11
Inventory Models
11.1 INTRODUCTION
Inventory is defined as any idle resource of an enterprise. It is stock of goods, commodities or economic resources that are kept for the purpose of future business affairs. Inventory may be kept in any form—raw materials in process, finished product, packaging, spares and others—stocked in order to meet the expected demand. It is important to maintain some inventories for the smooth functioning of an enterprise. Suppose a firm is not maintaining inventories; if a sales order comes, it has to purchase the raw materials required, wait till these arrive and then start production. This increases the waiting time of customers to get the delivery goods.
The disadvantages of not maintaining inventories are: raw materials are purchased at a high price because of piecemeal buying; production costs would be high because of not being able to take advantage of batching, and so on.
Meaning of Inventory Control
Inventory control may be defined as ‘the function of directing the movement of goods through the entire manufacturing cycle from the requisitioning of raw materials to the inventory of finished goods in an orderly manner to meet the objectives of maximum customerservice with minimum investment and lowcost plant operation.’
There are two basic functions.
 Maintaining an accounting record to handle the inventory transactions concerning each inventory item. For inventory transactions, a recordkeeping system called Kardex file is maintained for each inventory item.
 Deciding inventory replenishment decisions. There are two basic replenishment decisions:
 When is it necessary to place an order (or produce) to replenish inventory?
If the demand of an item is independent of that of other items, then the recorder point technique can be used to know the time of replenishment.
 How much is to be ordered (or produced) in each replenishment?
 When is it necessary to place an order (or produce) to replenish inventory?
The decision about the number of units to order (or produce) for replenishment depents on the types of inventory costs.
Reasons for Carrying Inventories
It is essential for any firm to have inventory because of the following reasons:
 It provides adequate service to customers.
 It helps in smooth and efficient running of business.
 It reduces the possibility of duplicating orders.
 Timely shipment of customers’ orders will improve cash flow.
 It takes care of economic fluctuations.
 It helps in minimising the loss due to deterioration, obsolescence, damage, and so on.
 It acts as a buffer stock when raw materials are received late and shop rejections are too many.
 It takes advantages of price discounts by bulk purchasing.
 It improves the manpower, equipment and facility utillisation because of better planning and scheduling.
Maintenance of inventories costs money by means of expenses on stores, equipment, personnel, and so on. So excess inventories are undesirable. This calls for controlling the inventories in the most profitable manner.
Types of Inventories
Basically, there are five types of inventories.
 Fluctuation Inventories (Buffer Inventories): These are inventories to meet uncertainties of demand and supply. Buffer inventories in excess of those necessary to meet the average demand during lead time (the time lapsing between placing an order and having the goods in stock ready for use) and held for protecting against the fluctuations in demand and lead time, are termed as safety stocks or reserve stocks.
 Anticipation Inventories: These are built up in advance for the season of large sales, a promotion programme, or a plant shut down period. It keeps men and machine ready for future needs. For example, keeping crackers well before Diwali or air coolers or air conditioners before summer.
 Cycle (Lot Size) Inventories: These are built up in advance because the purchases are usually made in lots rather than for the exact amounts needed at a point of time.
 Transportation Inventories (or Pipeline Inventories): Such inventories exist because of the transportation of inventory items to various distribution centres and customers from the production centres. This type of inventory is also called process inventory where the significant amount of time is consumed in the transshipment of items from one location to another. To meet the demand it is essential to hold extra stock at various work stations. The amount of transportation inventory depends on the time taken in transportation and the nature of demand.
 Decoupling Inventories: If various production stages operate successively, then the breakdown of one or many may affect the entire system. This kind of interdependence is not only expensive but also disruptive for the entire system. The inventories used to reduce the interdependence of various stages of the production system are known as decoupling inventories. These inventories may be classified as:
 Raw materials and component parts: It is used to decouple the producer from the suppliers. That is, raw materials and component parts inventory could
 act as a buffer to take care of delays on the part of suppliers.
 guard against seasonal variations in the demand of final product.
 Workinprocess inventory: As it takes time to convert raw material into finished product, workinprocess inventory is developed. This inventory takes the form of orders waiting to be transported between machines or of orders waiting to be processed on a particular machine. The level of such inventory can be increased by changing the production process, lot sizes, and so on.
 Finished goods inventory: It is the inventory of final products which could be released for sale to the customers. The size of this inventory depends upon the demand and the ability of the firm to sell its products.
 Spare parts inventory: These are the parts which are used in the production process but do not become part of the product.
 Raw materials and component parts: It is used to decouple the producer from the suppliers. That is, raw materials and component parts inventory could
Operating Constraints: The stock level of various items in the inventory is governed by various constraints such as limited warehouse space, limited budget available for inventory, customer service level to be achieved, and management attitude about the individual items in the inventory.
Operating Decision Rules: Two type of decisions need to be made in managing inventories:
 How much (size) is to order when the inventory of that item is to be replenished?
 When to place an order (or set up production) to replenish inventory?
These may be classified as follows:
The size and timing of replenishment of orders for an item can be decided using any one of the four inventory control decision rules given below:
Inventory Decision Rules 

