Exchange Curve

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Exchange Curve

Exchange curve (or optimal policy curve) is an effective technique to look at the inventories at an aggregate level in the organization. It is a plot between the total number of orders (TO) per year and the total investment in inventories (TI) per year. The rationale is that for an optimal inventory policy the trade-off between total inventory and total procurement effort as indicated by the total number of replenishment orders per year must be made such that if total number of orders is prescribed, we minimize total investment in inventories. Alternatively, if the total investment in inventories (TI) is prescribed then a rational inventory policy must aim at minimizing (TO).

Uses of Exchange Curve

Exchange curve is an effective instrument for aggregate inventory analysis to quickly determine the rationality (or otherwise) of our existing stock provisioning policies. We first plot the exchange curve by computing the value of K for a chosen group of items. Then we determine the total number of orders (TO) and total investment in inventories (TI) under current practice.


If the current practice is at point C (in Figure V) above the exchange curve then it shows that our present procurement policies are not rational. If we want to rationalize these then there are two possible paths-AC or BC; so that we reduce inventory to B for the same ordering effort or reduce number of orders to A for the same inventory. Thus, an exchange curve is a useful device at macro-level.

Deterministic Inventory Models


Classical EOQ Model

In this section we discuss some elementary inventory models with deterministic demand and lead time situations. The purpose is to provide an illustration of the mathematical analysis of inventory systems. The most classical of the inventory models was first proposed by Harris in 1915 and further developed by Wilson in 1928. It is very popularly known as EOQ (Economic Order Quantity) model or ‘Wilson’s Lot Size formula’.


When dealing with stocked items, the two important decisions to be made are-how much to order and when to order. EOQ attempts to provide answer to former while the Reorder point (RoP) provides the answer to the latter.


The following assumptions are made in the standard Wilson lot size formula to obtain EOQ:

  • Demand is continuous at a constant rate
  • The process continues infinitely.
  • No constraints are imposed on quantities ordered, storage capacity, budget etc.
  • Replenishment is instantaneous (the entire order quantity is received all at one time as soon as the order is released).
  • All costs are time-invariant.
  • No shortages are allowed
  • Quantity discounts are not available.


The inventory status under EOQ-RoP policy is continuously reviewed. Figure VI (a) shows the behavior of such a simple system whereas Figure VI (b) shows the total system cost behavior highlighting the conflicting trend of ordering and inventory carrying costs. EOQ aims at minimizing total system cost. ~

Let us use the following notation in developing the classical EOQ model: Inventory Management D = Demand rate; unit per year.

A = Ordering cost; Rs./order.

C = Unit cost, Rs. per unit of item.

r = Inventory carrying charge per year.


H = Annual cost of carrying inventory/unit item = r.c.

TC = Total annual cost of operating the system Rs./year (objective function).

Q = Order quantity, Number of units per lot (decision variable).

Since demand is at uniform rate average inventory is Q/2 throughout the year and the total number of orders are (D/Q) per year. Thus total annual cost of operating the systems consisting of carrying cost and ordering cost.

Some interesting insight may be obtained using this classical system:


  • If ordering cost is of high tendency, the optimal policy is to have high EOQ thus raising average inventory level.
  • If r or care high leading to high value of H, the tendency will be to go for smaller lot sizes.


r may vary from 0.15-0.30 and will depend on the nature of item, A the ordering cost should be marginal ordering cost while H should be based on total purchased cost of the items.

Finite Replenishment Rates

We will now relax the assumption (d) of the classical EOQ model and permit finite replenishment rate (staggered deliveries). When the rate of procurement is P in units/year and the demand rate is D, in units/year, the buildup of inventory is at a rate (P-D) due to simultaneous consumption. It is obvious that P> .D for inventory to build up. Figure VII shows the inventory behavior with finite supply rate. The stock builds up to a maximum level I during supply period ts, after which stock depletion takes place at rate D

Some interesting observations can be made about the behaviour of such systems. These are:

  • Q* under finite replenishment rates are higher than Q* under classical EOQ model for the same values of other parameters.
  • Total system cost under optimal Q* is lower than corresponding total system cost and EOQ model.
  • Thus, staggering the supplies always reduces inventory level and total operating system cost provided other cost parameters remain the same.
  • As P–>∞,Q* and TC* obtained are same as in standard Wilson’s formula of instantaneous replenishment.
  • At P = D, Q*.–>∞ TC->0. Thus, if we can have a fully devoted reliable supplier, then placing a single supply order of large size but matching supply rate with the demand rate is the optimal decision. Under such a system, no. stocks are built, no replenishments are made, and no shortages are incurred.

