Fixed-Order Quantity Model and Reorder Point

Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. Your start using them up over time. 3. When you reach down to

a level of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

Cost Minimization GoalCost Minimization Goal

Ordering Costs

HoldingCosts

Order Quantity (Q)

COST

Annual Cost ofItems (DC)

Total Cost

QOPT

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

Basic Fixed-Order Quantity (EOQ) Model

Formula

Basic Fixed-Order Quantity (EOQ) Model

Formula

H 2

Q + S

Q

D + DC = TC H

2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

Deriving the EOQDeriving the EOQUsing calculus, we take the first derivative

of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Q = 2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPTQ =

2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPT

Reorder point, R = d L_

Reorder point, R = d L_

(constant) timeLead = L

(constant) demanddaily average = d_

We also need a reorder point to tell us when to place an order

We also need a reorder point to tell us when to place an order

EOQ Example (1) Problem DataEOQ Example (1) Problem Data

Annual Demand = 1,000 unitsDays per year considered in average

daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15

Given the information below, what are the EOQ and reorder point?

Given the information below, what are the EOQ and reorder point?

EOQ Example (1) SolutionEOQ Example (1) Solution

Q = 2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 unitsQ =

2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 units

d = 1,000 units / year

365 days / year = 2.74 units / dayd =

1,000 units / year

365 days / year = 2.74 units / day

Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

EOQ Example (2) Problem DataEOQ Example (2) Problem Data

Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15

Determine the economic order quantity and the reorder point given the following…

Determine the economic order quantity and the reorder point given the following…

EOQ Example (2) SolutionEOQ Example (2) Solution

Q =2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 unitsQ =

2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 units

d =10,000 units / year

365 days / year= 27.397 units / dayd =

10,000 units / year

365 days / year= 27.397 units / day

R = d L = 27.397 units / day (10 days) = 273.97 or _

274 unitsR = d L = 27.397 units / day (10 days) = 273.97 or _

274 units

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Economic Production Quantity (EPQ)

Economic Production Quantity (EPQ)

Production done in batches or lotsCapacity to produce a part

exceeds the part’s usage or demand rate

Assumptions of EPQ are similar to EOQ except orders are received incrementally during production

Production done in batches or lotsCapacity to produce a part

exceeds the part’s usage or demand rate

Assumptions of EPQ are similar to EOQ except orders are received incrementally during production

Assumptions:Assumptions:

Only one item involvedAnnual demand is knownThe usage rate is constantUsage occurs continually, but

production occurs periodicallyThe production rate is constantLead time does not varyThere are no quantity discounts

Only one item involvedAnnual demand is knownThe usage rate is constantUsage occurs continually, but

production occurs periodicallyThe production rate is constantLead time does not varyThere are no quantity discounts

Economic Production Quantity Model

Economic Production Quantity Model

Formulas:Formulas:No. of runs per year = D/QAnnual setup cost = (D/Q)STCmin = Carrying Cost + Setup Cost = (

)H+(D/Qo)S

Qo = ; where p = production or delivery rate

u = usage rateCycle Time = Run Time = Imax = (p-u) and

Iaverage =

No. of runs per year = D/QAnnual setup cost = (D/Q)STCmin = Carrying Cost + Setup Cost = (

)H+(D/Qo)S

Qo = ; where p = production or delivery rate

u = usage rateCycle Time = Run Time = Imax = (p-u) and

Iaverage =

Example 1Example 1

A toy manufacturer uses 48,000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. The firm operates 240 days per year.

A toy manufacturer uses 48,000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. The firm operates 240 days per year.

Determine thea.Optimal run size.b.Minimum total annual cost for

carrying and setup.c.Cycle time for the optimal run size.d.Run time.

Determine thea.Optimal run size.b.Minimum total annual cost for

carrying and setup.c.Cycle time for the optimal run size.d.Run time.

Given:Given:

D = 48,000 wheels per yearS = $45H = $1 per wheel per yearp = 800 wheels per dayu = 48,000 wheels per 240 days, or 200 wheels per day

D = 48,000 wheels per yearS = $45H = $1 per wheel per yearp = 800 wheels per dayu = 48,000 wheels per 240 days, or 200 wheels per day

a. Qo =

Qo = = 2,400 wheels

b. TCmin = ( )H+(D/Qo)SThus, you must first compute Imax:

Imax = (p-u) = (800-200) = 1,800 wheels

TC = ( )($1) + ( )($45) = $1,800

Solution:

c. Cycle Time = = = 12 days

Thus, a run of wheels will be made every 12 days.

d. Run Time = = = 3 days

Thus, each run will require 3 days to complete.

Example 2Example 2

The Dine Corporation is both a producer & a user of brass couplings. The firm operates 220 days a year & uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 200 per day. Annual storage cost is $2 per coupling, & machine setup cost is $70 per run.

a.Determine the economic run quantity.b.Approximately how many runs per year will there be?c.Compute the maximum inventory level.

Solution

a. Qp = =

≈ 1,013 units

b. No. of runs = D = 11,000 = 10.86 ≈ 11per year Q0 1,013

c. Imax = Qp (p - u) = 1,013 (200 - 50)

pImax = 759.75 or 760 units

Quantity Discounts• Quantity Discounts, also called Price-Break

Models, are price reductions for large orders offered to costumers to induce them to buy in large quantities.

• The buyer must weigh the potential benefits of reduced purchase price and fewer orders that will result from buying in large quantities against the increase in carry costs caused by higher average inventories.

Quantity Discounts

• The buyer’s goal with quantity discounts is to select the order quantity that will minimize total cost.

