Prof. Upendra Kachru Topic 3 Inventory Models Operations Management (2)

Prof. Upendra Kachru

Topic 3

Inventory Models

OperationsManagement (2)

All organizations have inventory Can be a sizable organizational

asset Influences sales (revenue

generation and customer relations)

Influences production/ operations costs

Large amounts reduces ROI Costs of having inventory Frequently the largest single

expenditure Excesses can result in losses


WHY STUDY INVENTORY?

Inventory is the stock of any item or resource used in an organization.


WHAT IS INVENTORY?

Working Stock Safety Stock Anticipation Stock Pipeline Stock Decoupling Stock Psychic Stock


Inventory Categories

Functions ofInventory


Functions

To protect against variations (fluctuations) in demand and supply

To take advantage of batches and longer production runs

To facilitate intermittent production

To provide flexibility to allow changes in production plans in view of changes in demand etc.

To take advantage of price discounts by bulk purchases

To meet anticipated and current demand

Functions of Inventory

To smooth production requirements from seasonality

Capacity Requirement

without Inventory

Capacity Requirement with

Inventory

Excess Demand from Inventory

J F M A M J J A S O N D

Dem

and

Excess Production to

Inventory

Functions of Inventory

Department 1 Department 2 Department 3 Department 4

Interstage

Inventory D1

Interstage

Inventory D2

Interstage

Inventory D3

To decouple different components of the internal inventory-distribution system.


INVENTORY FUNCTIONS

Finished Goods

(Production)

In-Process Goods

(Production)

Raw Materials & Supplies

(Purchasing)

Supply

(Marketing)

Demand

Anticipation Stock

Decoupling Stock

To meet anticipated demand

Psychic Stock

Safety Stock

Significance of Inventory

FunctionalArea

Functional Responsibility

Inventory GoalInventoryInclination

Marketing

Sell the ProductMaximize customer service

High

Production Make the Product Efficient lot sizes High

Purchasing

Buy required materials Low cost per unit High

Finance

Provide working capital Efficient use of capital Low

Engineering

Design the product Avoid obsolescence Low



Area

Marketing / Sales

Production

Purchasing

Finance

Warehousing

Typical Response

I can’t sell without adequate stocks. I can’t keep our customers if we continue to stockout and there is not sufficient product variety

If I can produce larger lot sizes, I can reduce per unit cost and function efficiency.

I can reduce our per unit cost if I buy large quantities in bulk.

Where am I going to get the funds to pay for the inventory? The levels should be lower.

I am out of space. I can’t fit anything else in the building.

Marketing revenue generation customer relations

Production efficiencycost of operations cost of operations

Financeliquidity return on investment

LogisticsHandling CapacitySpace


Inventory Objectives

MaximizeCustomer

Service

OperatingEfficiency

MinimizeInventory

Investment

Balancing Objectives1. Provide customer service

2. Support plant efficiency

3. Minimize inventory investment


INVENTORY PLANNING

AND CONTROL

INPUTS OUTPUTS

CONSTRAINTSMgt. policies

Working capitalSpace

Plant capacity

DECISION RULES1. What to order?2. When to order?3. How much?4. From whom?

OPERATIONS PLANNINGForecastsDemand ratesProduction ratesStock-on-handBackordersLead timesProduct structures

PurchaseOrder / Set-up

HoldingStockout

COSTS

Inventory – PLANNING & CONTROL


Inventory Costs

Inventory Costs are additive


Inventory Costs• Holding (or carrying) costs• Ordering Costs/ Setup (or

production change) costs• Shortage or Stock-out Costs


Stock-out Costs

External Shortage1. Present Profit Loss (potential sales)

2. Backorder Costs3. Future Profit Loss (goodwill erosion)

Internal Shortage1. Lost Production (idle people / machines)

2. Substitute Cost (alternate)3. Overtime / Extra Shift Cost4. Delay Project Completion Date

Average Inventory Investment: The rupee value of a company’s average level of inventory is one of the most common measures of inventory.