Order frequency  Order quantity 

Fixed (Q)  Variable (S)  
Variable (R)  (Q, R)  (S, R) 
Fixed (t)  (Q, t)  (S, t) 
The above involves placing orders for either a fixed or a variable quantity (size) with either a fixed or a variable time between successive orders. For example, under the (Q, R) rule, an order for a fixed quantity (Q) for replenishment is placed when the inventory balance drops to a certain point (reorder level R). This policy is called a fixed order quantity or reorderpoint policy. Similarly, under (S, t) rule, order of replenishment are placed after every fixed interval (t) for an amount equal to the difference between the current stock level in hand and a predetermind maximum level. This policy is known as the fixedinterval policy or orderperiod policy or periodic review policy.
The success of any inventory decision rule depends on the accurate determination of its parameters Q, R.
Steps of Inventory Model Building
The steps to build up a suitable inventory model and then to derive decision rules are as follows:
Step 1: First take physical stock of all the inventory items in an organisation.
Step 2: Classify the stock of items into various categories.
Step 3: After classification of inventories, each item should be assigned a suitable code.
Step 4: Build up a mathematical model to achieve the objective function of minimising the total inventory costs subject to changes in inventory reorder policy and constraints of limited resources. The model would either be an unconstrained optimisation model or a mathematical programming model depending upon whether constraints are imposed or not.
Step 5: Derive an optimal inventory policy (i.e economic order quantity) by using an appropriate solution procedure to ensure balance among inventory costs.
11.2 COST INVOLVED IN INVENTORY PROBLEMS
Various costs involved in inventory problems may be generally classified as follows:
Setup Cost (C_{S} or C_{3}): These include the fixed cost associated with obtaining goods through placing of orders or purchasing or manufacturing or setting up a machinery before starting production. The costs include ordering of raw meterials for production purposes, advertisements, postage, telephone charges, travel expenditure and so on. These are also called order costs or replenishment costs per production run. These are assumed to be independent of the quantity ordered or produced. Ordering cost may be calculated as:
Ordering cost = (Cost per order) × (Number of orders)
Purchase or Production Cost: The cost of purchasing (or producing) a unit of item is known as purchase (or production) cost. Purchase cost per unit item is affected by the quantity purchased due to quantity discounts or price breaks.
Purchase cost = (Price per unit item) × (Demand per unit time)
When price break or quantity discounts are available for bulk purchase of a specified quantity, the unit price becomes smaller as size of order Q exceeds a specified quantity level. In such cases the purchase cost becomes variable and depends on the size of the order. In this case purchase cost is given by
Carrying or Holding Cost (C_{1} or C_{h}): The cost associated with carrying or holding goods in stock is called holding or carrying cost per unit of item for a unit of time. It is assumed to vary directly with the size of inventory as well as the time the item is held in stock. This cost generally includes:
 Invested Capital Cost: This is the interest charged on the capital investment.
 Recordkeeping and Administrative Cost: This shows that there is a need of funds for maintaining the records and necessary administration.
 Handling Costs: These include all costs associated with movement of stock, cost of labour, and so on.
 Storage Costs: These involve the rent for storage space or depreciation and interest even if own space is used.
 Depreciation, Deterioration, Obsolescence Costs: These costs arise due to the items in stock being out of fashion or the items undergoing chemical changes during storage, date expiring and so on.
 Taxes and Insurance Costs: These costs require careful study and generally amounts to 1 per cent to 2 per cent of the invested capital.
Shortage (or Stock out) Cost (C_{2}): The shortage of items occurs when actual demand cannot be fulfilled from the existing stock. These costs arise due to shortage of goods, and may cause loss of sales. Goodwill may be lost either by a delay in meeting the demand or being unable to meet the demand. The shortage can be looked at in two different ways:
 The supply of items is awaited by the customers, that is, the items are back ordered and therefore there is no loss in sale.
 Customers are not ready to wait and, therefore, there, is loss of sale. In the case of shortage, cost shall be measured in terms of goodwill lost and lost profit on the unit which were demanded but were not available.
Shortage cost may be calculated as:
Average number of units short
Salvage Cost (or Selling Price): When the demand for commodity is affected by the quantity stocked, the decision model of the problem depends upon the profit maximisation criterion and includes revenue from selling. Generally, salvage value may be combined with the cost of storage and not considered independently.
Revenue Cost: When it is assumed that both the price and the demand of the product are not under the control of the organisation, the revenue from the sales is independent of the company’s inventory policy. It may be neglected expect when the organisation cannot meet the demand and the sale is lost. The revenue cost may or may not be included in the study of inventory.
Total Inventory Cost: If the unit price of an item depends on the quantity purchased, that is, price discount is available, then we formulate an inventory policy that considers the purchase cost of the items held in stock. The total inventory cost is given as
Total inventory cost = Ordering cost + Carrying cost + Shortage cost
Total Variable Inventory Cost: When price discounts are not offered, the purchase cost remains constant and is independent of the quantity purchased. Hence,
Total variable inventory cost = Ordering cost + Carrying cost + Shortage cost.
Other Factors in Inventory Analysis: The factors which play an important role in the study of inventory problems are:
Demand: Demand is the number of units required per period and may be known exactly or in terms of probabilities or be completely unknown. If demand is known, it may be either fixed or variable per unit time. Problems in which the demand is known and fixed are called deterministic problems. If the demand is assumed to be a random variable, then those problems are called stochastic or probabilistic problems.
Lead time: The time gap between placing of an order and its actual arrival in the inventory is known as lead time. The level of inventory of an item depends upon the length of its lead time. The longer the lead time, the higher is the average inventory. Lead time has two components, namely administrative lead time—time from the initiation of procurement action to the placing of an order, and delivery lead time—time from placing of an order to the delivery of the ordered material.
Order cycle: The time period between placement of two successive orders is referred to as an order cycle. The order may be placed on the basis of following two types of inventory review systems.
 Continuous review: The record of the inventory level is checked continuously until a specified point is reached where a new order is placed.
 Periodic review: The inventory levels are viewed at equal time intervals and orders are placed at such intervals. The quantity ordered each time depends on the available inventory level at the time of review.
Stock replenishment: Actually, the replacement of stock may occur instantaneously or uniformly. Instantaneous replenishment occurs in case the stock is purchased form outside sources whereas uniform replenishment occurs when the product is manufactured by the company.
Time horizon: The time period over which the inventory level will be controlled is called the time horizon.
Recorder level: The level between maximum and minimum stock, at which purchasing (or production) activities start for replenishment is called reorder level.
Variables in Inventory problem: The variables used in any inventory model are of two types.
 Controlled variables
 Uncontrolled variables
 Controlled Variables: The following variables are controlled separately or in combination.
 How much quantity to buy (purchase, production, so on).
 The frequency or timing of acquisition—how often or when to replenish the inventory?
 The completion stage of stocked items.
 Uncontrolled variables: These include the holding costs, shortage or penalty costs, set up costs, demand, lead time, and supply of goods.
11.3 EOQ MODELS
11.3.1 Economic Order Quantity (EOQ)
The concept of economic ordering quantity was first development by F. Harries in 1916. The inventory problems in which the demand is assumed to be fixed and completely predetermined are usually referred to as economic order quantity. When the size of order increases, the ordering costs (cost of purchase, etc.) will decrease, whereas the carrying charges (cost of storage, insurance etc.) will increase. Hence, there are two opposite costs in the production process, one encourages increase in the order size and the other discourages. Hence, Economic Ordering Quantity (EOQ) is that size of order which minimises total annual (or any other time period as determined by individual firms) costs of carrying inventory and cost of ordering.
Note that by order quantity we mean the quantity produced or procured during one production cycle.
11.3.2 Determination of EOQ by Tabular Method
This method involves the following steps.
Step 1: Select a number of possible lot sizes to purchase.
Step 2: Determine the total cost for each lot size chosen.
Step 3: Finally, select the ordering quantity which minimises total cost.
11.3.3 Determination of EOQ by Graphical Method
The data calculated in tabular method can be graphed as in Fig 11.1.
FIGURE 11.1
The minimum total cost occurs at the point where the ordering costs and inventory carrying costs are equal.
A disadvantage of the graphical method is that without specific costs and values, an accurate plotting of the carrying costs, ordering costs, and total costs is not feasible.
Deterministic Inventory Models
The deterministic inventory models are categorised into four models:
Model I: Purchasing model with no shortages
Model II: Manufacturing model with no shortages
Model III: Purchasing model with shortages
Model IV: Manufacturing model with shortages
11.3.4 Model I: Purchasing Model with no Shortages
Assumptions of the Model
 Demand rate is uniform.
 Production rate is infinite.
 There is no shortage (i.e., shortage cost is infinite).
Here, the manufacturer has to order for supply of goods at a uniform rate D per unit time. He starts production run every t time units. Let the setup cost per production run be C_{3}. Since the replacement is instantaneous the production time is negligible. The holding cost per unit time is taken as C_{1}.
Objective of the Model
 How frequently should the manufacturer make a production run?
 How many units should be produced per run?
The diagrammatic representation of model I is shown in Fig. 11.2.
FIGURE 11.2 EOQ problem with uniform demand
If a production run is made at intervals t, a quantity q = Dt must be produced in one run. Since the stock in small time dt is Dtdt, the stock in period t will be given by
Thus the cost of holding inventory per production run = C_{1}Dt^{2}
Since the setup cost per production run is C_{3}, the total cost is given by C_{1}Dt^{2} + C_{3} (per production run).
Thus the average total cost per unit time C(t)
For C(t) to be minimum, and is positive. Differentiating C(t) with respect to t and equating to zero, we get
which gives
Now , which is positive for the value of t given by the above equation.
Thus, C(t) is minimum for optimum time interval .
Now optimal order quantity or economic order quantity or optimal lot size
The resulting minimum average cost per unit time is calculated as follows:
Minimum cost
Formulae of Model I
Example 1
An aircraft company uses rivets at an approximate customer rate of 2,500 kg per year. Each unit costs 30 per kg and the company personnel estimate that it costs 130 to place an order, and that the carrying cost of inventory is 10 per cent year. How frequently should orders for rivets be placed? Also, determine the optimum size of each order.
Solution:
Given
D = 2,500 per year
C_{1} = (cost of each unit) × (Inventory carrying cost)
= 30 × (0.1)
= 3.
C_{3} = 130 per order
Then,
= 465.47 units
= 0.186 year = 22.232 months
n = number of orders =
= 5.37 orders per year
Example 2
The production department for a company requires 3,600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is 36 and the cost of carrying inventory is 25 per cent of the investment in the inventories. The price is 10 per kg. The purchase manager wishes to determine an ordering policy for the raw material.
Solution: Given,
D = 3,600 kg year
C_{3} = C_{S} = 36 per order
C_{1} = 25 per cent of the investment in inventories
= 10 ×
= 2.5 per kg/year
So,
= 321.99 kg per order
= 0.08894 year
Minimum total annual inventory cost is
Minimum yearly total cost = C(q) + DC
= 804.98 + (3,600 kg) (10/kg)
= 36,804.98 per year.
A manufacturing company purchases 9,000 parts of a machine for its annual requirements, ordering one month’s requirement at a time. Each part costs 20. The ordering cost per order is 15 and the carrying charges are 15 per cent of the average inventory per year. You have been assigned to suggest a more economical purchasing policy for the company. What advice would you offer, and how much would it save the company per year?
