Description
t explains different inventory models like economic lot size model with constant demand, economic lot size model with finite replenishment, economic lot size with shortages, economic lot size with constant demand and variable order cycle time.
Inventory Control Models
1
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Lot-Size (or Cycle) Inventories • Lot-size inventories exist whenever one produces (or buys) in larger quantities than are needed to satisfy immediate requirements. • The amount of such inventory depends upon the production lot size, economical shipment quantities, replenishment lead times, price-quantity discount schedules and inventory carrying cost. Pipeline (or Transit) Inventories • Since movement cannot be instantaneous, inventories arise due of shipment from one point to another. • The amount of pipeline inventory depend on the time required for shipment and the nature of the demand
2 2
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Safety (or Buffer) Stocks • Some amount of stock is created as a protection against uncertainties of demand and the lead time. • Demand and lead time are variables with probable deviations. • This may cause unpredictable shortage with a high penalty cost. • To prevent the losses due to future uncertainty additional stocks have to be held in
addition to the regular stock called as safety stock.
• Level of safety stock is determined by desirable trade-off between protection against demand and supply uncertainties and the level of investment in safety stock.
3 3
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Seasonal Inventories • Seasonal inventory is needed for products whose markets exhibit seasonal patterns of demand and whose production (or supply) is not uniform • Seasonal inventories are built up on advance to meet the demand. • The amount of seasonal inventories should be determined by balancing the holding (or carrying) and expiry (if any) costs of seasonal inventories.
4 4
ADVANTAGES OF CARRYING INVENTORY
Interdependence : • One of the major advantage of keeping inventory is to link various manufacturing stages within the firm so that down time in one stage does not affect the entire manufacturing process. (Decoupling) • It helps the business going on by acting as a buffer between successive stages Irregular Supply and Demand : • When the supply or demand for an inventory item is irregular, storing certain amount of an item in inventory can provide service to customers at various locations by maintaining a adequate supply to meet their immediate and seasonal needs. Quantity Discounts : • Inventory of items is carried to take advantage of price-quantity discount because many suppliers offer discounts for large orders. Avoiding Stock-outs : • An important function of inventory is to ensure an adequate and prompt supply of items to the customers. • Loss of goodwill can be an expensive price to pay for not having the right item at the right time
5
GLOBAL
FEATURES OF INVENTORY SYSTEM
• Relevant Inventory Costs • Demand for Inventory items
• Replenishment Lead Time
• Length of Planning period • Constraint on the inventory system
6
GLOBAL
RELEVANT INVENTORY COSTS
• The costs that are affected by the firm’s decision to maintain a particular level of stock are called relevant costs. These costs typically are
? Purchase Cost
? Carrying (or Holding) Cost ? Ordering (or Set-up) Cost ? Shortage (or Stock-out) and Customer-service cost.
7
GLOBAL
PURCHASE COST
• This cost consists of the actual price paid for the procurement of items. • If the unit price ‘C’ of an item is independent of the size of the quantity ordered or
purchased, the purchase cost is given by:
?Purchase Cost = (Price per Unit) * (Demand per unit time) =C*D • When price-break or quantity discounts are available for bulk purchase above 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:
?Purchase Cost = (Price per unit when order size is Q) * (Demand per unit time)
= C(Q) * D
8
GLOBAL
CARRYING (OR HOLDING) COST
• • The carrying cost is associated with carrying (or holding) inventory. This cost arises due to many factors which include: ? Storage cost incurred for providing warehouse space to store the products ? Handling cost incurred for payment of salaries to employees to handle inventory ? Insurance cost against possible loss from fire or other form of damage ? Interest paid on investment of capital ? Obsolescence and deterioration costs incurred when a portion of inventory become either obsolete or is lost or pilfered. Carrying cost can be determined by two different ways: 1. Carrying Cost = (Cost of Carrying one unit of an item in the inventory for a given length of time) * (Average number of units of an item carried in the inventory for a given length of time) 2. Carrying Cost = (Cost to carry one rupee’s worth of inventory per time period) * (Rupee value of units carried)
9
•
GLOBAL
ORDERING (OR SET-UP) COST
• • Ordering cost is associated with the cost of placing orders for procuring items from outside suppliers or producing the items setting up of machinery. Cost per order generally includes ? Requisition cost of handling of purchase orders, invoices, stationery, payments etc, ? Cost of services include cost of mailing, telephone calls and other follow up actions ? Material handling cost incurred in receiving, inspecting and storing the items included in the order. For practical purposes, ordering (set-up) cost is independent of the size of the order, rather it varies with the number of orders placed during a given period of time. Thus if a large number of orders are placed, more money will be required for procuring the items. Ordering cost is calculated as: ? Ordering Cost = (Cost per order or per set-up) * (Number of orders or set-ups in the inventory planning period) When items are produced internally, a set-up cost is incurred and has the same meaning as the ordering cost.