Inventory Turnover Ratio: It is a ratio that measures how many times during a year the inventory turns around.

Inventory Metrics


Inventory turnover = annual cost of goods sold/average inventory investment

Days of Inventory: This measure is an indication of approximately how many days of sales can be supplied solely from inventory.

Inventory Metrics


Days of inventory = avg. inventory investment/ (annual cost of gods sold/days per year)

Days of inventory = days per year/ inventory turnover rate

The inventory of a medium sized business organization would comprise thousands of items, each item with different usage, price, lead time and specifications. There could be different procurement and technical problems associated with different items.

In order to escape this quagmire many selective inventory management techniques are used.

Inventory Control by Classification Systems


The ABC classification is based on focusing efforts where the payoff is highest; i.e. high-value, high-usage items must be tracked carefully and continuously.

Typically only 20 percent of all the items account for 80 percent of the total rupee usage, while the remaining 80 percent of the items typically account for remaining 20 percent of the rupee value.

The large value items constitute only 20 percent, the ABC analysis makes the task relatively easier.


Vilfredo Pareto’s 80-20 rule.

TYPICAL ABC INVENTORY ANALYSIS

A = HIGH VALUE ITEMSB = MEDIUM VALUE ITEMSC = LOW VALUE ITEMS


0

20

40

60

80

100

120

0 20 40 60 80 100

CBA

PE

RC

EN

T O

F T

OTA

L D

OL

LA

R U

SA

GE

PERCENT OF TOTAL ITEMS


TYPICAL ABC INVENTORY ANALYSIS

40

20

0

20

40

60

80

60

A

BCP

ER

CE

NT

OF

R

UP

EE

VA

LU

EP

ER

CE

NT

OF

IT

EM

S

RELATIVE ANALYSIS OF ABC CLASSIFICATIONS


Item Degree of

Control

Type of Records Lot Sizes Frequency of

Review

Size of Safety

Stocks

A Tight Accurate / Complete Low Continuous Small

B Moderate Good Medium Occasional Moderate

C Loose Simple Large Infrequent Large

1. Difficult Procurement Items

2. Short Shelf Life

3. Large Storage Space Requirements

4. Item’s Operational Criticality

5. Likelihood of Theft

6. Difficult Forecast Items


ABC EXCEPTIONS

Other Classification Systems


Title Basis Main Uses

ABC (Level of Usage) Value of consumptionraw material components and work-in progress inventories

HML (High, medium, low usage)

Unit price of the material Mainly to control purchase.

FSND (Fast, Slow moving, Non moving, Dead )

Consumption pattern of the component

Control obsolescence.

SDE (Scarce, difficult, easy to obtain items)

Problems faced in procurement

Lead time analysis and purchasing strategies

GOLF (Government, Ordinary, Local, Foreign)

Source of the material Procurement strategies

VED (Vital, Essential, (Desirable)

Criticality of the component

To determine the stocking levels of spare parts.

SOS (Seasonal, Off-seasonal)

Nature of suppliersSeasonal items like agriculture products

XYZ ( Value of Stock) Value of items in storageTo review the inventories and their use scheduled intervals.

Inventory systems are predicated on whether demand is derived from an end item or is related to the item itself.

There are two types of models that are used in the case of independent demand:

◦ Single Period Models, and◦ Multiple Period Models.


INVENTORY MODELS

E(1)

Independent Demand

Dependent Demand

subassemblies, raw materials, etc)

Finishedproduct

Component parts

SINGLE PERIOD MODELS



Single-Period Inventory Models are a special case of periodic inventory systems.

One time purchasing decision (Example: vendor selling food at Siababa temple)

Seeks to balance the costs of inventory overstock and under stock

It is used for a wide variety of service and manufacturing applications

Single-Period Inventory

Model

After prayers at the Siababa temple on Thursdays, people go to a vendor to eat food. The vendor has collected data over a few months that show, on an average, 100 meals were sold with a standard deviation of 10 meals.