Solution:
Given,
D = 9,000 parts per year
C_{3} = 15 per order
C_{1} = 15 per cent of the average inventory per year
= 20 × = 3 each part per year
Optimal lot size q = = 300 parts
Optimal order cycle = t = = 12 days (approx.)
Minimum average yearly cost =
=
= 900
But, if the company follows the policy of ordering every month, then the annual ordering cost is 12 × 15 = 180, and lot size of inventory each month = = 750 parts.
Average inventory at any time = Q/2 = 375 parts
Storage cost at any time = 375 C_{1} = 375 × 3 = 1,125.
Total annual cost = 1,125 + 180 = 1,305.
Hence, if the company orders 300 parts after every 12 days instead of 750 parts after every month, there will be a net saving of (1,305 − 900) = 405 per year.
11.3.5 Model II: Manufacturing Model with No Shortages
Assumptions
 Demand rate is uniform.
 Production rate is finite.
 Run sizes are constant and new run will be started whenever inventory is zero.
Notations
q, t, C_{1}, C_{3}, D as in Model I.
K: Number of items produced per unit time
FIGURE 11.3 Inventory level with finite rate of replenishment
Here, the time interval t is divided into two parts t_{1} and t_{2}. During the time period t_{1} (production rate K per unit time and utilisation D per unit time), the inventory is building up at the rate of K − D units per unit time. During the time period t_{2}, there is no production (or supply) and the inventory is decreasing at a constant rate D per unit time (Fig 11.3).
Let I_{m} be the maximum inventory available at the end of time t_{1}, which is expected to be consumed during the remaining period t_{2} at the demand rate D.
Thus
I_{m} = (K − D) t_{1}
or
Now the total quantity produced during time t_{1} is q and the quantity consumed during the same period is Dt_{1}; therefore, the remaining quantity available at the end of time t_{1} is
or
Now holding cost per production run for time period
Setup cost per production run = C_{3}
Thus total average cost per unit time
which gives
Thus the optimum lot size
Optimum time interval
Optimum average cost per unit time
Remark:
 If K = D, C = 0 and so there will be no holding cost and setup cost.
 If K = ∞, that is, if the production rate is infinite, then model II reduces to model I.
Example 1
A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He finds that when he starts a production run, he can produce 25,000 bearings per day. The cost of holding a bearing in stock for one year is 2 paise and the setup cost of a production run is 18. How frequently should the production run be made?
Solution:
Given,
C_{1} = 0.02 per bearing per year
= 0.000055 per bearing per day
C_{3} = 18.00 per production run
D = 10,000 bearings per day
K = 25,000 bearings per day
Therefore,
Length of production cycle = = 40 days (approx.)
Thus, the production cycle starts at an interval of 10.4 days and production continues for 4 days so that in each cycle a batch of 1,04,447 bearings is produced.
Example 2
 At present a company is purchasing an item X from outside suppliers. The consumption is 10,000 units/year. The cost of the item is 5 per unit and the ordering cost is estimated to be 100 per order. The cost of carrying inventory is 25%. If the consumption rate is uniform, determine the economic purchasing quantity.
 In the above problem assume the company is going to manufacture the item with the equipment that is estimated to produce 100 units per day. The cost of the unit thus produced is 3.50 per unit. The setup cost is 150 per set up and the inventory carrying charge is 25%. How has your answer changed?
Solution: (a) Given
D = 10,000 units per year
C = 5 per unit.
C_{3} = 100 per order
C_{1} = 25% of 5 = × 5
= 1.25 per year
EOQ q = = 1.265 units
(b) Given, K = 100 units per day
C = 3.50 per unit.
C_{3} = 150 per set up
C_{1} = 25 % of 3.50 = 0.875 per year
D = = 40 units per day (assuming 250 working days in the year).
EOQ for each production run is
The increase in EOQ in case (b) may be due to the following reasons:
 Increased item cost (c) and procurement cost (i.e. setup cost C_{S})
 Consumption of inventory simultaneous with production reduces average inventory from Q/2 to .
Example 3
An item is produced at the rate of 50 items per day. The demand occurs at the rate of 25 items per day. If the setup cost is 100 per setup and holding cost is 0.01 per unit of item per day, find the economic lot size for one run, assuming that shortages are not permitted. Also find the time of cycle, minimum cost for one run, manufacturing time and maximum inventory.
Solution: Given
D = 25 items per day
C_{1} = 0.01 per unit per day
C_{3} = 100 per setup
K = 50 items per day
11.3.6 Model III: Purchasing Model with Shortages
Assumptions
 Demand rate is uniform.
 Production rate is infinite.
 Shortages are allowed.
 Lead time is zero.
Given D, C_{1}, C_{3}, q = Dt, and C_{2} = shortage cost per item per unit time. I_{m} denotes the inventory level.
FIGURE 11.4 EOQ problem with shortages
Divide the total time T into n equal parts, each of value t. The time interval t is further divided into two parts, namely t_{1} and t_{2}, where t_{1} is the time interval during which items are drawn from inventory and t_{2} is the time interval during which the items are in shortage (Fig. 11.4).
From ΔAOB and ΔAEC, we have
That is,
⇒
⇒
⇒
Optimum value of I_{m} is
Similarly gives
C_{1}I_{m}^{2} − C_{2}(q − I_{m})[q − (q − I_{m})] + 2C_{3}D = 0
⇒
C_{1}I_{m}^{2} − C_{2}(q − I_{m})[q + I_{m}] + 2C_{3}D = 0
⇒
C_{1}I_{m}^{2} − C_{2}[q^{2} − I_{m}2] + 2C_{3}D = 0
⇒
C_{1}I_{m}^{2} − C_{2}q^{2} + C_{2}I_{m}2 + 2C_{3}D = 0
⇒
That is, and
Total inventory during time t = area of ΔOAB = I_{m}t_{1}
Inventory holding cost during time t = C_{1}I_{m}t_{1}
Total shortage during time t = area of ΔBCD = (q − I_{m})t_{2}
Shortage cost during time t = C_{2}(q − I_{m})t_{2}
Setup cost during time t = C_{3}
So, total cost during time t = C_{1}I_{m}t_{1} + C_{2}(q − I_{m})t_{2} + C_{3}
(or) total average cost per unit time C(I_{m}, q)
Total average cost per unit time C(I_{m}, q), being a function of two variables I_{m} and q, has to be partially differentiated with respect to I_{m} and q separately and equated to zero.
⇒
or
Note:
 If the penalty cost C_{2} = ∞, then
 If the penalty costs are negligible, that is, C_{2} → 0, with C_{1} > 0, then optimum lot size q → 0.
 If inventory costs are negligible, then C_{1} → 0, C_{2} > 0; in this case, optimum lot size q → ∞.
 When inventories and shortages are equally likely, that is, when C_{1} = C_{2}, then .
In this case, optimum lot size
 If time is fixed, setup cost is not considered because the time of one production run is fixed and so optimum inventory level = .
A commodity is to be supplied at a constant rate of 200 units per day. Supplies of any amounts can be had at any required time, but each ordering costs 50.00. Cost of holding the commodity in inventory is 2.00 per unit per day while delay in the supply of the items induces a penalty of 10.00 per unit per delay of 1 day. Find the optimal policy (q, t) where t is the recorder cycle period and q is the inventory level after reorder. What would be the best policy, if the penalty cost became ∞?
Solution: Here,
D = 200 units per day
C_{3} = 50 per order
C_{1} = 2.00 per unit per day
C_{2} = 10.00 per unit per day
Thus, an optimal order quantity of 109.5 units must be supplied after every 1/2 day.
If the penalty cost C_{2} = ∞, then
and,
Example 2
A commodity is to be supplied at a constant rate of 25 units per day. A penalty cost is being charged at the rate of 10 per unit per day late. The cost of carrying the commodity in inventory is 16 per unit per month. The production process is such that each month (30 days) a batch of items is started and is available for delivery any time after the end of the month. Find the optimal level of inventory at the beginning of each month.
Solution: Given,
D = 25 units/day
C = 16/30 = 0.53 per unit/per day
C_{2} = 10 per unit/per day
t = 30 days
The optimal inventory level is given by
The demand for an item is 18,000 units per year. The holding cost per unit time is 1.20, the cost of shortage is 5.00, and the production cost is 400. Assuming that replenishment rate is instantaneous, determine the optimal order quantity.
Solution: Given D = 18,000, C_{1} = 1.20, C_{2} = 5.00 and C_{3} = 400.
q* optimum lot quantity
= 1.113 × 3,464
= 3,856 units
Optimum time interval
Example 4
A dealer supplies the following information in connection with a product.
Annual demand = 5,000 units
Buying cost = 250 per order
Inventory carrying cost = 30% per year
Price = 100 per unit
The dealer is considering the possibility of allowing back orders to occur for the product. He has estimated that the annual cost of back ordering (allowing shortages) the product will be 10 per unit.
 What should be the optimum number of units of the product he should buy in one lot?
 What quantity of the product should he allow to be back ordered?
 How much additional cost will he have to incur on inventory if he does not permit back ordering?
Solution: Given D = 5,000, C_{3} = 250 per order, C_{1} = 100 × 0.30 = 30 per unit, C_{2} = 10 per unit,
 Optimum order quantity
 Shortages = q* − S, where
= 147 units
Quantity to be back ordered = 588 − 147 = 441 units
 Optimum of C(I_{m}, q), where C(I_{m}, q) is the total average cost per unit time, is given by . So
If back orders are not permitted, then optimum of C(I_{m}, q) = 8660. So additional cost when back orders are not permitted = [8,660 − 4,330] = 4,330
11.3.7 Model IV: Manufacturing Model with Shortages
Assumptions
 Demand rate D is uniform.
 Lead time is zero.
 Production rate is finite (k units per unit time).
 Shortages are allowed.
 Each production run of length t consists of two parts, say t_{1} and t_{2}, such that
 the inventory is building up at a constant rate of (k − D) units per unit of time during t_{1}; k > D
 there is no replenishment (or production) during time t_{2} and the inventory is decreasing at the rate of D per unit of time
 the average inventory level would not only be determined by the lot size q but also be affected by production rate k and depletion (demand) D
The inventory system is shown in Fig. 11.5.
FIGURE 11.5 Inventory cycleproduction is finite
Stocks start at zero and increase for a period t_{1} and then decrease for a period t_{2} until they again reach zero at the point where a backlog piles up for the period t_{3}. At the end of t_{3}, production starts and backlog diminishes for the time t_{4} when the backlog reaches zero. The cycle repeats itself after total time (t_{1} + t_{2} + t_{3} + t_{4}). Now,
Holding cost = C_{1} × Δ OBC = C_{1} × q_{1}(t_{1} + t_{2})
Shortage cost = C_{2} × Δ EFC = C_{2} × q_{2}(t_{3} + t_{4})
and the setup cost is C_{3}. Hence average cost per unit time
The inventory is zero at O and during the period t_{1} and amount kt_{1} is produced. But because orders are filed at a rate D, the net increase q_{1} in inventory during t_{1} is given by
After time t_{1}, the production is stopped and the stock q_{1} is used up during t_{2}, and because the rate of use is D, we have
From (11.3.2) and (11.3.3), we have
During period t_{3}, shortages accumulate at the rate D,
So,
During period t_{4}, production rate is k and demand rate is D, so that the net rate of reduction of shortage become k − D, and thus we have
Form (11.3.5) and (11.3.6), it follows that
Finally, because the production cycle (t_{1} + t_{2} + t_{3} + t_{4}) and production q is just sufficient to meet the demand at a rate D, then
Now, eliminating t_{1}, t_{4}, q_{1} and q_{2} from (11.3.1) by using relations (11.3.4), (11.3.7), we have
The optimum , values of t_{2} and t_{3} after differentiating partially with respect to t_{2} and t_{3} and setting the results equal to zero is
and from (11.3.5), (11.3.8), the optimum order quantity is obtained as:
The minimum cost is given by
Remarks:
Production cycle time between setups
Optimum inventory level
Optimum production time =
If k = ∞, then
Example 1
The demand for an item in a company is 18,000 units per year, and the company can produce the item at a rate of 3,000 per month. The cost of one setup is 500 and the holding cost of 1 unit per month is 15 paise. The shortage cost of one unit is 20 per month. Determine (i) optimum production batch quantity and the number of strategies;(ii) optimum cycle time and production time; (iii) maximum inventory level in the cycle; and (iv) total associated cost per year if the cost of the item is 20 per unit.
Solution: Given,
C_{1} = 0.15 per month,
C_{2} = 20.00
C_{3} = 500.00, k = 3,000 units per month,
D = 18,000 units per year or 1,500 units per month.
 Optimum production batch quantity is given by
Number of shortages is given by
Optimum production time =
Optimum cycle time between setups =
Maximum inventory level is
Total associated cost is given by
11.4 EOQ PROBLEMS WITH PRICE BREAKS
Nowadays, discounts are offered for the purchase of large quantities of products, and discounts take the form of price breaks. If discount is not available, then the optimum value of q is
If a discount (price break) is available, then total cost per unit of the inventoty is
where, the cost of manufacturing (or purchasing) is K_{1} per unit and I denotes the holding cost per unit. The optimum value of q will be obtained by
Hence, the optimum value of
Case 1: EOQ problem with one price break
When there is only one price break (one quantity discount), and if
Range of quantity  Purchase cost per unit 