Planned Backlogging

Let us now consider the effect of relaxing assumption (f) of classical Wilson’s model by permitting backlogging (shortages or back ordering) at a unit shortage cost of Sin Rs. /unit short/year. In such a case negative inventory shows the backlogging position. The order quantity Q is partly used to clear the backlogging level B and (Q-B) is the maximum stock level. Figure VIII shows the inventory behavior under planned backlogging Inventory is maintained for duration it and demands remain backlogged for duration tb. Total cycle time of each replenishment cycle is

Some useful observations could be made about the behavior of inventory system with planned backlogging as follows:


  • Total system cost is lower with planned backlogging than the corresponding total system cost under classical Wilson’s lot size formula. Thus for a deterministic system with finite backlogging cost, it is economical to plan for backlogging. It can be seen that at S->∞, the model reverts to classical EOQ model.
  • EOQ under backlogging is higher and maximum stock level is lower than the corresponding values under classical Wilson’s lot size model.
  • If S = 0 then B* = Q* = co. This means that with no charge for back orders one would keep piling up unfilled demand until the backlog gets infinitely large. Then one single order would be released to satisfy all accumulated demand. However, considering intangible cost of backordering such as loss of goodwill etc. it is debatable whether there are situations when the unit cost of shortages (S) is really zero.

Model with Quantity Discounts

Frequently, the vendors offer quantity discounts on bulk purchases to encourage users to place orders in large quantities. Quantity discounts may be all unit discounts or incremental quantity discounts. In all unit discounts entire order quantity is purchased at lower unit price if order size is higher than or equal to the stipulated conditions. In incremental case only quantity exceeding the threshold point is charged at lower unit cost. The immediate reaction may be to avail the discount and place bulk orders but if we see the total system cost, our decision may be otherwise. There may be a single or multiple quantity discounts.

The broken lines show the total cost curves without price break whereas solid lines show the actual total cost if price break takes place. The larger the number of price breaks, the more difficult it becomes to analyze the situation as more alternatives are to be evaluated.

Sensitivity Analysis

It may not be operationally very convenient to stick to EOQ if it is an odd figure. Then one may like to know the repercussions on total system cost if one deviated either way from Q*. This is done through sensitivity analysis. If Qa is actual order quantity, Qa = b.Q* where b is sensitivity parameter. If b = 0.8 then actual Qa is 20% less than Q* and if b = 1.2 then Qa is 20% less of higher than Q*. Obviously the TC will increase over TC* in either case. If we substitute Qa as bQ* in total cost expressions in the classical EOQ model, we can easily get the following relationship.


Where TCa is actual cost with order size being Qa It can be seen that at b = 1, p = 1. If b is allowed to vary within 0.9 to 1.10 then p will be within 1.005 indicating that ±10% deviation in EOQ leads to less than half a per cent increase in TC Thus TC is not very sensitive to EOQ and for operational convenience we should be able to vary EOQ within ± 10% of Q* without adversely affecting total system cost.

Probabilistic Inventory Models

In the inventory models described in previous section we assumed that there was no uncertainty associated with the demand and replenishment-lead times. However, in reality there is always some uncertainty associated with the demand pattern and lead times. It can be shown that as the uncertainty (variability) of demand and lead times increases, extra stock in the form of safety stock (buffer stock) is required to account for these uncertainties. In the deterministic system, reorder point is very easy to determine as it is the demand during the lead time

Safety Stock and Service Levels

In the determination of safety stock, the factor K obtainable from normal distribution tables for normally distributed lead time demand depends upon the risk of shortage we are prepared to accept. Higher value of K means less risk of shortage (or high service level) and vice-versa.