• •
•
•
10
GLOBAL
Total Inventory Cost
• The total inventory cost (TC) is then given by:
? Total Inventory Cost = Purchase cost + Ordering Cost+ Carrying Cost+
Shortage Cost
Total Variable Inventory Cost
• Total Variable Inventory Cost = Ordering Cost+ Carrying Cost+ Shortage Cost
11
GLOBAL
Demand for Inventory Items
• • • • The understanding of the nature of demand (i.e. its rate, size and pattern) for the
inventory items is essential to determine an optimal inventory policy.
Size of demand is referred to the number of items required in each period. The size of demand may be either deterministic or probabilistic. In the deterministic case, the quantities needed over a period of time are known with certainty. This can be fixed (Static) or can vary (dynamic) from period to period. • In the probabilistic case, the quantities needed over a period of time are not known
with certainty but the nature of such requirement can be described by a known
probability distribution.
12
GLOBAL
Order Cycle
• An ordering cycle is the time period between two successive placement of orders.
•
The ordering cycle may be determined in one of the two ways:
? Continuous Review: In this case the level of inventory is updated continuously as current level is reached at which point (also called reorder point) a new order is placed. This is also referred to as the two-bin system, fixed order size system, Q-system or (Q,R) system ? Periodic Review: In this case the orders are placed at equal interval of time, but the size of the order may vary with the variations in demand. This is also
referred to as fixed order interval system, P-system, (S,t) system.
13
GLOBAL
Lead Time or Delivery Lag
• When an order is placed, it may require some time before delivery is reached. The
time between the placement of an order and its receipt is called delivery lag or lead
time. • Lead time may be deterministic or probabilistic
Constraints in the inventory system
• This could be: • Warehouse space constraints
•
• •
Investment constraints
Production capacity constraints Transportation capacity constraints
14
GLOBAL
LIST OF SYMBOLS
• C = Purchase (or manufacturing) cost per unit (Rs / Unit)
•
• • • •
Cp= Ordering (or set-up) cost per order (Rs / Order)
Ch= Cost of carrying one unit in the inventory for a given length of time (Rs/Unit time) r = Cost of carrying one rupee’s worth of inventory per time period (usually expressed in terms of percent of rupee value of inventory) Cs= Shortage cost per unit per time (Rs / Unit-Time) D = Demand Rate, Units per time (Unit / Time). Also denoted by d
•
• • •
Q = Order Quantity i.e. number of units ordered per order (units)
ROL = Reorder Level i.e. the level of inventory at which an order is placed (units) LT = Replenishment Lead Time. n = number of orders per time period (orders / time)
15
GLOBAL
LIST OF SYMBOLS
• • • • • t = reorder cycle time i.e the time interval between successive orders to replenish
(time)
tp= Production period (time) p = Production rate i.e the rate at which quantity Q is added to inventory (quantity / time) TC = Total Inventory Cost (Rs) TVC = Total Variable Inventory Cost (Rs)
16
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
• The inventory control system in this case can be described in terms of the following
assumption:
? Demand (D) rate is constant and known throughout the reorder cycle time. ? Production rate (rp) is infinite (the entire order quantity Q is received at one time as soon as the order is placed) ? Lead Time (LT) is constant and known with certainty (No shortage of inventory) ? Purchase price (or Cost, C) per unit of the given item is constant i.e quantity
discount is not available
? Unit costs of carrying inventory (Ch) and ordering (Cp) are known and constant
17
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
1. Q (EOQ)
=
2 D Cp
Ch
=
2 X Annual Demand X Ordering Cost Carrying Cost