0 ≤ q_{1} < b  k_{11} 
b ≤ q_{2}  k_{12} 
where b is that quantity at and beyond which the quantity discount applies and k_{12} < k_{11}.
The procedure to obtain EOQ is:
Step 1: Compute . That is, find the optimum order quantity for the highet discount and compare it with the quantity b.
If ≥ b, then place orders for quantities of size and obtain discount, otherwise go to the next step.
Step 2: If < b, then compare the total inventory for Q = (for price k_{11}) with Q = b.
The values of C() and C(b) are:
If C()> C(b), then q* = b, otherwise q* =
Case 2: EOQ problems with two price breaks
Where there are two price breaks (two quantity discounts), and if
Range of quantity  Purchase cost per unit 

0 ≤ q_{1} < b_{1}  k_{11} 
b_{1} ≤ q_{2} < b_{2}  k_{12} 
b_{2} ≤ q_{3}  k_{13} 
where b_{1} and b_{2} are those quantities which determine the price breaks.
The procedure to obtain EOQ is:
Step 1: Compute and compare it with b_{2}.
Step 2: If ≥ b_{2}, the optimum order quantity is .
Step 3: If < b_{2}, compute . As < b_{2}, we have < b_{2}, as < < … So, either < b_{1} or b_{1} ≤ < b_{2}.
Step 4: If < b_{2}, b_{1} ≤ < b_{2}, then proceed as in the case of only one price break.
Step 5: If < b_{2}, < b_{1}, then compute .
Now, < b_{1} compare C() with C(b_{1}) and C(b_{2}) in order to get optimum purchase quantity.
Case 3: EOQ problem with n price breaks
When there are n price breaks, we have
Range of quantity  Purchase cost per unit 

0 ≤ q_{1} < b_{1}  k_{11} 
b_{1} ≤ q_{2} < b_{2}  k_{12} 
⋮  ⋮ 
b_{n−1} ≤ q_{n}  k_{1n} 
where b_{1}, b_{2}, …, b_{n − 1}, are those quantities which determine the price breaks.
The procedure to obtain EOQ is:
Step 1: Compute and compare it with b_{n − 1}.
Step 2: If ≥ b_{n−1}, the optimum order quantity is .
Step 3: If < b_{n−1}, compute . If > b_{n−2}, proceed as in the case of one price break.
Step 4: If < b_{n−2}, compute . If > b_{n−3}, proceed as in the case of one price break.
Step 5: If < b_{n−2}, < b_{n−3}, then compute . If ≥ b_{n−4}, then compute C() with C(b_{n − 3}), C(b_{n − 2}) and C(b_{n − 1}).
Compute this process until b_{n−(j+1}; 0 ≤ j ≤ n − 1 and then compare C() C(b_{n – j − 2}), …C(b_{n−1}).
Example 1
Find the optimum order quantity for a product for which the prices are as follows:
Quantity  Unit Cost () 

0 ≤ q_{1} < 500  10.00 
500 ≤ q_{2}  9.25 
The monthly demand for the product is 200 units, the cost of storage is 2% of the unit cost and the cost of ordering is 350.00
Solution: Given
C_{3} = 350.00, D = 200, I = 0.02
k_{11} = 10.00, k_{12} = 9.25
Step 1:
Since q_{2} = 870 > b_{2} = 500, the optimum purchase quantity is q* = = 870 units.
Example 2
The annual demand of a product is 10,000 units. Each unit costs 100 if the order placed is in quantities below 200 units. But for orders of 200 or above the price is 95. The annual inventory holding cost is 10% of the value of the item and the ordering cost is 5 per order. Find the economic lot size.
Solution: Given D = 10,000 units, C_{3} = 500, I = 10% of price of an item = 0.10, k_{11} = 100, k_{12} = 95.
The optimum order quantity is based on price k_{12} = 95.
Now,
Since < b(= 200), calculate
As < b (= 200), therefore, comparing C() and C (b = 200), we have
Since C() < C(b), the optimal order quantity is q* = = 100 units.
Example 3
Annual demand for an item is 6,000 units. Ordering cost is 600 per order. Inventory carrying cost is 18% of the purchase price/unit/year. The price break up is as shown below:
Quantity  Price 