• Total cost is the sum of carrying cost, ordering cost and purchasing cost.

Price-Break Model FormulaPrice-Break Model Formula

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

iC

2DS = QOPT

Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:

i = percentage of unit cost attributed to carrying inventoryC = cost per unit

Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

Total Cost = Carrying Cost + Ordering Cost + Purchasing Cost

TC = (Q/2)H + (D/Q)S + PD

Where:Q = Quantity P = Unit PriceD = DemandS = Ordering Cost

Example1Example1

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 1.004,000 or more .98

SolutionSolution

units 1,826 = 0.02(1.20)

4)2(10,000)( =

iC

2DS = QOPT

Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4

First, plug data into formula for each price-break value of “C”

units 2,000 = 0.02(1.00)

4)2(10,000)( =

iC

2DS = QOPT

units 2,020 = 0.02(0.98)

4)2(10,000)( =

iC

2DS = QOPT

Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98

Interval from 0 to 2499, the Qopt value is feasible

Interval from 2500-3999, the Qopt value is not feasible

Interval from 4000 & more, the Qopt value is not feasible

Next, determine if the computed Qopt values are feasible or not

SolutionSolutionSince the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

0 1826 2500 4000 Order Quantity

Total annual costs

So the candidates for the price-breaks are 1826, 2500, and 4000 units

So the candidates for the price-breaks are 1826, 2500, and 4000 units

Because the total annual cost function is a “u” shaped function

Because the total annual cost function is a “u” shaped function

SolutionSolution

iC 2

Q + S

Q

D + DC = TC iC

2

Q + S

Q

D + DC = TC

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break


TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

Example2Example2

Consider the following case, where

D = 10,000 units (annual demand)

i = 20% of cost (annual carrying cost, storage, interest, obsolescence, etc.)

S = $20 to place an order

C = Cost per unit (according to order size; 0-499 units, $5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9 per unit)

Consider the following case, where

D = 10,000 units (annual demand)

i = 20% of cost (annual carrying cost, storage, interest, obsolescence, etc.)

S = $20 to place an order

C = Cost per unit (according to order size; 0-499 units, $5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9 per unit)

What quantity should be ordered?What quantity should be ordered?

SolutionSolution

units 632.46 = 0.2(5)

20)2(10,000)( =

iC

2DS = QOPT

Annual Demand (D)= 10,000 unitsCost to place an order (S)= $20

First, plug data into formula for each price-break value of “C”

units 666.67 = 0.2(4.50)

20)2(10,000)( =

iC

2DS = QOPT

units 716.11 = 0.2(3.90)

20)2(10,000)( =

iC

2DS = QOPT

Carrying cost % of total cost (i)= 20%Cost per unit (C) = $5, $4.5, $3.9

Interval from 0 to 499, the Qopt value is feasible

Interval from 500-999, the Qopt

value is not feasible

Interval from 1000 & more, the Qopt value is not feasible

Next, determine if the computed Qopt values are feasible or not

SolutionSolution

iC 2

Q + S

Q

D + DC = TC iC

2

Q + S

Q

D + DC = TC



TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5) = $50,632.46TC(500-999)= $45,600TC(1000&more)= $39,558.57

TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5) = $50,632.46TC(500-999)= $45,600TC(1000&more)= $39,558.57

Finally, we select the least costly Qopt, which in this problem occurs in the 1000 & more interval. In summary, our optimal order quantity is 1000 units

Finally, we select the least costly Qopt, which in this problem occurs in the 1000 & more interval. In summary, our optimal order quantity is 1000 units

A-B-C ClassificationA-B-C ClassificationClassifies Inventory items according

to some measure of importance, Usually annual dollar value, and then allocates control efforts accordingly

Classifies Inventory items according to some measure of importance, Usually annual dollar value, and then allocates control efforts accordingly

Three Classes of ItemsThree Classes of ItemsA (Very Important Items)

Account for 10%-20% of the number of items in inventory 60%-70% of the dollar inventory of an inventory

B (Moderately Important) Account for 20%-30% of the number of items in inventory 20%-50% of the dollar inventory of an inventory

C (Least Important) Account for 50%-60% of the number of items in inventory 10%-15% of the dollar inventory of an inventory

A (Very Important Items) Account for 10%-20% of the number of items in inventory 60%-70% of the dollar inventory of an inventory

B (Moderately Important) Account for 20%-30% of the number of items in inventory 20%-50% of the dollar inventory of an inventory

C (Least Important) Account for 50%-60% of the number of items in inventory 10%-15% of the dollar inventory of an inventory

A typical A-B-C breakdown in relative annual dollar value of items and number of items by category.

A typical A-B-C breakdown in relative annual dollar value of items and number of items by category.

ExampleExample

The annual dollar value of 12 items has been calculated according to annual demand and unit cost. The annual dollar values were then arrayed from highest to lowest to simplify classification of items.

The annual dollar value of 12 items has been calculated according to annual demand and unit cost. The annual dollar values were then arrayed from highest to lowest to simplify classification of items.

Item Number

Annual Demand

Unit Cost Annual Dollar Value

Classification

8 $1,000 $4,000 $4,000,000 A

5 $3,900 $700 $2,730,000 A

3 $1,900 $500 $950,000 B

6 $1,000 $915 $915,000 B

1 $2,500 $330 $825,000 B

4 $1,500 $100 $150,000 C

12 $400 $300 $120,000 C

11 $500 $200 $100,000 C

9 $8,000 $10 $80,000 C

2 $1,000 $70 $70,000 C

7 $200 $210 $42,000 C

10 $9,000 $2 $18,000 C

Inventory systems

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