Name: Chris Mickael D.

Lacson

Year & Section: BSBA 1-G

Subject: Inventory management and control

Chapter 7

Inventory Control And Models

Objectives:

 To introduce basic inventory control and models:

 Discuss how inventory controls and models works.

1.M. KEYNES differentiated three motives for holding money which can be applied to inventory
problems.

1. The Transaction Motive

- Since outflows are not synchronized with inflows, stocks are needed to bridge these
discrepancies. Usually, incoming goods arrive in greater quantities and in longer time
intervals than outgoing ones.

2. The Precautionary Motive

- If an order is placed, one must maintain reserve stocks in order to satisfy the demand
while awaiting delivery.

3. The Speculative Motive

- If prices are expected to rise, it pays to keep stocks on hand In operations research
(OR), inventory is typically geared towards the first two motives. The third is
occasionally treated in linear optimization as the so-called warehousing problem.

Inventory theory belongs to the first and, therefore, classical application areas of
Operations Research. It was strongly supported a the 1950s, primarily by the US Navy. Scientists
of the caliber of OSKAR MORGENSTERN, JAKOB MARSCHAK KENNETH ARROW,
HERBERT SCARF, THOMAS WHITIN, JACK KIEFER and others have intensively worked
with the application of OR and Statistics on inventory problems Its beginnings, however, go back
to the mythical WILSON much earlier a about the turn of the century. At that time, the question
of optimal inventory control strategies was a long disputed issue. This led to the development the
Theory of Dynamic Programming (RICHARD BELLMAN)

ECONOMIC ORDER QUANTITY (EOQ)

The standard case of the ECQ (or optimal lot size) problem is a trading firm which orders
a good to store up stocks. Customer demand is satisfied by stocks on hand. We assume a constant
rate of demand.
Inventory Cost:

It consists of interest costs, handling costs and rental costs for storage (even if one is the
owner of the warehouse, in this case, the rental cost is an opportunity cost, the possibility of using
the warehouse for other purposes is given up). Moreover, cost of wastage (in India, ¼ of the grain
harvest is eaten up by rats), depreciation and obsolescence can also arise. All these costs are
summed up in the inventory cost.

Shortage Cost:

Shortages occur in case stocks are very low and, therefore, demand cannot be fully
satisfied. These are charged penalty costs. Usually, shortage costs are assumed to be proportional
to the amount of shortage, Shortage costs can also be thought of as being independent of the size
of the deficit.

Cost Structure of the Inventory Model

Ordering Cost: We assume a linear relationship for the ordering cost:

K: fixed ordering cost. This covers administrative costs (e.g. P100-500; a business letter costs
approximately 170), customer complaints, etc.

A: Proportional ordering cost, e.g. transportation cost, cost for controlling incoming goods, in our
model, it is mainly the buying price.

Shortages can arise if the stock is not permanently recorded (Periodic Inspection), if the
stock is ordered too late or when the ordered quantity a delivered late
The cost structure described here is very simple. A detailed discussion of differentiated
cost considerations can be found, however, in business
Economics literature. The WILSON Lot Size Formula (also ANDT.FR”s Formula or
HARRIS Formula”).

The lot size model of Wilson (or Harris or Andler) has the assumptions on these models
are of the simplest type, the derived formula for the optimal lot size proves to be rather robust in
marry practical situations, e.g. in transition from a constant rate of demand to the Poisson
demand. The

Average Range of Coverage

An important reference value is the inventory-sales ratio. It tells something about the
long-term efficiency of a stock control system.

Research has shown that, in spite of operations research, the average inventory holdings
in the last two decades have increased. There are two reasons for this
 1. Wages and salaries have increased sharply so that the rate k/h increased in spite of
increasing interest costs (h increases) and the decrease in part of the fixed costs
through EDP.

2. Decentralization has increased the number of warehouses Moreover, type variants

(product variations) have increased such that the demand rate per variant and stock
location has decreased which according to (2-4) leads to an increase in the average
range of coverage.