0 ≤ q_{1} < 2,000  20 
2,000 ≤ q_{2} < 4,000  15 
4,000 ≤ q_{3}  9 
Find the optimal order size.
Solution: Given D = 6,000/year, C_{3} = 600, I = 0.18, k_{11} = 20, k_{12} = 15, k_{13} = 9.
Step 1:
Since < b_{2} (= 4,000), proceed to next step.
Step 2: Compute . Here, the highest discount available is k_{12} = 15. Now,
Here, < b_{1} (= 2000), so go to next step.
Step 3: Compute . Here, k_{11} = 20. Now,
Since < b_{1}, compare C() with C(b_{1}) and C(b_{2}) to get the optimum purchase quantity.
The least cost is 58,140, hence optimal order quantity is b_{2} = 4000 units.
Example 4
The annual demand for a product is 500 units. The cost of storage per unit per year is 10% of the unit cost. The ordering cost is 180 for each order. The unit cost depends upon the amount ordered.
The range of amount ordered and the unit cost price are as follows:
Quantity  Price 

0 ≤ q_{1} < 500  25 
500 ≤ q_{2} < 1,500  24.80 
1,500 ≤ q_{3} < 3,000  24.60 
3,500 ≤ q_{4}  24.40 
Find the optimal order quantity.
Solution: Given D = 500 units, C_{3} = 180, I = 0.10, b_{1} = 500, b_{2} = 1,500, b_{3} = 3,000, k_{11} = 25, k_{12} = 24.80, k_{13} = 24.60, k_{14} = 24.40.
Step 1:
Since < b_{3}, we compute .
Step 2:
Since < b_{2} (= 1,500), we calculate .
Step 3:
Since < b_{1}, we go to next step.
Step 4:
Now, compute C() and compare with C(b_{1}), C(b_{2}) and C(b_{3}) in order to get optimal order quantity.
Since C() < C(b_{1}) < C(b_{2}) < C(b_{3}), = 268 units is the optimum order quantity.
Example 5
 Mini Computer Company purchases a component of which it has a steady usage of 1,000 units per year. The ordering cost is 50 per order. The estimated cost of money invested is 25% per year. The unit cost of the component is 40. Calculate the optimal ordering policy and total cost of inventory system, including purchase cost of the components.
 If in the above, the component supplier agrees to offer price discounts of minimum lot supplies as per schedule given below, reassess the decision on the optimal ordering policy and the total cost.
Lot Size  Price 

Upto 149  40 
150–499  39 
500 or more  38 
Solution: (a) Given D = 1,000 units, C_{3} = 50, I = 0.25, k_{11} = 40
Now,
(b) When quantity discounts are allowed
Since < 149 units, it is not feasible.
Total cost
Also,
Since < 150 units, it is not feasible.
Now,
Since C(150) < C(q_{1}), the optimum purchase quantity = 150 units.
Saving = 41,000.00 − 40,064.58
= 935.42.
11.5 REORDER LEVEL AND OPTIMUM BUFFER STOCK
In this section we study inventory models in which demand cannot be completely predetermined. It is uncertain and can fluctuate either way. In many situations, it is found that neither the consumption rate nor the lead time is constant throughout the year. In order to overcome these difficulties, extra stock is maintained to meet the demands, if any. This extra stock is termed as Buffer Stock or Safety Stock.
Determining Buffer Stock and Reorder Level
To determine the buffer stock, we approximate the estimated maximum lead time and normal lead time for a particular item. Let B denote the buffer stock, L_{d} denote the difference between the maximum and normal lead times, r the consumption rate during the lead time. Then,
B = L_{d} × r
If we do not maintain a buffer stock, then the total requirements for inventory during lead time L will become Lr. Thus, as soon as the stock reaches the level Lr, a quantity Q should be ordered. This point is called reorder level (ROL). However, due to uncertainty in supply, the policy of ROL may result in shortages. To avoid this, a buffer stock B is added and an order placed when stock level reaches B + Lr. Hence, the reorder level is given by
ROL = Lead time demand + Buffer stock
= B + Lr
= L_{d}r + Lr = (L_{d} + L)r
For example, if the monthly consumption rate for items is 150 units, the normal lead time is 15 days, and the buffer stock is 200 units. Then, ROL units 200 + × 150 = 275 units.
If we take t days for reviewing the reorder system, then on the assumption of uniform consumption rate, we get
Determining Optimum Buffer Stock
When the buffer stock maintained is very low, the inventory holding cost is low but shortages will occur frequently and the cost of shortages will be high. Suppose the buffer stock maintained is very high, then shortages would be rare, and the cost of shortages would be low, but the inventory costs would be high.
Hence, it becomes necessary to strike a balance between the cost of shortages and the cost of inventory holding to arrive at the optimum buffer stock.
Example 1
The average monthly consumption for an item is 300 units and the normal lead time is one month. If maximum consumption has been up to 370 units per month and maximum lead time is 1 months, what should be the buffer stock for the item?
Solution: Max. lead time demand = (Max. lead time) × (Max. demand rate) = × 370 = 555 units.
Normal lead time demand = 1 × 300 units
Buffer stock = 555 − 300 = 255 units
Example 2
A firm uses 12,000 units of a raw material every year costing 1.25 per unit. Ordering cost is 15 per order and the holding cost is 5% per year of average inventory.
 Find the economic order quantity.
 The firm follows EOQ purchasing policy. It operates for 300 days per year. Procurement time is 14 days and safety stock is 400 units. Find the reorder point, the maximum inventory and the average inventory.
Solution: (i)
(ii) ROL = Buffer stock + Consumption during normal lead time
(iii) Maximum inventory = q* + B = 2,400 + 400 = 2,800 units
(iv) Minimum inventory = B = 400 units
(v) Average inventory = = 1,600 units
The following information is provided for an item:
Annual usage = 12,000 units, ordering costs = 60 per order, carrying cost 10%, unit cost of item = 10 and lead time =10 days.
There are 300 working days a year. Determine the EOQ and the number of orders per year. In the past two years the use rate has gone as high as 70 units per day. For a reordering system based on the inventory level, what should be the safety stock? What should be the reorder level at this safety stock? What would be the carrying costs for a year?
Solution: Given C_{3} = 60/order, D = 12,000 units/year,
Therefore,
No. of orders/year =
Average usage =
Maximum usage = 70 units/day
Therefore, safety stock = (70 − 40) × 10 = 300 units
ROL = Average lead time demand + Safety stock
= 40 × 10 + 300 = 700 units
Average inventory =
Hence, inventory carrying cost/year =
Example 4
An automobile has determined that 16 spare engineers will result into a stockout risk of 25% while 20 will reduce the risk to 15% and 24 to 10%. If the lead time is 3 months and the average usage is 6 engines per month, what should be the ROL to maintain 85% service level?
Solution: Lead time demand = 3 × 6 = 18 engines
Safety stock for 85% service or 15% disservice = 20 engines
ROL = 18 + 20 = 38 engines.
Example 5
A firm has normally distributed forecast of usage with mean absolute deviation (MAD) = 60 units. It desires a service level which limits stock outs to one order cycle per year.
 How much safety stock should be kept if the order quantity is normally a week’s supply?
 What will be the safety stock if the order quantity is 4 weeks’ supply?
Solution: (a) No. of orders/year = 52. Since there is one stock out per year, service level =
Therefore, safety (service) factor Z = 2.05 (by normal table),
Therefore,
(b) No. of orders per year =
Therefore, service level =
Safety factor Z from normal table = 1.43
Example 6
The average demand for an item is 120 units/year. The lead time is 1 month and the demand during lead time follows normal distribution with average of 10 units and standard deviation of 2 units. If the item is ordered once in 4 months and the policy of the company is that there should not be more than 1 stock out of every two years, determine the reorder level.
Solution: No. of orders in 2 years = (2 × 12) = 6%. Service = × 100 = 83.33. Therefore, Z (from normal table) = 0.96. Now Z = ⇒ Safety stock = Z × SD = (0.96) × 2 = 1.96.
Also, ROL = 1 × 10 + 1.92 = 11.92 units
Example 7
A look at the past records gives the following distribution for lead time and daily demand during lead time.
What should be the buffer stock?
Solution: The computation of average lead time.
Lead time (1)  Frequency (2)  (1) × (2) 

1  0  0 
2  1  2 
3  2  6 
4  3  12 
5  4  20 
6  4  24 
7  3  21 
8  2  16 
9  2  18 
10  1  10 
22  129 
Average lead time = 5.56 days
Computation of average demand rate
Lead time (1)  Frequency (2)  (1) × (2) 