If our vendor wants to be 90 percent sure of not running out of food each Thursday, how many meals should he prepare?


Problem and Solution

If we assume that the distribution is normal and the vendor prepared food for exactly 100 persons, the risk of food running out would be 50 percent. The demand would be expected to be less than 100 meals 50 percent of the time, and greater than 100 the other 50 percent.

To be 90 percent sure of not falling short, he needs to prepare more food.

From the “standard normal distribution“, we can find out that he needs to have additional food to cover 1.282 standard deviations.

In order to ensure that he is 90 percent sure having sufficient food:The number extra food required would be 1.282 x 100 = 128.2, or 129 meals.

In order to ensure that he is 90 percent sure having sufficient food:The number extra food required would be 1.282 x 100 = 128.2, or 129 meals.

z

f(z)F(0.9)= ŷ+1.282σ

z*

If Co = Cost per unit of demand overage, Cu = Cost per unit of demand underage, The probability that the unit will be sold is ‘P’;


Single-Period Inventory Model

The expected marginal cost equation can be represented as:

P * Co < (1-P) * CuSolving for P, we obtainP < [Cu / (Co +Cu)]

Here (1-P) is the probability of the service/ product not being sold.

A newspaper vendor is faced with the problem of deciding how many newspapers to order daily so as to maximize the daily profit.

Daily demand (d) for newspapers is a random variable.

No reordering is possible during a day, ◦ If the newsvendor orders fewer papers than customers

demand he or she will lose the opportunity to sell some papers.

If supply exceeds demand, the vendor will be stuck with papers which cannot be sold.


The Classical Newsvendor’s Problem

Based on observations over several weeks, the vendor has established the following probability distribution of daily demand:

The vendor purchases daily papers at Rs.2 and sells them at Rs. 5 apiece. Leftover papers are valueless and are discarded (i.e. no salvage value).


Demand Data

Demandd

ProbabilityP(d)

Cumulative Prob.F(d) = P(D d)

35 or less36373839404142434445

46 or more

0.000.050.070.080.150.150.200.150.100.030.020.00

0.000.050.120.200.350.500.700.850.950.981.001.00

The vendor identifies two penalty costs which he/she will incur, regardless of his/her decision:

Cost of Overage

CO = Purchase Price - Salvage Value = c - s

For each paper overstocked the newsvendor incurs a penalty cost of:

CO = Rs. 2.00 – Rs.0.00 = Rs. 2.00

Cost of Underage

CU = Selling Price - Purchase Price = p - c

For each paper understocked the newsvendor incurs a penalty (opportunity) cost of:

CU = Rs. 5.00 – Rs. 2.00 = Rs. 3.00


Assume that there is already a policy in place to order a certain number of papers daily, say 38.

Consider the decisions:

D1 : Continue the present policy: Stock 38 papers.

D2 : Order one more paper: Stock 39 papers.

The possible events are:

E1 : The 39th paper sells (i.e. demand 39 = demand > 38).

E2 : The 39th paper does not sell (i.e. demand 39 = demand 38).


To Stock orNot to Stock!

Item 39 will not sell on a given day only if demand on that day is for 38 or fewer items:


P(D 38) = F(38) = 0.20.

The probability that an item will not sell is the cumulative probability associated with the previous item. Item 39 will sell on a given day only if demand on that day is for 39 or more items:

P(D 39) = 1 - P(D 38) = 1 - F(38) = 1 - 0.2 = 0.80.

The expected payoff is:Rs. 3(0.8) + (- Rs. 2)(0.2) = Rs. 2.

This implies an increase in profit of Rs. 2.00 as compared to the alternative decision which has a payoff of Rs. 0.00. He should stock the 39th paper.

This implies an increase in profit of Rs. 2.00 as compared to the alternative decision which has a payoff of Rs. 0.00. He should stock the 39th paper.