Economic Order Quantity (EOQ)

EOQ is a mathematical formula designed to minimize the combination of annual holding costs and ordering costs. There is a lot of hype about just in time inventory systems (JIT), which achieve smaller inventories through very frequent orders, but frequent ordering can often result in an over-spending on ordering costs. Even though companies often miscalculate their ordering costs, which make frequent ordering seem costly, EOQ is an important tool for determining what inventory should be


Safety Stock

First of all, here’s the formula so you don’t have to dig through my well-written article for it. Safety Stock: {Z*SQRT (Avg. Lead Time*Standard Deviation of Demand^2 + Avg. Demand^2*Standard Deviation of Lead Time^2}.



Basics of Production Inventory Management

Production inventory management differs from general warehouse management because it involves the determination of how quickly to produce a particular product. The factors involved in many cases are similar, though there are some variances in making the final decision as to how quickly manufacturing should push items through the production line.


Available Materials

Of course, the first concern in production inventory management is on the front end of the process. If you don’t have the materials required for production, then you can’t move forward in providing the products to others. You must make certain that you have all the supplies you need, from raw materials to factory workers, to complete the production process.


Supply and Demand

You must determine the current demand for the product on the market. Good production inventory management occurs when you produce just enough material to satisfy customers’ needs without overextending the production line and manufacturing too many of any given product


You don’t want an incredible amount of back stock lying around, as this detracts from your net profit. On the other hand, you don’t want to be in short supply when a large order comes in, so having a little extra on hand is a great idea, and making sure you are prepared to make a production run for such orders is vital.


Quality Control

Never simply assume that everything manufactured will be flawless. An important consideration in production inventory management is to allow room for error. In other words, calculate a sufficient


Amount of product to assume that, even with flaws that get past quality control efforts, there is sufficient stock of the product required.


Cost Analysis

In many instances, even the best production inventory management strategies fail in the long run due to the cost of the production process being overlooked as a factor. It is important to maintain a cost effective production process, and this includes making sure that your inventory is not an overwhelming factor. This comes back to not MR Producing any items that come off the assembly lines. Doing so is a waste of time and materials, costing you excess money to create.

Inventory Classifications:

Inventory is idle resources that have future economic value. It indicates that it may be available in different forms depending upon the production cycle stage it is in. Classification of inventory is done on this basis and thus the different classifications of inventory are as follows:

Raw materials – Raw materials are input goods intended for combination and/or conversion through the manufacturing process into semi-finished or finished goods. They change their form and become part of the finished product.


Components and Parts – Just as raw materials are converted to finished goods in a manufacturing operation, components and parts are assembled into finished goods in an assembly operation. Maintenance, repair and operating inventories (MRO) – These include parts, supplies, and materials used in or consumed by routine maintenance and repair of operating equipment, or in support of operations.


In-process goods – These are goods in the process of manufacturing and only partially completed. They are usually measured for accounting purposes in between significant conversion phases. In-process inventories provide the flexibility necessary to deal with variations in demand between different phases of manufacturing.

Finished goods – These represent the completed conversion of raw materials into the final product.

They are goods ready for sale and shipment.

Resale goods – These are goods acquired for resale. Such goods may be purchased by a wholesaler for resale to distributors, or by distributors for resale to consumers, etc.

Capital goods – These are items (such as equipment) that are not used up or consumed during a single. Operating period, but have extended useful lives and must be expensed over multiple operating periods. Tax laws require that such an item be capitalized, and a predetermined percentage of its cost be recognized as an expense each operating period, over a predetermined time frame, according to equipment classes.

Construction materials – These are raw materials and components for construction projects such as a building, bridge, etc.

Hard goods/soft goods What one identifies as hard goods and soft goods will vary depending on the industry involved. For example, in data processing, hard goods include apparatus such as computers and terminals, while soft goods include software, data storage media, and the like.