0  3  0 
1  5  5 
2  4  8 
3  5  15 
4  2  8 
5  3  15 
6  2  12 
7  1  7 
25  70 
Average demand rate = = 2.8
Average lead time demand = 5.86 × 2.8 = 16.4 units
Max. lead time demand = Max. lead time × Max. demand rate
= 10 × 7 = 70 units
Buffer stock = 70 − 16.4 = 53.6 units
Example 8
 A company uses 24,000 units of a raw material annually which costs 1.25 per unit. Placing each order costs 22.50 and the carrying cost is 5.47 per year of the average inventory, find the EOQ and total inventory cost.
 Should the company accept the offer made by the supplier of a discount of 5% on the cost price on a single order of 24,000 units?
 Suppose the company works for 300 days a year. If the procurement time is 12 days and safety stock is 400 units, find the reorder point, the minimum, maximum and average inventory.
Solution:
 Give D = 24000 units/year, C = 1.2/unit, C_{3} = 22.50/order
 Under the discount offer, annual fixed cost (Capital cost) = (1.25 × 0.95 × 24,000) = 28,500 inventory carrying cost = × 24,000 × (0.0675 × 0.95) = 769.50
Ordering cost = 22.50. Total annual cost = (28,500 + 769.50 + 22.5) = 29,292.
Since the total annual cost under discount conditions is lower than that under optimal order size, discount offer should be accepted.
 Since the company works for 300 days, the daily demand = 50 days, lead time = 12 days. Since t_{0} > lead time,
ROL = 80 × 12 + 400 = 1,360 units.
Minimum inventory = 400 units, Maximum inventory = 400 × q* = 4,400 units
Average inventory = (400 + 4,400) = 2,400 units.
Example 9
A company uses 50,000 units of an item annually each costing 1.20. Each order costs 45 and inventory carrying costs are 15% of the annual average inventory value.
 Find EOQ.
 If the company operates 250 days a year, the procurement time is 10 days and safety stock is 500 units, find reorder level, maximum, minimum and average inventory.
Solution: Given D = 50,000 units, C_{3} = 45 per order
C = 1.20 per unit
I = 0.15
 The company operates 250 days a year, requirement per day .
Lead time demand = r = 10 × 200 = 2,000 units
Safety Stock = B = 500 units
Hence, ROL = 2,000 + 500 = 2,500 units
Maximum inventory = 5,000 + 500 = 5,500 units
Minimum inventory = 5,000 units
Average inventory = × 5,000 + 500
= 3,000 units.
11.6 PROBABILISTIC INVENTORY MODELS
The inventory models discussed in earlier sections seems to be unrealistic. In some reality, the demand of an item is not known exactly but it will follow some probability distribution. In probabilistic systems we minimise the total expected costs rather than actual costs.
Consider the case of a perishable item, for example, newspaper. Assume that the demand of the newspaper does not follow a fixed pattern. A probability distribution can be obtained for the same. The probability distribution may be a discrete distribution, a uniform distribution or continuous distribution.
For example, for each unsold newspaper, there will be a penalty which is given by the formula
Marginal cost of surplus/unit C_{1} = Purchase price/unit − Salvage value/unit
For each shortage unit, there will be a penalty which is given by the formula
Marginal cost of shortage/unit C_{2} = Selling price/unit − purchase price/unit.
Let the generalised probability distribution of the demand of the item be a discreate distribution as shown below:
The optimal order size D_{i} is determined by the relation
where P_{i} is the cumulative probability of having demand up to D_{i}.
11.6.1 Model V: Instantaneous Demand, No Setup Cost, Stock in Discrete Units
The following assumptions should be made in this case:
 t is the constant interval between orders (t may be considered as unity, e.g. daily, weekly, monthly, etc.).
 q is the stock (in discrete units) for time t.
 r is the estimated (random) demand at a discontinuous rate with probability p(r).
 C_{1} is the holding cost per item per t time unit.
 C_{2} is the shortage cost per item per t time unit.
 Lead time is zero.
In this model, it is assumed that the total demand is filled at the beginning of the period. Thus, depending on the amount r demanded, the inventory position just after the demand occurs as a surplus or shortage. These two cases are shown in Fig.11.6(a) and 11.6 (b).
Case 1: When demand r does not exceed the stock q, that is, r ≤ q
Case 2: When demand r exceeds the stock q, that is, r > q,
FIGURE 11.6
Suppose r ≤ q; the holding cost becomes
Suppose r > q; the shortage cost per unit time is
To get expected cost, multiply the cost by given probability p(r). To get the total expected cost, sum over all expected costs, that is, the costs associated with each possible value of r. Hence, the total expected cost per unit time is given by the cost equation
To find the optimum value of q so as to minimise C(q), the following conditions must hold
C(q + 1) − C(q) > 0 and C(q) − C(q − 1) < 0
i.e.,
ΔC(q − 1) < 0 < ΔC(q)
Now,
ΔC(q) = C(q + 1) − C(q)
For minimum of C(q), ΔC(q) > 0. Hence,
⇒
Similarly, ΔC(q−1) = C(q)−C(q−1) implies
Hence,
Example 1
The daily demand of bread at a bakery follows a discrete distribution as follows:
The purchase price of the bread is 8 per packet. The selling price is 11 per packet. If the bread packets are not sold within the day of purchase, they are sold at 4 per packet to hotels for secondary use. Find the optimal order size of the bread.
Solution: Given purchase price/packet = 8
Selling price/packet = 11
Salvage price/packet = 4
Marginal cost of surplus C_{1} = 8 − 4 = 4
Marginal cost of shortage C_{2} = 11 − 8 = 3
Therefore, cumulative probability
Next, we find the cumulative probability of demand
From the Table it follows that
That is,
0.41 < 0.43 < 0.50
Therefore, the optimal order size is D_{4}, which is equal to 28 breads.
11.6.2 Model VI: Instantaneous Demand and Continuous Units
Here, the instantaneous demand is a continuous random variable rather than a discrete one. The probability distribution p(r) are replaced by the probability density function f(r). In this case and f (r) ≥ 0. The cost matrix for this model becomes
The optimal value of q is obtained by equating to zero the first derivative of C(q).
That is,
We know that if
Differentiating the equation (11.6.1), we get
In order that , we have
⇒
Further,
Hence, the optimum value of q satisfies
Example 2
A baking company sells cake by the kilogram. It makes a profit of 50 paise a kilo on every kilogram sold on the day it is backed. It disposes of all cakes not sold on the date it is baked at a loss of 12 paise a kilogram. If demand is known to be rectangular, between 2,000 and 3,000 kg, determine the optimum daily amount baked.
Solution: Given,
C_{1} = 0.12 (sale of left overs);
C_{2} = 0.50 (profit on sale)
Demand r is rectangular between 2,000 and 3,000 kg means the distribution f (r) is continuous, given by
We know
So,
⇒
⇒
⇒
q = 2,806.45 kg.
11.6.3 Model VII: Uniform Demand, No Set up cost
Assumptions
 Demand is uniform over a period. Let it be r units per time period.
 Reorder time is fixed and known. Thus, the setup cost is not included in the total cost.
 q is the stock level to which the stock is raised at the end of every period t.
 The production is instantaneous.
 C_{1} is the carrying cost per quantity per unit time.
 C_{2} is the shortage cost per quantity per unit time.
 Lead time is zero.
 f (r) is the probability distribution for demand r which is known.
The demand occurs uniformly rather than instantaneously during the period as shown in Fig. 11.7(a) and Fig. 11.7(b).
FIGURE 11.7 Uniform demand no set up cost model
The problem is to determine the optimal order level where r ≤ q or r > q. For r ≤ q, no shortages occur. For r > q, shortage occurs.
Now, for r ≤ q: No shortage and only carrying cost is involved (see Fig. 11.7(a)).
For r > q: Both costs are involved (see Fig. 11.7(b)).
The carrying cost is the area of ΔOAD and the shortage cost is the area of ΔABC
By the property of similar triangles of ΔDEB and ΔDOA, we get
Similarly, we get
Hence, total shortage cost
The total expected cost is given by the cost equation
The optimum value of q is obtained by minimising (C(q)). In that case, we have
C(q + 1) − C(q) > 0 and C(q) − C(q − 1) < 0
i.e.
Δ C(q − 1) < 0 < ΔC(q)
Now,
ΔC(q) = C(q + 1) − C(q)
But ΔC(q) > 0, for minimum C(q). Hence,
By using the condition ΔC(q − 1) < 0 for minimum C(q), we obtain the relation
Hence,
Using the above relation we can find the range of optimum value of q.
Example 3
The probability distribution of monthly sales of a certain item is as follows:
The cost of carrying inventory is 10.00 per unit per month. The current policy is to maintain a stock of four items at the beginning of each month. Assuming that the cost of shortage is proportional to both time and quantity short, obtain the imputed cost of a shortage of one item for one unit of time.
Solution: Given,
C_{1} = 10 per unit per month
q = Optimum stock = 4 units
Since the demand is uniformly distributed over the month, the least value of shortage cost C_{2} can be determined using the relation.
Now,
The least value of C_{2} is given by
or,
C_{2} (1 − 0.92) = 9.2 or, C_{2} = 115
Similarly, the greatest value of C_{2} is obtained by considering
Hence, the range of shortage is given by
115 < C_{2} < 390.
11.6.4 Model VIII: Uniform Demand and Continuous Units
The cost equation for this model is similar to the above model. Replace p(r) by f(r) and the summation by integration. Hence, the total expected cost is
Now,
For minimum cost, = 0, therefore,
⇒
Further,
Hence, gives the condition for finding the optimum value of q.
Example 4
Let the probability density of demand of a certain item during a week be
This demand is assumed to occur with a uniform pattern over the week. Let the unit carrying cost of the item in inventory bè2.00 per week and unit shortage cost bè8.00 per week. How will you determine the optimal order level of the inventory?
Solution: Since f (r) = 0.1, 0 ≤ r ≤ 10, C_{1} = 2.00, C_{2} = 8.00, the optimum value of q can be obtained by the relation
or,
(0.1)[q + p[log 10 − log q] = 0.8
or,
(0.1)[q − q[logq + (2.3)q] = 0.8
3.3q − qlogq − 8 = 0
The solution of this equation q = 4.5 is obtained by trial and error method.
11.7 SELECTION INVENTORY CONTROL TECHNIQUES
Inventories in an organisation cannot be controlled with equal importance. Some inventory items may be very important and some may not be that important. Efficient management calls for an understanding of the nature of inventories and accordingly they give priorities and extent of control required in respect of each item. For this purpose, items may be classified into groups depending upon their utility and importance. Such a classification is termed the principle of selective control, and is used to control the inventories. Now, we see techniques like ABC, VED, HML, FSN, XYZ, SOS, SDE of selective control.
ABC Analysis (Always, Better, Control)
An ABC analysis consists of separating the inventory items into three groups A, B, C, where A−high consumption value items, B−moderate consumption value items, C−low consumption value items. Note that
 Category ‘A’ items: Most costly and valuable items are classified as ‘A’. Such items have large investment but not much in number. The items of this category are ordered frequently but in small number. A periodic review policy should be followed to minimise the shortage percentage of such items. Closed control is required for these category.
 Category ‘B’ items: The items having average consumption value are classified as ‘B’. These items have less importance than ‘A’ class items, but are much costly to pay more attention on their use. These items cannot be overlooked and require lesser degree of control then those in category ‘A’. Statistical sampling is generally useful to control these.
 Category ‘C’ items: The items having low consumption value are classified as ‘C’. Nearly 75 per cent of inventory items account only for 10 per cent of the total invested capital. Such items can be stocked at an operative place where people can help themselves with any requisition formality. Not much control is needed for the ‘C’ category. This will increase their investment cost.
After dividing the items into various categories, the usage value of each item is plotted in a graph. Such a graph is known as the ABC distribution curve.
The following procedure of classifying in A, B, C categories may be observed.
Step 1: Determine the number of units sold or used in the past one year period.
Step 2: Determine the unit cost standard for each item.
Step 3: Compute the annual consumption value (in rupees) for each consumed item by multiplying annual consumption (of units) with the unit price.
Step 4: Arrange the items in descending order according to their usage value computed in Step 3.
Step 5: Prepare a Table showing unit cost, annual consumption and annual usage value for each item.
Step 6: Calculate the cumulative sum of the number of items and the usage value for each item obtained in Step 3.
Step 7: Find the percentage of the values obtained in Step 6 with respect to the grand total of the corresponding columns.
Step 8: Draw a graph by taking percentage of items on the Xaxis and the corresponding usage value on the Yaxis. After plotting the various points on the graph draw a curve.
Step 9: Identify cutoff points X and Y where the curve sharply changes its shape. This provides three segments A, B and C. (See Fig. 11.8)
FIGURE 11.8 ABC distribution curve
Limitation of ABC Analysis
ABC analysis does not permit precise consideration of all relevant problems of inventory control. For example, a never ending problem in inventory management is that of adequately handling thousands of low value ‘C’ items.
Low value purchases frequently require more items, and consequently reduce the time allowance available and purchasing personnel for value analysis, vendor investigation and other ‘B’ items.
 If ABC analysis is not updated and reviewed periodically, the real purpose of control may be defeated. For example, ‘C’ items like diesel oil will become highvalue items during a power crisis and, therefore, it require more attention.
 ABC classification can lead to overlooking the needs of spare parts whose criticality is high but consumption value is low.
Example 1
The following information is known about a graph of items. Classify the items as A, B, and C.
Solution: Number of items sold in the past 12 months as well as the unit cost standard for each item are given. Multiplying annual consumption of each item by its unit cost and then ranking the items in the descending order of the usage values thus obtained, the following Table is obtained:
Compute the cumulative total number of items and their usage values and convert the accumulated total into percentage of the grand total.
The following classification is thus obtained.
The cutoff points have been determined from the Fig. 11.9.
FIGURE 11.9
VED Analysis: It is based on the criticality of items. If the items are arranged in the descending order of their criticality, then the more attention is to these type of items.
V—Vital items: Those items which when required are not available, which make the whole system inoperative (e.g. clutchwire used for scooters, motor cycles, etc.)
E—Essential items: Items which are not available when demanded, reduce the efficiency of the system, e.g., telephone.
D—Desirable item: The items which neither stop the system nor reduce its efficiency, but it will be good if they are present in the system.
This analysis is useful in controlling the inventory of spare parts.
Both ABC and VED are combined to control the stocking of spare parts. The control actions are:
XYZ Analysis: The classification is based on the closing value of items in storage. Items whose inventory values are high and moderate are classified as X items and Y items, while Z items are those having low inventory value.
XYZ classification is usually performed once in a year during annual stocktaking. It helps us in identifying items which are being stocked extensively. It can also be combined with ABC analysis and the controls of the items are shown in the Table below:
FNSD Analysis: Based on the consumption (or usuage) rate of items, it is classified as
F–Fast moving items
N–Normal moving items
S–Slow moving items
D–Dead items
F–items require close attention whereas
D items are send to the disposal cell. This method is used to control obsolescence in all types of inventories.
XYZ and FNSD can be combined to control obsolete items which are useful in the timely prevention of obsolescence.
SDE Analysis: It is based on the nature of procurement (or availability) of items. Here S represents scarce items, D represents difficult items which are available and but not always traceable and E represents easy to obtain items.
HML Analysis: Based on the unit price of items, HML classification separates inventory items, such as high price, medium price and low price. This controls overconsumption of various items of inventory.
The suitability of selective control techniques ABC, VED, XYZ, and FNSD depends on the nature of inventories carried by an organisation. The following Table gives the classification and their uses.
Selective control techniques  Basis of classification  Main uses 