Inventory Management Game


Product Price Sell Don’t Sell (Overage)

Stockout (Underage)

Burger 18.00 7.00 9.00 11.00

Pizza 23.00 12.00 23.00 17.00

Patties 8.00 4.00 4.00 6.00

Samosa 6.00 2.00 6.00 3.00

Sandwich 10.00 5.00 5.00 7.00

Pastry 7.00 3.00 3.50 5.00

Hotdog 11.00 4.00 11.00 6.00

MULTI PERIOD INVENTORY MODELS


◦ Fixed-Order Quantity Models: Event triggered (Example: running out of stock)

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


Multi-Period Inventory Models


Time T1 T2

Inve

nto

ry L

eve

l

‘Q

’

Fixed order Quantity and Fixed-Time Period Differences

Feature Fixed-order quantity Model Fixed-Time Period Model

Order quantity The same amount ordered each time

Quantity varies each time order is placed

When to place order Reorder point when inventory position dips to a predetermined level

Reorder when the review period arrives

Record keeping Each time a withdrawal or addition is made

Counted only at review period.

Size of inventory Less than fixed-time period model Larger than fixed-order quantity model

Time to maintain Higher due to perpetual record keeping

Type of items Higher-priced, critical, or important items.


Fixed Order Quantity ModelsEconomic Order Quantity (EOQ) models, due to simplicity and versatility, are fixed order quantity models used for material planning.

When independent demand is the most important issue, the EOQ model provides a solution to the problem.


The Inventory Cycle

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Placeorder

Lead time

Reorderpoint

Receive order

Receive order

Usage rate

Time

The inventory cycle determines when an order should be placed and how much should be ordered so as to minimize average annual variable costs.


Prof. Upendra KachruProf. Upendra Kachru 41

The basic assumptions in the EOQ Model are as follows: The rate of demand for the item is deterministic

and is a constant ‘D’ units per annum independent of time.

Lead time is zero or constant and it is independent of both demand as well as the quantity ordered.

Price per unit of product is constant Inventory holding cost is based on average

inventory Ordering or setup costs are constant

The EOQ Model


COST MINIMIZATION GOAL

42

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 average inventory level:

The holding cost per unit:

The setup cost per unit:

The production cost per unit:

2

Q

D

HQ

D

rvQ

22

Q

A

P


BASIC FIXED-ORDER QUANTITY (EOQ) MODEL FORMULA

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =DemandP =Cost per unitQ =Order quantityA =Cost of placing an order or setup costR =Reorder pointL =Lead timeH = v*r =Annual holding and storage cost per unit of inventory

TC=Total annual costD =DemandP =Cost per unitQ =Order quantityA =Cost of placing an order or setup costR =Reorder pointL =Lead timeH = v*r =Annual holding and storage cost per unit of inventory

TC = P*D + D*A / Q + Q*v*r / 2


The EOQ

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual

H

2DA =

rv

2DA = QOPT

Reorder point, R = d L_

d = average daily demand (constant)

L = Lead time (constant)

_

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


A company, for one of its class ‘A’ items, placed 8 orders each for a lot of 150 numbers, in a year. Given that the ordering cost is Rs. 5,400.00, the inventory holding cost is 40 percent, and the cost per unit is Rs. 40.00. Find out if the company is making a loss in not using the EOQ Model for order quantity policies.

What are your recommendations for ordering the item in the future? And what should be the reorder level, if the lead time to deliver the item is 6 months?

‘D’ = Annual demand = 8*150 = 1200 units

‘v’ = Unit purchase cost = Rs. 40.00

‘A’ = Ordering Cost = Rs. 5400.00

‘r’ = Holding Cost = 40%

EOQ Model Problem


Using the Economic Order Equation:

QEOQ = √ (2*A*D /r*v) = 900 units.

