inventory control in operation research

7/25/2019 Inventory control in Operation Research

1/23

Dr Valentina Plekhanova

CISM02: Decision Support for Management

Unit 8

Inventory Systems

Unit 8: Learning Outcomes

2CISM02 Decision Support for Management Unit 8

1. To understand the inventory problems

2. To be able to distinguish between a variety of the major stock costs

3. To understand the nature of demands

4. To understand the simple Economic Order Quantity (EOQ) model

5. To construct an algebraic model for a simple inventory system

6. To be able to apply inventory models to the practical problems

7. To be able to explain the limitations of inventory control models


2/23

Inventory

3

Inventory: Stock of items for internal or external demand.

Tools, machinery , equipment

Raw materials

Working capital

Finished goods

Thepurpose of inventory managementis to determine the amount

of inventory to keep in stock.

CISM02 Decision Support for Management Unit 8

Inventory Management

4

The word inventory was first recorded in 1601. The French term inventaire, or

"detailed list of goods," dates back to 1415.

Inventory management is primarily about specifying the size and placement of

stocked goods.

Inventory management is required at different locations within a facility or

within multiple locations of a supply network to protect the regular and planned

course of production against the random disturbance of running out of

materials or goods.

The scope of inventory managementalso concerns the fine lines between

replenishment lead time, carrying costs of inventory, asset management,

inventory forecasting, inventory valuation, inventory visibility, future inventory

price forecasting, physical inventory, available physical space for inventory,

quality management, replenishment, returns and defective goods and demand

forecasting.



3/23

Basic Inventory Problem

5

Any inventory problem gives an answer to the following questions:

How much to order?

When to order?


Inventory Control

6

The problem of inventory exists in all companies whether big or small.

Exceeding inventory has contributed to the failure of many companies.

If stock is taken regularly to honour a demand at any time, a company

will never be out of stock and thereby lose the sales and goodwill of its

customers. But the cost of investment and keeping up the inventory

(holding cost) will increase.

Less inventory leads to frequent orders and stock out and costs

increase as a result.

The ideal inventory policy is to find out the break-even of these costs.

Also the policy should be adjustable to suit the existing system.



4/23

Buffer Stocks

7

Many organisations keep an additional amount of inventory called

safetyorbuffer stocksfor the following reasons:

Variations in demand

High seasonal demand

Variations in supplier deliveries

Volume discounts

High ordering price

etc.


Inventory Costs

8

Inventory must be sufficient to provide high quality customer service.

Inventory Costs:

Carrying costs/holding costsare the costs of holding an item in

inventory (rent, heating, cooling, lighting, security, refrigeration,

record keeping, deterioration, taxes, etc.)Ordering costsincrease/decrease with the number/size of orders

Shortage costsinsufficient inventory for customer demand



5/23

Inventory Models:Economic Parameters

9

Set-up cost: This involves the fixed charge associated with theplacement of an order or with the initial preparation of a production

facility. The set-up cost is usually assumed independent of the quantity

ordered or produced.

Purchase price or production cost: This parameter is of special interest

when quantity discounts or price breaks can be secured or when large

production runs may result in a decrease in the production cost. Under

these conditions, the order quantity must be adjusted to take advantage

of these price breaks.


Inventory Models:Economic Parameters

10

Holding cost: This represents the cost of carrying inventory in storage.

It includes the interest on invested capital, storage costs, handling

costs, depreciation costs, etc. Holding costs usually are assumed to

vary directly with the level of inventory as well as the length of time

the item is held in stock. As the inventory increases, this cost

increases.

Shortage cost or penalty cost: These are the penalty costs incurred asa result of running out of stock when the commodity is needed (i.e.

when the demand is fulfilled). They generally include costs due to

loss in customersgoodwill and potential loss in income.



6/23

Inventory Models: Terminology

11

Replenishment: An inventory system may operate with delivery lags, the

actual replenishment of stock may occur instantaneously or uniformly.

Lead Time or time lag or delivery lag: The time between placing an

order and its arrival in stock. If the lead time is known and is not equal to

zero, and if demand is known, all that one needs to do is to order in

advance by an amount of time equal to the lead time. If, however, it is a

variable that is known only probabilistically, the question of when to

order is more difficult one. If either the demand or the lead time is known

only probabilistically, the amount and timing of replenishment if found by

considering expected costs of holding and storage over the lead time

period.