ABC  Consumption value  Control raw material components and workinprogress inventories. 
VED  Criticality of the items  Determining the inventory levels of spare parts. 
XYZ  Value of items in storage  Reviewing the Inventories and other uses. 
FNSD  Consumption rate (or movement) of the item  Controlling obsolescence. 
EXERCISES
Sections 11.1 and 11.2
1. What are the types of inventory? Why are they maintained? Explain the various costs related to inventory.
2. Explain in detail what constitutes ordering cost and carrying cost.
3. What are the categories of costs associated within developing a sound inventory model? What are the components of cost under each of them?
4. What are the advantages and disadvantages of increase in inventory? Briefly explain the objectives of holding inventory.
5. Describe the basic characteristics of an inventory system.
6. Distinguish between deterministic and stochastic models.
7. Explain the significance of lead time and safety stock in inventory control.
8. Explain the terms lead time, reorder point, stockout cost and setup cost.
Section 11.3
9. An oil engine manufacturer purchases lubricants at the rate of 42 per piece from a vendor. The requirement of these lubricants is 1,800 per year. What should be the order quantity per order, if the cost per placement of an order is 16 and inventory carrying charge per rupee per year is only 20 paise?
[Answer: Optimum inventory quantity of lubricant is 83.]
10. A manufacturer has to supply his customers 600 units of his product per year. Shortages are not allowed and the shortage cost amounts tò0.60 per unit per year. The setup cost per run is 80. Find the optimum run size and the minimum average yearly cost.
[Answer: q = 346 units, C(q) = 173, t* = 11.5 days.]
11. A shopkeeper has uniform demand of an item at the rate of 600 items per year. He buys from a supplier at a cost of 8 per item and the cost of ordering is 12 each time. If the stockholding costs are 20% per year of stock value, how frequently should he replenish his stocks and what is the optimal order quantity?
Answer: q* = 95 units, t* = 57 days.]
12. For an item, the production is instantaneous. The shortage cost of one item is 1 per month and the setup cost is 25 per run. If the demand is 200 units per month, find the optimum quantity to be produced per setup and hence determine the total cost of storage and setup per month.
[Answer: q* = 100 units, t° = 15 days, C(q) = 125.]
13. An item is produced at the rate of 50 items per day. The demand occurs at the rate of 25 items per day. If the setup cost is 100 and holding cost is 0.01 per unit of item per day, find the economic lot size for one run, assuming that shortages are not permitted. Also, find the time of cycle and minimum total cost for one run.
[Answer: q* = 1,000 units, t* = 40 days, C(q) = 200.]
14. A contractor has to supply 20,000 units per day. He can produce 30,000 units per day. The cost of holding a unit in stock is 3 per year and the setup cost per run is 50. How frequently, and of what size, should the production runs be made?
[Answer: q* = 1414 units, t* = 1.68 hours.]
15. The demand for a certain item is 16 units per period. Unsatisfied demand causes a shortage cost of 0.75 per unit per short period. The cost of initiating purchasing action is 1 per purchase and holding cost is 15% of average inventory valuation per period. Item cost is 8.00 per unit. (Assume that shortages are being back ordered at the above mentioned cost). Find the minimum cost of purchase quantity.
[Answer: q* = 33.3 units, minimum cost = 14.88 (nearly)].
16. A commodity is to be supplied at a constant rate of 200 units per day. Supplies for any amounts can be had at any required time, but each ordering costs 50.00: costs of holding the commodity in inventory is 2.00 per unit per day while the delay in the supply of the item induces a penalty of 10.00 per unit per delay of 1 day. Formulate the average cost function of this situation and find the optimal policy q (t) where t is the reorder cycleperiod and q is the inventory level after reorder. What should be the best policy if the penalty cost becomes infinite?
[Answer: q* = 109.5 units, t* = days; when C_{2} → ∞, q* = 100 units.]
17. A dealer supplies you the following information with regard to a product handled by him: Annual demand = 5,000 units, buying cost = 250 per order, inventory carrying cost = 30% per year, price = 100 per unit.
The dealer is considering the possibility of allowing some backorders to occur for the product. He has estimated that the annual cost of backordering (allowing shortage) of the product that will bè10 per unit.
 What quantity of the product should he allow to be backordered?
 What should be the optimum number of units of the products he should buy in one lot?
 How much additional cost will he have to incur on inventory if he does not permit backordering?
[Answer: (i) 577.35 (ii) 433 units (iii) 4,330.13.]
18. A company has a demand of 12,000 units/year for an item and it can produce 2,000 such items per month. The cost of one setup is 400 and the holding cost per unit month is 0.15. The shortage cost of one unit is 20 per year. Find the optimum lot size and the total cost per year, assuming the cost of 1 unit as 14. Also, find the maximum inventory, manufacturing time and total time.
[Answer: q* = 3,413 units, C(q) = 51,336.]
19. The demand of an item is uniform at a rate of 25 units per month. The fixed cost is 15 each time a production run is made. The production cost is 1 per item, and the inventory carrying cost is 0.30 per item per month. If the shortage cost is 1.50 per item per month, determine how often to make a production run and of what size it should be?
20. An automobile factory manufactures a particular type of gear within the factory. This gear is used in the final assembly. The particulars of this gear are: demand rate D = 14,000 units/year, production rate k = 35,000 units/year.
Setup cost C_{3} = 500 per setup and carrying cost C_{1} = 15/unit/year. Find the economic batch quantity and cycle time.
,
Therefore, the cycle time is
The number of setups per year .]
Section 11.4
21. Find the optimum order quantity for the following: annual demand = 3,600 units, ordering cost = 50, cost of storage = 20% of the unit cost
Price break Quantity  Unit cost () 