Returns to scale can also be seen from. The inventory-sales ratio of a company becomes
more favorable with increasing sales volume. This, however, says something about the cost. The
following section deals with cost considerations

COSTS AND SENSITIVITY

Costs and sensitivity are describe and shown that these are decreasing with scale when
the ordering rule is optimal, i.e., with increasing turnover, the cost per unit of inventory becomes
smaller. Under decentralization, this effect of increasing returns to scale is partially lost. The
appropriate formulas ate derived. Using sensitivity analysis, the effects of various parameters on
the total expected cost are discussed; first, when the rate of demand or the various costs are
wrongly estimated; secondly, when an optimal order quantity is not realizable due to special
packaging units or container size conditions, or, thirdly, when the desired period length between
orders is predetermined because of internal business organizational reasons or because of
prescribed delivery dates as in the case of the pharmaceutical Industry.

As is to be expected, the cost of an order cycle per unit time increases with increasing
sales volume. The increase, however, is sublinear and it decreases, therefore, with increasing
turnover.

This also shows the effect of increasing returns to scale (an advantage of bigger firms).

The reason for this is INDIVISIBILITY, in our case, the indivisibility of the fixed order
costs. Whether the order quantity is small or large, the fixed order costs, remain the same.

In many large firms, however, inventory is decentralized. Decentralization has often been
justified in the course of corporate history and, hence an energetic drive is needed to break out of
the imposing structures and to reorganize the logistics. A similar stimulus in the form of high
interest rates happened at the beginning of the eighties as one tried hard to rationalize the sales
Income of a corporation to free liquid resources This resulted in the centralization of inventory in
many firms. One should not ignore, however, the advantage of decentralization: improved
customer service.

RM-SYSTEMS (ABC ANALYSIS)

 The abbreviation RM stands for the latin “roductio and maximun”. In an RM-system, the
goods are arranged according to importance. The importance of good is considered according to
its sales volume A-a, measured in terms of its buying price (and not by its selling price for which
we measure costs). In our model, the buying price is the proportional ordering cost. This gives the
theoretical justification to use as a measure for the cost which causes one to hold inventory.

The purpose of an ABC classification is to save inventory management costs. Only goods
of class A (highest importance) are handled according to the best possible methods. Note: In this
case, continuous stock control is often necessary! The simplest inventory models are used for
goods in classes Band C. One normally relies on the rule of thumb.

PRODUCT-MIX DECISION

An ABC-Analysis can lead to the decision to straighten out the product mix and to
discard certain articles, such as goods with high costs or low demand, the so-called slow movers
A systematic product mix adjustment is often done, for example, in the case of
booksellers. If the rate of sales falls below a critical value, the books will not be republished and
the remaining stocks will be sold cheaply To avoid the danger of selling out too early, it is
important to know exactly

STOCK-OUTS ALLOWED :

1. LOST SALES CASE. Unsatisfied demand is lost,

2. BACKORDER CASE. Unsatisfied demand is deferred until supply is again available.

We consider the BACKORDER CASE. In practice, under conditions of certainty, this can
only be done by a firm without competition, Le, a monopolist (with the attitude “the public be
damned”). It is, however, different with stochastic demand.

BUDGET RESTRICTION

In a multi-product warehouse, the individual goods compete fi storage space. In a small

storage room, one cannot expect that the total area will cover storage of the optimal lot size for
each good. As a rule one must get by with a fraction of lot size. This leads to an inventory model
with capacity restrictions. Instead of limiting the storage area, the available capital could also be
limited: either the current inventory asset or the current account limited by the available credit
line, in case all orders within an inventory cycle are paid for at the same time.

KNOWN BUT VARYING DEMAND

Until now the demand rate was considered as constant over an infinite time horizon.
For commercial warehouses this occurs only in exceptional cases There is a similar
situation with manufacturing inventories, for example, where purchased parts are kept in stock to
be used as components in serial production. A constant demand rate is also rare in that case.
 Application 7.1

Automobile Factory

Depending on customer’s preferences, a vehicle can be equipped with a wood grain

steering wheel. The daily requirements for the steering wheel are determined from the bill of
materials for the vehicles prepared from the preceding quarter. These are ordered from a supplier.
What is the optimal lot size?

There are two possibilities for this model:

1. Either one knows how the problem behaves in the future, Le, one can, at least, give
probabilities for future demands. Such models are treated later.

2. Or, one considers it as an infinite horizon problem, One asks how long an order will
suffice. The time period T is chosen such that the average costs are minimized. This procedure is
repeated each time the stock sinks to zero.

Surely, this method is not optimal but is more practical than the exact method because
one need not wait for a new decision until the whole planning horizon has passed.

The objective function is to "minimize the average cost of the first order cycle.