JNTUK III Mechanical, Operations Research Notes

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UNIT6
INVENTORY
Introduction – single item – deterministic models – purchase inventory models with one price
break and multiple price breaks – shortages are not allowed – stochastic models – demand
instantaneous and supply instantaneous.
 The term inventory refers to any kind of resource that has economic value and is
maintained to fulfill the present and future needs of an organization.
 According to Fred Hasman, inventory refers is an idle resource of any kind provided
such resource has economic value.
TYPE OF ORGANIZATION TYPES OF INVENTORY OF
MATERIALS HELD
Manufacturer Raw Materials, semi-finished goods, finished
goods, spare parts etc
Hospital Beds, stocks of medicines, other medical
supplies
Bank Cash reserves
Airline Company Seating capacity, spare parts etc.
How much inventory should be held?
 Inventory of resources is held to provide desirable product or service to customers
(users) and hence to achieve sales targets.
 Since investment in inventory represents a substantial portion of the total capital
investment in any business, therefore investment in inventories beyond a certain level
affect organization’s cash flow and working capital.
 Hence, to ensure minimum inventory cost, it is essential to balance the advantage of
having inventory of resources and the cost of maintaining them.
INVENTORY CONTROL
(1) Items to be stocked
 Since physical storage of inventory items is expensive, therefore a control is needed to
ensure that inventory level remains as low as possible.
 This implies that:
(a) inventory level of existing items is kept at a reasonable level;
(b) unnecessary items are not added to the inventory;
(c) items which have not been used for a long time are removed from inventory.
(2) Time to replenish (to build up) inventory
 There are two different approaches to replenish the stock of inventory items

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(i) Periodic review system, where orders are placed at fixed intervals of time. The
quantity ordered varies, depending on the inventory in hand and consumption level at the
time of ordering.
(ii) Fixed order quantity system, where stock level of inventory items is monitored
regularly and when it comes down to a specific level, a replenishment order for a fixed
quantity is placed.
(3) Quantity of replenishment order
 Every time an order is placed, there are certain costs incurred on account of
administration, transportation, inspection etc.
 If large and infrequent orders are placed, it increases the average stock of inventory
items.
 If small and frequent orders are placed, it increases cost of ordering and delivery but
the average cost of inventory becomes low.
 Thus, an optimal inventory control policy is required to minimize the total inventory
cost.
 The order quantity usually depends on:
(i) Demand pattern;
(ii) Price of an item, discount options, total budget and warehouse space.
(iii) Lead time.
INVENTORY COST COMPONENTS
The inventory costs are affected (i.e., increase or decrease) by the policies of an organization
to maintain a particular level of inventory.
(1) Purchase Cost (Cost of the material)
 Purchase cost/year= (Price per unit) X (Annual Demand) = C X D
Where C = Price per unit. D = Annual Demand
 When price breaks (or quantity discounts) are available on purchase of material, the
unit price becomes smaller as the size of the order exceeds a specified quantity level.

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(2) Carrying cost (Holding cost)
 The inventory cost incurred for carrying (or holding) inventory items in the godown is
referred to as the carrying cost.
 The carrying cost includes:
(i) storage cost for rent paid for warehouse space;
(ii) inventory handling cost for payment of supervisors and security;
(iii) insurance premia against fire or other forms of damage;
(iv) opportunity cost of the money invested in inventory;
(v) obsolescence costs, deterioration costs, lost or pilfered costs; and
(vi) depreciation cost.
 Carrying cost is indicated
(i) as cost of carrying one unit per a time period (usually one year) or
(ii) as a percentage of the cost of the material
(3) Ordering (or set up cost)
 The inventory cost incurred each time an order is placed for procuring items from the
vendors is referred to as ordering cost.
 This cost per order generally includes:
(i) requisition cost of handling invoices, stationery, payments etc;
(ii) cost of services which include cost of mailing, telephone calls, transportation, and
other follow up actions;
(iii) materials handling costs incurred in receiving, sorting, inspecting and storing
items included in the order;
(iv) accounting and auditing etc.
 When an item is produced in-house, ordering cost is referred to as set-up cost, which
includes both paper work costs and physical preparation costs.
 Ordering (or set up) cost does not vary with size of the order (or production in case of
in-house production), but varies with the number of orders (or set-up in case of in-
house production) placed during a given period of time.
 Ordering Cost / year = Cost per order (or set up) X Number of orders (or set ups)
placed in the year.