Inventory control is concerned with minimizing the total cost of inventory. In the U.K. the term often used is stock control. The three main factors in inventory control decision making process are:

The cost of holding the stock (e.g., based on the interest rate).

The cost of placing an order (e.g., for row material stocks) or the set-up cost of production. The cost of shortage, i.e., what is lost if the stock is insufficient to meet all demand. The third element is the most difficult to measure and is often handled by establishing a “service level” policy, e. g, certain percentage of demand will be met from stock without delay.


The ABC Classification: The ABC classification system is to grouping items according to annual sales volume, in an attempt to identify the small number of items that will account for most of the sales volume and that are the most important ones to control for effective inventory management.


Reorder Point: The inventory level R in which an order is placed where R = D.L, D = demand rate (demand rate period (day, week, etc.), and L = lead time.

Safety Stock: Remaining inventory between the times that an order is placed and when new stock is received. If there are not enough inventories then a shortage may occur.

Safety stock is a hedge against running out of inventory. It is an extra inventory to take care on unexpected events. It is often called buffer stock. The absence of inventory is called a shortage.


Quantity Discount Model Calculation Steps:

  • Compute EOQ for each quantity discount price.
  • Is computed EOQ in the discount range
  • If not, use lowest cost quantity in the discount range.
  • Compute Total Cost for EOQ or lowest cost quantity in discount range.
  • Select quantity with the lowest Total Cost, including the cost of the items purchased.
  • The following This JavaScript compute the optimal values for the decision variables based on currently available information about the above factors.


Enter the needed information, and then click the Calculate button.

In entering your data to move from cell to cell in the data-matrix use the Tab key not arrow or enter keys.


  • The Classical Model
  • Shortage Permitted Model
  • Production & Consumption Model
  • Production & Consumption with Shortage Model
  • EOQ with shortage & lead Model
  • The ABC Classification
  • Inventory Control & uncertain Demand


Inventory System valuation, terminology

Inventory Control Systems: Inventory control is concerned with minimizing the total cost of inventory. The three main factors in inventory control decision making process are:


  • The cost of holding the stock (e.g., based on the interest rate).
  • The cost of placing an order (e.g., for row material stocks) or the set-up cost of production.
  • The cost of shortage, i.e., what is lost if the stock is insufficient to meet all demand.


The third element is the most difficult to measure and is often handled by establishing a “service level” policy, e. g, certain percentage of demand will be met from stock without delay.


In designing an inventory control system, we really provide answers to the three questions:


How often should the inventory status be determined?

This is an internal check system to ascertain that timely action is being taken to replenish the stock

When should a replenishment order be placed?

This shows the actual action to be taken to replenish the stock.


How large should the replenishment order be?

A replenishment order should have a rational about its size. The real problem is to determine the inventory level at which money invested in inventory produces a higher rate of return than it would were it invested in some other phase of the business.


Designing Inventory control systems:

The Demand pattern (D) happens to be the soul of any Inventory control mechanism. Basically inventory control is an attempt to balance the consumption and replenishment of stock of an item in an optimum manner. Obviously, the Demand pattern (D) sets the tone for devising any control measure and therefore while designing inventory control systems the nature of demand pattern viz. independent or dependent needs to be determined first. The Control approaches differ on this differentiation.

Inventory control systems under Independent demand scenario:

Also called the Order-point control systems, Independent demand patterns for an item occur when future demand for it is not related to and is unaffected by its previous demand.

For inventories exhibiting independent demand pattern control is exercised based on predetermined order points. Such systems are so designed that whenever a predetermined point in inventory level or in time is reached action to re-order is taken.

There are two basic systems of managing or controlling Inventory under the independent demand pattern:

  • Cyclical ordering or Fixed period system (Time Based)
  • Order point or fixed order quantity system

Cyclical ordering or fixed period system (Time Based):

Fixed Period based systems (also called “cyclical systems”) are designed so that each inventory item is reviewed and reorders are placed after a predetermined time interval (i.e. every 2 weeks, every 30 days, etc.).