2. Optimal length of inventory replenishment cycle time (t) i.e optimal interval between 2 successive orders • Q = Annual Demand x Reorder Cycle time = D X t • t = Q /D • t = 1/D x 2 D Cp = 2 Cp 2 x Ordering Cost = Ch D Ch Annual Demand x Carrying Cost
18
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND 3. Optimal number of orders (N) to be placed in the given time period (assumed as one year) • N = Annual Demand /Optimal Order Quantity = D/Q = D x Ch 2 x Cp = Annual Demand x Carrying Cost 2 x Ordering Cost
19
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND 4. Optimal (minimum ) Total Variable Cost (TVC) = 2 D C pC h
=
2 x Annual Demand x Ordering Cost x Carrying Cost
20
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Problem: 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 Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. The purchase manager wishes to determine an ordering policy for raw material. Calculate : 1. Optimal Lot Size 2. Optimal Order Cycle Time 3. Minimum yearly variable Inventory Cost
4. Minimum yearly total cost or Total Inventory Cost
21
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Given Data: • D = 3,600 kg per year
•
•
Cp= Rs. 36 per order
Ch= 25 percent of the investment in inventories = Rs. 10 x 0.25 = Rs. 2.5 per kg /year
Calculation of Optimal Lot Size:
EOQ, Q
=
2 X Annual Demand X Ordering Cost Carrying Cost
22
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Calculation of Optimal Lot Size:
Q
=
2 X 3,600 X 36
2.5 = 322 kg per order
2. Optimal Order Cycle Time:
• t = Q/D = 322/3600 = 0.0894 year = 32.6 days
23
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Calculation of Optimal Lot Size:
3. Minimum Yearly Variable Inventory Cost: • TVC = 2DCpCh
=
2 x 3,600 x 36 x 2.5 = Rs. 805 per year
4. Minimum Yearly Variable Inventory Cost: • TC = TVC + (D*C) = Rs. 805 + (3,600 kg) (Rs. 10 / kg) = Rs. 36,805 per year
24
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
This modes is also based on the assumptions in model I (a) except that of instantaneous replenishment. This is because of the fact that in many situations, the amount ordered is not delivered all at once, but available at a finite rate per unit of time.
Economic Order Quantity = 2DCp p Q (EOQ) P-d Ch Total Minimum Inventory Variable Cost TVC
=
2DCpCh 1 – d/p
25
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Optimal Order Cycle Time
t = Q/D
=
2Cp DCh
p P-d
Optimal number of production runs per year
N = D/Q = DCh 2 Cp
26
p-d p
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Problem
A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He
finds that, when he starts production run, he can product 25,000 bearings per day. The cost
of holding a bearing in stock for a year is Rs. 2 and the set up cost of a production run is Rs. 1800. Assuming 300 working days in a year, calculate • • Economic Order Quantity Order Cycle Time
27
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution
Given Data: Cp = Rs. 1800 per production run Ch=Rs. 2 per year per bearing p = 25,000 bearings per day d = 10,000 bearings per day D = 10,000 * 300 = 30,00,000 units / year
28
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution Q (EOQ) = =
2DCp p P-d Ch
2 X30,00,000 x 1800
2
X 25,000 25,000 – 10,000
= 94,868 bearings
29
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution
Frequency of Production runs is given by
t = Q/D = 94, 868 /30,00,000 = 0.031 year = 9.486 days
30
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH SHORTAGES (BACKORDERING)
This model is based on all the assumptions of model I (a), except that the inventory
system run out of stock for a certain period of time, i.e. shortages are allowed. Usually,
two types of situations occur when an inventory system runs out of stock. •Customers are not ready to wait and therefore any sale that would have resulted is lost.