Minimum Total Annual Cost (TC) = √ 2*A*D*r*v = Rs. 14,400.00

The Total annual Cost under the present system = Rs. 45,000.00

The loss to the company = Rs. 45,000 – Rs. 14,400 = Rs. 30,600.00

Reorder Level = Ro = L*D = (6/12)* 1200 = 600 units

The company should place orders for economic lot sizes of 900 units in each order.

It should have a reorder level at 600 units.

TC= = √ 2*5400*1200*0.40*40

QEOQ = √ (2*5400*1200)/(0.40*40)

= Rs. (1200*5400/150 + 0.40*40*150/2) = Rs. (43,800 + 1200)

Total Costs with Purchasing Cost

Co

st

EOQ

TC with Purchasing Cost

TC without Purchasing Cost

Purchasing Cost

0 Quantity

Adding Purchasing costdoesn’t change EOQ


Total Cost with Constant Carrying Costs

OC

EOQ Quantity

Tota

l Co

st

TCa

TCc

TCbDecreasing Price

CC a,b,c


ProblemNovelty Ltd carries a wide assortment of items for its customers. One item, Gaylook, is very popular. Desirous of keeping its inventory under control, a decision is taken to order only the optimal economic quantity, for this item, each time. You have the following information. Make your recommendations:

◦ Annual demand : 1,60,000 units◦ Price per unit : Rs.20◦ Carrying cost : Re.1 per unit or 5 per cent◦ Cost per order : Rs. 50

Determine the optimal economic quantity.



Order per year

Size Average inventory

Carrying cost (Re.1)

Ordering cost (Rs.50

per order)

Total cost per year

1 1,60,000 80,000 80,000 50 80,000

10 16,000 8,000 8,000 500 8,500

40 8,000 4,000 4,000 1,000 5,000

80 4,000 2,000 2,000 2,000 4,000

100 2,000 2,000 1,000 4,000 5,000

1,600 800 800 5,000 5,800

Solution

The optimum economic quantity (lot size) for this item is 4,000 numbers.

What-if

Show that changing the order quantity by a small amount has very little effect on the cost.



Quantity Discounts

Quantity discounts, which are price incentives to purchase large quantities, create pressure to maintain a large inventory.

For any per-unit price level, P, the total cost is:

Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials

C = (H) + (A) + PDQ2

DQ

Total cost curves with purchased materials added

Quantity Discounts

EOQs and price break quantities

C for P = Rs.4.00C for P = Rs.3.50C for P = Rs.3.00

PD forP = Rs.4.00 PD for

P = Rs.3.50 PD forP = Rs.3.00

EOQ 4.00

EOQ 3.50

EOQ 3.00

First price break

Second price break

Tota

l cos

t (R

upee

s)

Tota

l cos

t (R

upee

s)

Purchase quantity (Q)0 100 200 300

Purchase quantity (Q)0 100 200 300

First price break

Second price break


Step 1. Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its

price. Step 2. If the first feasible EOQ found is for the

lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible

EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.

Finding Q with Quantity Discounts


Annual demand (D) = 936 unitsOrdering cost (A) = Rs. 45

Holding cost (H) = rv = 25% of unit price

Order Quantity Price per Unit

0 – 299 Rs. 60.00300 – 499 Rs. 58.80500 or more Rs. 57.00

A supplier for Apollo Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is:

Problem


EOQ 57.00 =2DS

H2(936)(45)0.25(57.00)

= = 77 units

EOQ 58.80 =2DS

H2(936)(45)0.25(58.80)

= = 76 units

EOQ 60.00 =2DS

H2(936)(45)0.25(60.00)

= = 75 units

This quantity is feasible because it lies in the range corresponding to its price.

Not feasible

Not feasible

Feasible

Step 1: Start with lowest price level:

= Rs. 56,999

Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels (Rs.58.80 and Rs.57.00):

C = (rv) + (A) + PDQ2

DQ

C75 = [(0.25)(Rs. 60.00)] + (Rs. 45) + Rs. 60.00(936)752

93675

C75 = Rs. 57,284

C300 = [(0.25)(Rs. 58.80)] + (Rs. 45) + Rs. 58.80(936)300

2936300

= Rs. 57,382

C500 = [(0.25)(Rs.57.00)] + (Rs.45) + Rs. 57.00(936)500

2936500

The best purchase quantity is 500 units, which qualifies for the deepest discount.