Orders are placed for each item equal to the difference between current inventory level and a predetermined maximum. In cyclical systems, time between reorders is constant, but reorder quantity is variable.


Predetermined maximums are set with a consideration of order lead time. It involves scheduled periodic reviews of the Stock level of all inventory items as follows:

Fixed Schedules (calendar) for reviewing a group of items is drawn


Fixed Desired inventory level (DIL) of each item or group of items is calculated. In case stock level of an item is insufficient to sustain the production operation until the next scheduled review , order is placed to replenish the stock to DIL

Maintenance of perpetual inventory records

Procedure : First , all the inventory items are grouped in certain feasible categories or classes of items such as Pipes & Pipe fittings, Raw materials, Chemicals & Reagents, Oil and Lubricants etc.


Now, a calendar is drawn for all the classes so that depending upon the number of classes each class is reviewed for replenishment during certain specified time frame.


Besides, the DIL or MaxL for each group or individual item is fixed.


Depending upon the Review period, a class of items is reviewed w.e.f. its Stock position, Production plan, any dues in quantity against any previous order.


The DIL or MaxL= (RP+LT+SS) X D


During review and based on the Lead time , if the present stock of an item or group of items is not expected to last the next production plan then action for replenishment is taken by raising the material procurement requisition.


The Order quantity is decided by (MaxL- (Present stock + dues in)


Suitability of the system:

  • For materials whose purchases can be planned months in advance
  • For materials which exhibit an irregular or seasonal usage pattern
  • For items with volatile prices
  • For group of items purchased from and shipped together by one supplier


Disadvantages of the system:

  • Compels periodic reviews of all items
  • Not effective to combat stock out situations
  • Actual ordering quantities may deviate from optimum quantity
  • Tends to peak the purchasing work load around the review dates

There is no automatic trigger for reorder before the review time in the event of increased usage, which generally leads to increased inventory levels as a means of stock out prevention

System does not permit effective use of economic order quantities


Order point or fixed order quantity system:

Order point system / Fixed Order quantity system of inventory control is based on the (Re)Order point and Order quantity factors rather than on the time factor. The inventory policy, in this system, is drawn, defining the following

  • Fixed Order point / Re-Order Level (ROL) for each item
  • Fixed Maximum, Minimum levels for each item
  • Fixed Quantity to be ordered

Often called Min-Max systems, these involve both a maximum inventory level and a minimum at which reorders are generated. Basically, units of an item are issued until the level of that inventory reaches the predefined reorder point. An order is then triggered for a predetermined quantity (usually a calculated economic order quantity). In this system, the order quantity is constant and the time between orders s variable.


The different Inventory points (Levels) of stock for an item are :


Maximum level (Max.) , predetermined

Minimum Level (Safety stock, SS), predetermined

Lead time (LT) , predetermined

Monthly demand = D (often based on Moving average method)

MaxL.= (Review period + LT + SS) X D

Reorder level (ROL) = (LT + SS) X D

Order Quantity (OQ) = Max. – (Present stock + Pipeline dues)

Process: In course of consumption of an Inventory item, say, in the form of issue from Stores to the users, the stock level of the item starts depleting through its usage rate D.


As per the above definition, the stock goes up to the maximum level in the first replenishment and then, because of steady consumption, comes gradually down. In that process ,again as per the definition , it touches the ROL. As soon as the stock level touches the ROL fresh replenishment action is initiated.


It is presumed that the next lot shall arrive by the time the present depleting stock touches the Safety stock , keeping a stable Lead time and a stable usage rate D.


In some places the Order quantity is decided by the above formula whereas in some other places it is determined by the Economic Order Quantity (EOQ) concept. That’s whenever an order is to be placed the quantity shall be EOQ.