0 ≤ q_{1} < 100  20.00 
100 ≤ q_{2}  18.00 
[Answer: q* = 316.23 units.]
22. Anil’s company buys 2,000 bats annually. A fixed cost of 50 is incurred each time an order is placed. Inventory carrying cost is estimated at 20%. Supplier offers a 10% discount in price per bat of 100 if orders are placed for more than or equal to 150 bats at a time. In what order size should the company purchase?
[Answer: q* = 150 bats.]
23. Find the optimal order quantity for a product for which the price breaks are as follows:
Quantity  Unit cost () 

0 ≤ q_{1} < 500  10.00 
500 ≤ q_{2} < 750  9.25 
750 ≤ q_{3}  8.75 
The monthly demand for the product is 200 units, the cost of storage is 2% of the unit cost and the cost of ordering is 350.
24. The consumption of an item is known to be fixed at 4,800 units per year. The cost of processing an order for purchase of this item is 400 and the inventory carrying charges work out to 24% per annum of the cost of the item. The cost of the item depends on the purchase lot size as per the schedule given below. Determine the optimum ordering policy.
Quantity per order  Unit cost () 

Up to 999  20.00 
1,000 to 1499  18.50 
1,500 and over  17.00 
[Answer: q* = 1,000 units.]
25. A shopkeeper has a uniform demand of an item at the rate of 50 items per month. He buys from supplier at a cost of 6 per item and the cost of ordering is 10 each time. If the stockholding costs are 20% per year of stock value, how frequently should he replenish his stocks? Now, suppose the supplier offers a 5% discount on orders between 200 and 999 items, and a 10% discount on orders exceeding or equal to 1,000. Can the shopkeeper reduce his costs by taking advantage of either of these discounts?
[Answer: (i) q* = 100 items C(q) = 3,720.00
(ii) q* = 200 items C(q) = 3,564.00
Saving by adopting EOQ is 156.00.]
26. Annual demand for an item is 500 units, ordering cost is 18 per order. Inventory carrying cost is 15 per unit per year. Relationship between price and quantity ordered is as follows:
Specify optimal order quantity and the corresponding price of this item.
[Answer: q* = 16 units, C(q) = 6,142.50.]
Section 11.5
27. A factory uses 32,000 worth of a raw material per year. The ordering cost per order is 50 and the carrying cost is 20% per year of the average inventory.
If the company follows the EOQ purchasing policy, calculate the reorder point, the maximum inventory, the minimum inventory and the average inventory, given that the factory works for 360 days a year, the replenishment time is 9 days and the safety stock is worth 300. [Answer: ROL = 1,100, maximum inventory = 4,300, minimum inventory = 300, average inventory = 2,300.]
Section 11.6
28. A newspaper boy buys paper for 60 paise each and sells them for 1.40 each. He cannot return unsold newspapers. Daily demand has the following distribution:
If each day’s demand is independent of the previous day’s demand, how many papers should be ordered each day?
[Answer: No. of papers to be ordered is 28.]
29. Some of the spare parts of a ship cost 1,00,000 each. These spare parts can only be ordered together with the ship. If not ordered at the time when the ship is constructed, these parts cannot be available on need. Suppose that a loss of 1,00,00,000 is suffered for each spare that is needed as replacement during the life term of the class of the ship discussed are:
How many spare parts should be procured?
[Answer: 3 spare parts should be procured.]
30. A baking company sells one type of cake by weight. It makes a profit of 9.50 on every kg of cake sold on the day it is baked. It disposes of all cakes not sold on the date it is baked at a loss of 1.50 per kg. If demand known to be rectangular between 300 and 400 kg, determine the optimum amount to be baked?
[Answer: q* = 386.4kg (nearly).]
31. An icecream company sells one type of icecream by weight. If the product is not sold on the day it is prepared, it can be sold at a loss of 50 paise per pound. But there is an unlimited market for oneday old icecreams. On the other hand, the company makes a profit of 3.20 on every pound of icecream sold on the day it is prepared. Past daily orders form as distribution with
f(x) = 0.02 − 0.0002 x, 0 ≤ x ≤ 100.
How many pounds of icecream should the company prepare every day?
[Answer: q* = 63.3 pounds.]
Section 11.7
32. Perform ABC analysis for the following inventory
Item  Annual Consumption  Price per unit (in Paise) 

A  300  10 
B  2,800  15 
C  30  10 
D  1,100  5 
E  40  5 
F  220  100 
G  1,500  5 
H  800  5 
I  600  15 
J  80  10 
Answer:
33. A company purchases three items A, B, C. Their annual demand and unit prices are given in the following Table.
Item  Annual demand (Units)  Unit price () 

A  1,00,000  3 
B  80,000  2 
C  600  96 
If the company wants to place 40 orders per year for the three items what is the optimal number of orders for each item?
[Answer: 18 orders of A, 14 orders of B, and 8 orders of C.]
34. What is selective inventory control?
35. The following 30 numbers represents the annual value in thousand of rupees of some 30 items of materials selected at random. Carry out an ABC analysis and list out the values of ‘A’ items only