•Customers leave order (s) with the supplier and this backorder is filled on stock
availability Thus all of the costs -Costs of keeping backlog orders, cost of shipping the items to the customers, loss or goodwill etc are costs of back order. The backorder is expressed in
rupees per unit of time. This is denoted by Cs
31
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Q = 2 DCp Ch + Cs Ch Cs
Optimal Maximum Stock Level,
M =
2 DCp Ch
Cs Cs + Ch
32
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME Optimal Backorder Quantity
R =
Q
Ch Cs + Ch
Total Variable Cost
TVC = 2 DCpCh
Cs Cs + Ch
33
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Problem
A dealer supplies you the following information with regard to a product dealt in by him: • Annual Demand – 10,000 units • Ordering Cost – Rs. 10 per order • Price : Rs. 20 per unit • Inventory Carrying Cost – 20% of the value of inventory per year. The dealer is considering the possibility of allowing some backorder (stock-out) to occur. He has estimated that the annual cost of backordering will be 25% of the value of inventory. 1. What are the economic order quantities with backorder and without backorder? 2. What quantity of the product should be allowed to be back-ordered, if any? 3. Would you recommend to allow back-ordering? If so, what would be the annual cost saving in Total Variable Cost by adopting the policy of back-ordering?
34
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Given Data: D = 10,000 units / year Cp = Rs. 10 per order C = Rs. 20 per unit Ch = 20% of Rs. 20 = Rs. 4 per unit per year Cs = 25% of Rs. 20 = Rs. 5 per unit per year.
35
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Economic Order Quantity without back-ordering Q
=
2 X Annual Demand X Ordering Cost Carrying Cost
2 X 10,000 x 10
=
4
=
223.6 units
36
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Economic Order Quantity with back-ordering
Q = 2 DCp Ch + Cs Ch Cs
=
2 X 10,000 x 10
4+5
4
=
5
300 units
37
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Optimal Quantity to be back-ordered R =
Q
335
Ch Cs + Ch
4
=
4+5
= 149 units
38
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Minimum Total Variable Cost without backordering
TVC (223.6) =
2 x Annual Demand x Ordering Cost x Carrying Cost
=
2 x 10,000 x 10 x 4
= Rs. 894.272
39
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Minimum Total Variable Cost with backordering
TVC (335)
=
=
2 DCpCh
Cs Ch+ Cs
5/9
2 x 10,000 x 10 x 4
= Rs. 666.67
Since TVC (223.6) > TVC (666.67), dealer should accept the proposal for back ordering as this will save him Rs. 227 every year (894-666)
40
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Suppose the following price discount schedule is quoted by a supplier in which a price break (Quantity discount) occurs at quantity b1. This means: Quantity Price per unit
0 < Q1 < b1
C1
b1<=Q2 < b2
Step 1
C2 (< C1)
• First, calculate EOQ, Q2 using both C2, the lowest price.
• If Q2 lies in the range b1<=Q2 < b2, the Q2 is the Economic Order Quantity. •
41
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Step 2
• If, Q2 <b1, then taking quantity discount is not viable. Calculate EOQ with price C1 and
the corresponding minimum total cost, TC1.
42
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Problem The annual demand of a product is 10,000 units. Each unit costs Rs. 100 if orders placed
in quantities below 200 units but for orders of 200 or above the price is Rs. 95. The annual
inventory holding costs is 10% of the value of the item and the ordering cost is Rs. 5 per order. Find the economic lot size.
43
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Solution Given Data
• D = 10,000 units / year
•Cp = Rs. 5 per order
•R = 10 percent of price of an item = Rs. 0.10
•The unit cost for the range of quantities is given by: Quantity Price per unit
0 < Q1 < 200
100
200 <=Q2
44
95
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Solution Given Data
• The optimum order quantity based on price C2 = Rs 95 is given by: Q = 2 X 10,000 X 5 95 x 0.10 = 103 units
Since 103 < 200, the given discount will not minimize the Total Variable Inventory Cost.