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(4) Shortage Cost (Stock-out cost)
 Shortage occurs when inventory items cannot be supplied due to delay in delivery
or because demand becomes more than the expected demand.
 Shortage can be viewed in two different ways:
(i) Customers are ready to wait for supply of items (back ordering)
 In this case, there is no loss of sale, but the nature and magnitude of back ordering
cost, extra paper work and expenses incurred in processing the order are not exactly
known.
(ii) Customers are not ready to wait for supply of items
 In this case, an organization may suffer a loss of customer good will and a loss of
sale.
 The loss of good will is expected to increase in proportion to the length of the delay
and causes decline in the growth of business due to loss of potential revenue.
 Shortage Cost = (Cost of one unit of an item short) X (Avg number of units short)
 Avg number of units short = [ Minimum shortage + Maximum shortage] / 2

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TOTAL INVENTORY COST
Total Inventory Cost per year = Purchase Cost + Ordering Cost per year +
Carrying Cost per year + Shortage Cost per year
If there are no price discounts (i.e., the price is the same per unit irrespective of how much
many units are bought,
Total Variable Inventory Cost per year = Ordering Cost per year + Carrying
Cost per year + Shortage Cost per year
ECONOMIC ORDER QUANTITY
 Economic order quantity (EOQ) is the optimal replenishment order size (lot size) of
inventory items that achieves the optimum total inventory cost during the given period
of time.
 Thus, EOQ is the amount of inventory to be ordered at one time for purposes of
minimizing annual inventory cost.
INVENTORY CONTROL MODELS
(1) EOQ models without Price-Quantity Discount
(a) Deterministic demand EOQ models
(i) Static demand models
o EOQ models without shortage
o EOQ models with shortage
o EOQ models with constraints
(ii) Dynamic demand models
(a) EOQ model with variable demand and constant lead time
(b) EOQ model with variable lead time
 Fixed order Quantity EOQ model with variable lead time
 Fixed interval ordering EOQ model with variable lead
time
(b) Probabilistic demand EOQ models
(2) EOQ models with Price-Quantity Discount.

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SINGLE ITEM INVENTORY CONTROL MODELS
(WITHOUT SHORTAGES)
EOQ MODEL WITH CONSTANT RATE OF DEMAND
Assumptions
(1) The inventory system involves one type of item (or product).
(2) The demand is known and constant and is resupplied immediately.
(3) The inventory is replenished in single delivery for each order.
(4) Lead time (LT) is constant and known.
(5) Shortages are not allowed. There is always enough inventory on hand to meet the
demand.
(6) Purchase price and reorder costs do not vary with quantity ordered. (Quantity discounts
are not available).
(7) Carrying cost per year (as a fraction of the cost of the material) and ordering cost per
order are known and are constant.
(8) Each item is independent and money cannot be saved by substituting an item with other
items.
 At the beginning of the inventory cycle time, a maximum amount of inventory equal
to the order Q is available in the stock.
 The level of inventory on consumption drops at a constant usage rate.
 When it reaches a specific level called reorder point, enough inventory is available to
cover expected demand during the lead time.

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 At this stage, an order equal to Q is placed. It arrives at the end of the lead time, when
the inventory level becomes equal to zero.
 This amount is placed in stock all at once and the inventory level goes up to its
maximum level Q.
Total Variable Inventory Cost per year = Annual ordering cost +Annual carrying cost
Annual ordering Cost = (Number of orders placed per year) X (Ordering Cost per order) =
(D/Q) X Co
where D = Annual Demand; Q = Quantity per order and Co= ordering cost per order
Annual Carrying (holding cost) = Average inventory level X Carrying cost per unit per year =
(Q/2) X Cc
Where Cc = carrying cost per unit per year.
Therefore, Total Variable Inventory Cost = (D/Q) X Co + (Q/2) X Cc