Each item is procured in the most economical quantity An item is attended to only when it needs attention i.e. when its stock has reached the ROL Control can be exercised on Inventory w.e.f.. Max & Min levels

Applicability of Order Point system

  • Item must have a reasonable stable usage
  • Lead time should not have radical variation
  • Supplier should be able to accept irregularly timed and unscheduled orders


Limitations of the system

Needs continuous monitoring of stock level of each item Cumbersome to operate for items with unstable usage and lead time perpetual inventory records are required.


Inventory control systems under Dependent demand scenario:

Dependent demand occurs when the need for parts, supplies, or materials is dependent upon a predetermined usage or production schedule.

In such cases, a description and quantity of components needed and the exact date of each need is defined by a production schedule

Required delivery dates for each component will then be offset by lead time, and orders will be placed accordingly. For example, if a pen manufacturing company plans to produce 1000 nos. of a given pen model in a period, it will need 1000 nibs, 1000 caps, etc., and will need them at the rate they will be installed in the finished pens.


Such needs, with consideration for lead time, are considered in a dependent demand planned order schedule.


Material Requirement Planning is one example of a system specifically designed to manage dependent demand reorders.


Role of EQO Basic Inventory Models (Economic Order Quantity / EOQ Models) Economic Order Quantity (EOQ) models are the most basic models of inventory management. The approach in EOW models is essentially to trade-off various relevant costs and derives an order quantity and time for placing an order such, that the total costs are minimized. This note discusses the basic


EOQ model and the sensitivity of costs in EOQ model to various parameters. Later an extension of basic EOQ model is discussed in which case back-orders or shortage are allowed.

Functions of inventory

Though inventory is an idle resource, it is almost essential to keep some inventory in order to promote smooth and efficient running of business.

Cost of increasing production and employment

  • Employment and training
  • Additional staff and service activities
  • Added shifts
  • Overtime costs

Cost of decreasing production and employment

  • Employee compensation
  • other employee costs
  • Staff, clerical and service activities
  • Total time costs

In other words, the use of seasonal inventories can often after balance of these costs. Broadly, some other functions of inventories are to:

  • protect against unpredictable variations (fluctuations) in demand and supply
  • take the advantage of price discounts by bulk purchases
  • take the advantage of batches and longer production run.
  • Provide flexibility to allow changes in production plans in view of change in demands, etc. and
  • facilitate intermittent production.

Elementary Inventory Models (with Deterministic Demand)

Let us consider the inventory models in which demand is assumed to be fixed and completely pre-determined.


D – Annual demand rate

V – Unit purchase cost or unit cost of production (Rs./unit)

A – Ordering or set up cost (Rs/order)

R – Holding cost per Rs. per year (*Rs./Rs/year) (Inventory carrying charges factor)

B – Shortage cost per Rs. short unit time (Rs/ Rs/ year)

Q – Order quantity (to be determined

Inventory Cost

The heart of inventory analysis resides in the identification of relevant costs. Some of the important costs that apply to inventory situation are:

Ordering or set up costs

These are costs associated with ordering or manufacturing goods through purchasing or manufacturing and are known as set up costs or cost of ordering. Set up costs are generally assumed to be independent of the quantity ordered or produced.

Purchase cost or production cost

When large production runs are in process, these results in reduction of production cost per unit. Often, discounts are offered for the purchase of large quantities. In other words, often the unit cost of an item depends on the quantity procured or produced.

Inventory Holding Cost

The cost associated with carrying or holding the goods in stock are known as carrying or holding costs. These costs arise due to the storage costs, property taxes on the items in inventory, interest on the invested capital (interest on value of the inventory items, spillage of the inventory items, depreciation of the inventory items, transportation and handling of the items in inventory, etc.

Shortage or stock out costs

The costs that are incurred as a result of running out of stock are known as stock PUT TO SHORTAGE COSTS. As a result of shortages, sales or goodwill may be lost. If the unfulfilled demand for the items can be satisfied at a later date (back order case), in this case cost of back orders are assumed to vary directly with the shortage quantity (in rupee value) and the delaying time (Rs./ Rs.) . However, if the unfulfilled demand is lost (lost-sales case), in this case cost of shortages are assumed to vary directly with the shortage quantity (Rs./ unit shortage).

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