© 2007 Pearson Education

Decision Point:If the price per unit for the range of 300 to 499 units is reduced to Rs. 58.00, the best decision is to order 300 catheters.


This shows that the decision is sensitive to the price schedule. A reduction of slightly more than 1 percent is enough to make the difference in this example.

Annual Demand 936Ordering Cost Rs. 45.00Holding Cost 25%

Price EOQ Inventory Cost Order Cost Purchase Cost Total Cost

Rs. 60.00 75 Rs. 562.50 Rs. 562.50 Rs. 56,160 Rs. 57,284

Rs. 58.00 300 Rs. 2175 Rs. 140.40 Rs. 54,288 Rs. 56,603

Rs. 57.00 500 Rs. 3563 Rs. 84.24 Rs. 53,352 Rs. 56,999

Lower unit cost Higher holding costs

Lower ordering costs Larger inventory investment

Fewer stockouts Older stock

Price increase hedge Slow inventory turnover

QUANTITY DISCOUNTS

Advantages Disadvantages

In many retail merchandising systems, a fixed-time period system is used. Sales people make routine visits to customers and take orders. Inventory, therefore, is counted only at particular times.

Fixed-time period models generate order quantities that vary from period to period, depending on the usage rates.

Fixed-Time Period Models


Fixed Period Model


IT L)(dQ

L = Lead Time

usage period Averaged T = Time between orders

I = Existing Inventory

Q = Order Size

IT LddQ

A T d Quantity Order Average

Total Annual Cost = Purchase Cost + Ordering Cost + Holding Cost

Q/2)(H A/Q)DDPCT (


Order Quantity = Average demand over the vulnerable period + safety stock - Inventory currently on hand

LT zILTQ )(dWhere

z = Number of standard deviations for a specified service probability

σT + L= Standard deviation of demand over the review and lead time

Accounting for Safety Stock:

SS IT L)(dQ

Fewer orders are placed

Purchase discounts more likely

Lower shipping and freight costs


Fixed Time Period System - Advantages

Consumes capital Requires storage space Incurs taxes Requires insurance Can become lost, stolen,

damaged, outdated, or obsolete

Must be counted, sorted, verified, stored, retrieved, moved, issued, and protected


Ever - increasing storage space needs

Slow-moving materials Disposition of scrap, obsolete, &

surplus materials Transaction recording errors Misplaced materials


Classical Inventory Prblems


RAW MATERIALS INVENTORY PROFILE

Excess StockSurplus / Idle

Working Stock

Safety Stock

Outputs

Inputs

Nonproductive

Productive


IN-PROCESS INVENTORY PROFILE

In-ProcessInventory

Unreleased Orders

Orders in TransitOrders in Temporary Storage

Orders Waiting to be Worked

Orders Being Inspected

Orders Being Worked

Backlog

Nonproductive

Productive

Outputs

Inputs

Finished Goods

1. Standardize Stock Items

2. Reduce Lead Times

3. Reduce Cycle Times

4. Use Fewer Suppliers

5. Inform Suppliers of Expected Demand

6. Contract for Minimum Annual Purchases

7. Buy on Consignment

8. Consider Transportation Costs

9. Order Economical Quantities

10. Control Access to Storage Areas

11. Obtain Better Forecasts


Inventory System Improvement

12. Dispose of Excess Stock

13. Improve Record Accuracy (cycle count)

14. Improve Capacity Planning

15. Minimize Setup Times

16. Simplify Product Structures

17. Multishift operations

18. Continuous Improvement


Inventory System Improvement

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Prof. Upendra Kachru Topic 3 Inventory Models Operations Management (2)

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