Water Tank Analogy for Inventory

12

Supply Rate

Inventory Level

Demand Rate

Inventory LevelBuffers Demand Ratefrom Supply Rate



7/23

Two Forms of Demand

13

Dependent:

items used to produce final products (table top, legs, hardware, paint, etc.)Demand determined once we know the type and number of final products

Independent:items demanded by external customers (Kitchen Tables; Cars,appliances, computers, and houses are examples of independent demand

inventory)

Independent demand

Uncertain / forecasted

Continuous Review / Periodic Review

Dependent demand

Requirements/ planned

Materials Requirements Planning / Just in Time


Nature of Demand

14

Sometimes, the demand is known with certainty with its rate remaining the

same or not in relation to time. If demand is certain it is called a

deterministic demand.

Depending upon whether the rate of demand remains the same or not

with respect to time it is called staticor dynamicdemand.

Instead of a deterministic demand, sometimes a demand is expressed in

terms of probability distribution in which case it is termed a probabilistic

demand. In this case also the demand may change in relation to time.

If it does not change it is called stationary, and if it does as non-

stationary.



8/23

Reasons to Hold Inventory (1)

15

Meet variations in customer demand:Meet unexpected demandSmooth seasonal or cyclical demand

Pricing related:Temporary price discountsHedge against price increasesTake advantage of quantity discounts

Process & supply surprisesInternalupsets in parts of or our own processesExternaldelays in incoming goods

Transit


Reasons to Hold Inventory (2)

16

Meet unexpected demand

Smooth seasonal or cyclical demand

Meet variations in customer demand

Take advantage of price discounts

Hedge against price increases

Quantity discounts



9/23

Reasons to Not Hold Inventory

17

Carrying cost

Financially calculable

Takes up valuable factory space

Especially for in-process inventory

Inventory covers up problems

That are best exposed and solved


Inventory Control Systems

18

Functions of Inventory Management

Track inventory

How much to order

When to order

Prioritization (e.g. ABC classification/prioritization)

Inventory Management ApproachEOQ

Continuous / Periodic



10/23 1

Inventory Control Objectives

19

Inventory Control Objectives Twofold:

1. Maximize customer service

2. Minimize costs

Cost Objective: Minimize sum of relevant costs

Service Objective: desired customer service levels significantly impact

inventory levels.


Inventory Control Models

20

Service level may be defined in a number of ways, such as:

Inventory Control Models

Remindertwo important issues in inventory control:

order quantity and order timing.

Two general classes of models:

1. continuous review (fixed order quantity)

2. periodic review (fixed order period)



11/23

Inventory Control Systems:

Continuous & Periodic

21

In a continuous system, an order is placed for the same constant amountwhenever the inventory on hand decreases to a certain level, whereas in

a periodic system, an order is placed for a variable amount after specific

regular intervals.

Fixed-order-quantity system ( or Continuous Review or Q-systems) -

constant amount ordered when inventory declines to predetermined level

Fixed-time-period system (or Periodic review or P-systems) - order

placed for variable amount after fixed passage of time


Inventory Control Systems:Continuous

22

In a continuous inventory system (also referred to as a perpetual system

and a fixed-order-quantity system), a continual record of the inventory

level for every item is maintained. Whenever the inventory on hand

decreases to a predetermined level, referred to as the reorder point,a

new order is placed to replenish the stock of inventory.

The order that is placed is for a fixed amount that minimizes the total

inventory costs. This amount, called the economic order quantity.

A positive feature of a continuous system is that the inventory level is

continuously monitored, so management always knows the inventory

status. This is advantageous for critical items such as replacement parts

or raw materials and supplies. However, maintaining a continual record of

the amount of inventory on hand can also be costly.



12/23 1

Inventory Control Systems:

Continuous

23

Continuous Review or Fixed Order Quantity Systems (Q-systems)

Multi period models

1. Fixed order quantity, variable time between orders (EOQ, EPQ,

and Quantity Discount)

2. On-hand inventory balance serves as order trigger (R)

3. Perpetual inventory count

4. 2-bin system

Single period Model


Inventory Control Systems: Periodic

24

In a periodic inventory system (also referred to as a fixed-time-period

systemor a periodic review system), the inventory on hand is counted at

specific time intervals; for example, every week or at the end of each

month.

After the inventory in stock is determined, an order is placed for an

amount that will bring inventory back up to a desired level. In this system

the inventory level is not monitored at all during the time interval between

orders, so it has the advantage of little or no required record keeping.

The disadvantage is less direct control. This typically results in larger

inventory levels for a periodic inventory system than in a continuous

system to guard against unexpected stockouts early in the fixed period.

Such a system also requires that a new order quantity be determined

each time a periodic order is made.