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As can be seen from the graph above, TVC is minimum at the point where the annual
ordering cost is equal to the annual carrying cost.
That is, when (D/Q*) X Co = (Q*/2) X Cc
Or when Q* = √(2 D Co)/Cc
This formula is known as the EOQ formula (or) Wilson-Harris lot size formula.
Example 1
(1) Annual demand = 1200 units.
Ordering cost = Rs 150 per order. Carrying cost/unit/year = Rs 4.
Determine the Economic order quantity.
Solution
Given Annual demand = D =1200 units.
Ordering cost = Co = Rs 150 per order.
Carrying cost/unit/year = Cc = Rs 4.
= √[(2X 1200 X 150)/4] = 300 units.
Economic Order Quantity = 300 units
(2) A factory uses annually 24,000 units of raw material which costs Rs. 1.25 per unit.
Placing each order costs Rs. 25/- and the carrying cost is 6% per year of the average
inventory. Find the economic order quantity and the total inventory cost (including the cost of
material).
Solution
Annual Demand = D = 24,000 units per year.
Ordering cost = Co = Rs 25 Cost of the material = C = 1.25
Carrying cost/unit/year = 6% of the cost of the material = 0.06 * 1.25 = Rs 0.075
EOQ = Q* = √(2 X 24000X 25/0.075) = 4,000 units.

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Total Inventory Cost
(1) Cost of material = D X C = 24000 X 1.25 = Rs 30,000
(2) Ordering cost per year = (D/Q*) Co = (24000/4000) X 25 = Rs 300
(3) Carrying cost per year = (Q*/2) Cc = (4000/2) X 0.075 = Rs 300
Total Inventory cost = 30,000 + 150 + 150 = Rs 30,300.
(3) The production department of a company requires 3600 kgs of raw material for
manufacturing a particular item per year. It has been estimated that the cost of placing an
order is Rs 36 and the cost of carrying an inventory is 25% of investment in inventories. The
price of the raw material is Rs 10 per kg. Help the purchase manager to determine an ordering
policy for raw material.
Solution
Annual Demand = D = 3,600 kg per year.
Ordering cost = Co = Rs 36 Cost of the material = C = 10
Carrying cost/unit/year = 25% of the cost of the material = 0.25 * 10 = Rs 2.50
(a) EOQ = Q* = √[(2 D Co)/Cc] = √(2 X 3600 X 36/2.50) = 321.99 kg
(b) Optimal number of orders in a year = (D/Q*) = (3600/321.99) = 11.18
(c) Optimal order cycle time = (365/optimal number of orders in a year) = (365/11.18) =
32.65 days.
(d) Minimum total annual variable cost = Total annual ordering cost + Total annual carrying
cost
(e) Total annual variable inventory cost
Total annual ordering cost = (Optimal number of orders in a year X ordering cost per order)
= (11.18) X (36) = Rs 402.48
Total annual carrying cost = (Average inventory) X (Carrying cost/unit/year)
= (321.99/2) X 2.50 = Rs 402.48
Therefore, minimum total annual variable inventory cost = 402.48 + 402.48 = Rs 804.96
Cost of material per year = Annual demand X Price of material = 3600 X 10 = Rs 36,000.
(f) Total annual inventory cost
Minimum total annual inventory cost = minimum total annual variable inventory cost +
Cost of material per year = 804.96 + 36,000 = Rs 36804.96

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EOQ MODELS WITH PRICE DISCOUNTS
 The Economic Order Quantity (EOQ) model calculates the optimum quantity to order
using the assumption that price is fixed (regardless of the quantity ordered).
 But, it is common for suppliers to give discounts when the quantity ordered is high.
When discounts are considered, the EOQ may change.
 The optional order quantity model investigates the total annual inventory costs with
and without discounts and the optional order quantity (OOQ) is the one that
minimizes total annual inventory costs.
 The result is that when quantity discounts are offered, larger than usual economic
order quantities may be justified.
Procedure
(1) Calculate the EOQ at the lowest price
(2) Determine whether the EOQ is feasible at that price
- Will the vendor sell that quantity at that price?
- If yes, stop – if no, continue
(3) Check the feasibility of EOQ at the next higher price
(4) Continue until you identify a feasible EOQ
(5) Calculate the total costs (including total item cost) for the feasible EOQ model
(6) Calculate the total costs of buying at the minimum quantity required for each of the
cheaper unit prices
(7) Compare the total cost of each option & choose the lowest cost alternative
Example
Annual Demand = 5000 units
Ordering cost = $ 49
Annual carrying charge = 20%
Unit price schedule:
QUANTITY UNIT PRICE
0 to 999 $ 5.00
1000 to 1999 $ 4.80
2000 and above $ 4.75