13/23 1

Inventory Control Systems: Periodic

25

Periodic Review or Fixed Order Period Systems (P-systems)

1. Variable order quantity, fixed time between orders

2. Time serves as order trigger

3. Periodic count

4. Process: When a predetermined amount of time has elapsed, a

physical inventory count is taken. Based upon the number of units in

stock at that time, OH, and a target inventory of TI units, an order is

placed for Q = (TI-OH) units


Single Item Static Model

26

The simplest type of inventory model occurs when demand is constant

over time with instantaneous replenishment and no shortages.

Typical situations to which this model may apply are

Use of light bulbs in a building

Use of clerical suppliers, such as paper, pads, and pencils, in an office

Use of certain industrial suppliers such as blots and nuts.



14/23 1

Single Item Static Model:

Economic Order Quantity (EOQ) Model (1)

27

This model known as the economic order quantitymodel is the oldest

and best known inventory model which originated as early as 1915 with

Wilson and so the formula derived under this model to decide the

ordering quantity is called the Wilson formula (i.e. Wilsons economic

lot size).

It basically answers the question how much to order. Out of the four

components of inventory cost, here we are considering only two of

them, namely, theset-up costand theholding cost.

The other two costsshortage cost and purchasing cost, we adjust in

such a way that shortage is not allowed and the holding cost takes care

of purchasing cost which is possible to some extent with theassumption that there is no price break possible even if we order

quantity in big lots.


Single Item Static Model:Economic Order Quantity (EOQ) Model (2)

28

Our aim is to reduce thetotal costper unit(TCU) time. This is expressed

it terms of inventory as we are finding out the optimal inventory level

which reduces TCU.

With our assumptions TCU can be expressed as

TCU= set-up cost per unit time+holding cost per unit time

Let us denote

B = rate of demand or consumption

Y= quantity to be ordered

H= holding cost per unit per item

K = set-up cost per order



15/23 1

Single Item Static Model:Economic Order Quantity (EOQ) Model

29

Figure: Variation of the Inventory Level

Points in time at which orders are placed

Inventory Level

Time

SS

Average Inventory:Y/2

Y

t0=Y/B


Single Item Static Model:Economic Order Quantity (EOQ) Model

30

The smaller the order quantity Y, the more frequent will be the placement of new orders. However, the

average level of inventory held in stock will be reduced. On the other hand, larger order quantities

indicate larger inventory level but less frequent placement of orders.

Inventory Level

TimeSS

Low ordering frequency

High ordering frequency



16/23 1


31

Holding cost per unit time= Average inventory x Holding cost per unit per item ==Y x H / 2

Y/2becoming average inventory.

The cycle time is denoted byt0 and defined as

Set-up cost per unit time= K x Number of orders per unit time =K x B/Y =

= K/ (Y/B)

Reminder:TCU= set-up cost per unit time+holding cost per unit time

or TCU = K x B/Y + (Y/2) H

0

0

2( )

YK H t

TCU Y

t

0

Yt

B



32

Using the calculus method to find the optimal inventory we have to differentiate

TCUwith respect toY (assuming Y is a continuous variable)and equate to zero:

This will give us the value ofY as

(2 ) /Y KB H

2

( )0

2

dTCU Y KB H

dY Y

The above order quantity is usually referred as Wilsons economic lot size.



17/23 1

Single Item Static Model:an Example

33

The daily demand for a commodity is approximately100 units. Every time anorder is placed, a fixed cost of100is incurred. The daily holding cost per unit

inventory is0.02. If the lead time is12days, determine the economic lot size

and reorder point.

From the above formulas, the economic lot size is:

* 2 2 100 1001000 ( )

0.02

KBY units

H

Since the lead time is12 daysand the cycle length is10 days, reordering

occurs when the level of inventory is sufficient to satisfy the demand for two

(12-10)days. Thus, the quantityY*=1000is ordered when the level of inventory

reaches2 x 100 = 200 units. And the optimum costTCUis defined as

100 1000( ) 0.02 20 ( per day)

10002 2

100

K YTCU Y H

Y

B


Single Item Static Model:Remarks

34

The assumptions of the above model are usually rarely satisfied for real

situations, mainly because demand is probabilistic.

Nevertheless, a crudeprocedure has evolved among practitioners

which, while retaining the simplicity of applying the economic lot size

model, does not completely ignore the effect of probabilistic demand.



18/23 1

Quantity Discount Model

35

Form of an economic order quantity (EOQ) model that takes into accountquantity discounts. Quantity discountsare price reductions designed to induce

large orders.