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Step 1
Q (p = 4.75) = √[(2 X 5000 X 49)/(0.20 X 4.75 )] = 718 (not feasible)
Q (p = 4.80) = √[(2 X 5000 X 49)/(0.20 X 4.80 )] = 714 (not feasible)
Q (p = 5.00) = √[(2 X 5000 X 49)/(0.20 X 5.00 )] = 700 (feasible)
Since the total cost for 1000 units is the least, Economic Order Quantity =
1000 units.
700,25$500000.500.52.0
2
700
49
700
000,5
700 QTC
725,24$500080.480.42.0
2
1000
49
1000
000,5
1000 QTC
50.822,24$500075.475.42.0
2
2000
49
2000
000,5
2000 QTC

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PROBABILISTIC INVENTORY MODELS
In many cases, the demand is not known with certainty. Such models are called probabilistic
(stochastic) inventory models.
Procedure
 Suppose demand for a product has the following probability distribution
Demand
(units)
D1 D2 D3 D4 ---- Dn
Probability p1 p2 p3 p4 ----- pn
The retailer buys goods at a unit cost CP and sells them at a price SP. The unsold goods are
sold as scrap at a salvage price of SV per unit. How many goods should the retailer stock?
Step 1:
Calculate the critical ratio = (ML)/(ML +MP)
Where ML = Marginal Loss = Cost Price – Salvage Price = (CP- SV)
And MP = Marginal Profit = Selling Price – Cost Price = (SP-CP)
Step 2:
Calculate the Cumulative Probability P (D ≥ Q) for the demand distribution
Demand
(units)
D1 D2 D3 D4 ---- Dn
Probability p1 p2 p3 p4 ----- pn
Cumulative
Probability
P (D ≥ Q)
p1 +p2 +p3
+p4 + ---
+pn-1 +
pn
p2 +p3 +p4
+ ---+pn-1
+
pn
p3 +p4 + -
--+ pn-1 +
pn
p4 + ---+
pn-1 +
pn
pn-1 + pn pn
Then the retailer must stock items as long as Cumulative Probability P (D ≥ Q) is more than
the Critical ratio.
Example:
A newspaper boy is trying to decide how many newspapers he should stock each day. Each
newspaper costs him Rs 3. He sells each paper for Rs 5. He does not get any revenue if he
does not sell a newspaper. From past data, he knows that the probability of selling
newspapers is as follows:

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Number of
newspapers
2 3 4 5 6 7 8 9 10
Probability
of selling
0.05 0.05 0.10 0.20 0.30 0.15 0.10 0.03 0.02
How many newspapers should the newspaper boy stock to maximize his profit?
Solution
Cost Price = Rs 3 Salvage Value = Rs 0
ML = Marginal Loss = Cost Price – Salvage value = 3-0 = 3
Selling Price = Rs 5 Cost Price = Rs 3
MP = Marginal Profit = Selling Price – Cost Price = 5-3 = 2
Marginal Ratio = (ML)/(ML +MP) = 3/(3+2) = 0.60
Number of
newspapers
2 3 4 5 6 7 8 9 10
Probability
of selling
0.05 0.05 0.10 0.20 0.30 0.15 0.10 0.03 0.02
Cum Prob
(D≥ Q)
1.00 0.95 0.90 0.80 0.60 0.30 0.15 0.05 0.02
So, the newspaper boy must stock 5 newspapers as Cum P (D ≥ Q) is more than the critical
ratio for 5 newspapers.

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