If quantity discounts are offered, the buyer must weigh the potential benefits of

reduced purchase price and fewer orders against the increase in carrying costs

caused by higher average inventories. Hence, the buyer's goal in this case is to

select the order quantity that will minimize total costs, where total cost is the sum

of carrying cost, ordering cost, andpurchase cost:

Total Cost = Carrying Cost + Ordering Cost + Purchase Cost =

C x Q/2 + O x (D/Q) + P x D

where C= carrying cost per unit, O= ordering cost per order, D= annual

demand, P= unit price, and Q= order quantity.


Inventory Models

36

More models:

Single Item Static Model withSimultaneous Production and Sales

Single Item Static Model with Simultaneous Production and Sales

whereShortage is allowed

Single Item Static Modelwith Time LagSingle Item Static Model withPrice Breaks

Multiple-Item Static Model with Storage LimitationSingle-Item N-Period Dynamic Model



19/23 1

Prioritization (e.g. ABC classification/prioritization)

37

ABC Inventory Classification

Inventory control based on a form of Pareto analysis. The inventory

items are divided into three categories (A, B, and C), according to a

criterion such as revenue generation, turnover, or value.

Typically, Aitems represent 20 percent in terms of quantity and 75 to

80 percent in terms of the value.


ABC Analysis

38

ABC Analysis:Analysis of a range of items, from inventory levels to

customers and sales territories, into three groups:

A= very important

B= important

C= marginal significance.

The goal is to categorize items which would be prioritized, managed, orcontrolled in different ways.

ABC analysis is also called usage-value analysis.

Details of ABC classifications will be discussed in Unit 11.



20/23 2

ABC classification:

Applications in Inventory Control

39

ABC classification could be used to determine the importance rating fordifferent items and help the Stores in better inventory control. We could

have the following objectives for classifications:

Purchasing and Finance may want to classify the items in three

classes, i.e.A, Band Cbased upon last years consumption value.

Stores may want to classify the items in class High, Medium, and

Lowbased upon on-hand quantities for the purpose of doing a better

store space planning.

Inventory Controller may want to classify the items inHigh

movement, Medium movements and Low movements for

analysing the Min-Max Levels of different items.


Inventory Control: an Example (1)

40

A product is manufactured to meet the demand over the nextn periods.

At periodi, the demandri (units) may be filled from the productionxi

(units) in this period and/or the inventory carried forward from previous

periods. This means it is possible to produce more thanriunits in period

i, with the surplus being used to fill (some of) the demand for one or

more succeeding periods.

No shortages are allowed in any period so that the demand for all

periods must be filled. If the level of production in periodi is increased

over that in period (i-1), that is,xi> xi-1, a costaipounds per excess unitis incurred.

On the other hand, ifxi< xi-1, a costbipounds per decreased unit is

added for decreasing the level of production.

An inventory holding costci pounds is incurred for every unit held over to

the next period.



21/23 2


41

The objective of the modelis to determine the optimal production

schedule such that the total cost of the production-inventory system isminimised.

The balance equation at periodi is:

Ii+1= xi+ Iiri, I = 1,2,,n

whereI1is the initial inventory at the beginning of period1and In+1is

the final inventory at the end of periodn; the demandri (units); the

productionxi (units).

Generally,I1>=0, whileIn+1must be zero in any optimal solution. In

order to guarantee that the demand is satisfied in every period (no

shortage), Ii>=0, I = 1,2,, n-1.

Naturally, production quantities must be nonnegative so that

xi>= 0, i=1,2,,n



1 1

1 1

1

( ), if

( ), if

0, if =

i i i i i

i i i i i i

i i

a x x x x

y b x x x x

x x

42

The above restrictions define all the constraints of the problem. The

objective function includes three components:cost of holding inventory,

cost of increasing production, andcost of decreasing production.

The holding cost is the simplest, and it is given directly for period i by

ci Ii+1(notice the inventory holding cost is charged on units left at the

end of the period). The cost of increasing and decreasing production in

periodi is given by



22/23 2


1 1max{ ( ), ( )}i i i i i i iy a x x b x x

43

Equivalently,yimay be written as

This follows since if production is increased,xi-1xi< 0so that

bi(xi-1xi)0. Thusbi (xi-1xi)


23/23


1

1

1

0

0

, , 0, 1,2,...,i

i i i i i

i i i i i

i i i i

i i

a x a x y

b x b x y

x I I r

I x y i n

0 1

1

minimise ( )n

i i i

i

x c I y

45

The linear programming model of the production-inventory problem maybe summarised as

subject to

whereIn+1=0. The variableyiis nonnegative because it represents thecost of increasing or reducing production.


inventory control in operation research

Documents