• The correct optimum order quantity based on price C2 = Rs 95 is given by:
Q = 2 X 10,000 X 5
100 x 0.10
45
= 100 units
doc_627255137.pptx
t explains different inventory models like economic lot size model with constant demand, economic lot size model with finite replenishment, economic lot size with shortages, economic lot size with constant demand and variable order cycle time.
Inventory Control Models
1
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Lot-Size (or Cycle) Inventories • Lot-size inventories exist whenever one produces (or buys) in larger quantities than are needed to satisfy immediate requirements. • The amount of such inventory depends upon the production lot size, economical shipment quantities, replenishment lead times, price-quantity discount schedules and inventory carrying cost. Pipeline (or Transit) Inventories • Since movement cannot be instantaneous, inventories arise due of shipment from one point to another. • The amount of pipeline inventory depend on the time required for shipment and the nature of the demand
2 2
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Safety (or Buffer) Stocks • Some amount of stock is created as a protection against uncertainties of demand and the lead time. • Demand and lead time are variables with probable deviations. • This may cause unpredictable shortage with a high penalty cost. • To prevent the losses due to future uncertainty additional stocks have to be held in
addition to the regular stock called as safety stock.
• Level of safety stock is determined by desirable trade-off between protection against demand and supply uncertainties and the level of investment in safety stock.
3 3
GLOBAL
FUNCTIONAL CLASSIFICATION OF INVENTORIES
Seasonal Inventories • Seasonal inventory is needed for products whose markets exhibit seasonal patterns of demand and whose production (or supply) is not uniform • Seasonal inventories are built up on advance to meet the demand. • The amount of seasonal inventories should be determined by balancing the holding (or carrying) and expiry (if any) costs of seasonal inventories.
4 4
ADVANTAGES OF CARRYING INVENTORY
Interdependence : • One of the major advantage of keeping inventory is to link various manufacturing stages within the firm so that down time in one stage does not affect the entire manufacturing process. (Decoupling) • It helps the business going on by acting as a buffer between successive stages Irregular Supply and Demand : • When the supply or demand for an inventory item is irregular, storing certain amount of an item in inventory can provide service to customers at various locations by maintaining a adequate supply to meet their immediate and seasonal needs. Quantity Discounts : • Inventory of items is carried to take advantage of price-quantity discount because many suppliers offer discounts for large orders. Avoiding Stock-outs : • An important function of inventory is to ensure an adequate and prompt supply of items to the customers. • Loss of goodwill can be an expensive price to pay for not having the right item at the right time
5
GLOBAL
FEATURES OF INVENTORY SYSTEM
• Relevant Inventory Costs • Demand for Inventory items
• Replenishment Lead Time
• Length of Planning period • Constraint on the inventory system
6
GLOBAL
RELEVANT INVENTORY COSTS
• The costs that are affected by the firm’s decision to maintain a particular level of stock are called relevant costs. These costs typically are
? Purchase Cost
? Carrying (or Holding) Cost ? Ordering (or Set-up) Cost ? Shortage (or Stock-out) and Customer-service cost.
7
GLOBAL
PURCHASE COST
• This cost consists of the actual price paid for the procurement of items. • If the unit price ‘C’ of an item is independent of the size of the quantity ordered or
purchased, the purchase cost is given by:
?Purchase Cost = (Price per Unit) * (Demand per unit time) =C*D • When price-break or quantity discounts are available for bulk purchase above 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:
?Purchase Cost = (Price per unit when order size is Q) * (Demand per unit time)
= C(Q) * D
8
GLOBAL
CARRYING (OR HOLDING) COST
• • The carrying cost is associated with carrying (or holding) inventory. This cost arises due to many factors which include: ? Storage cost incurred for providing warehouse space to store the products ? Handling cost incurred for payment of salaries to employees to handle inventory ? Insurance cost against possible loss from fire or other form of damage ? Interest paid on investment of capital ? Obsolescence and deterioration costs incurred when a portion of inventory become either obsolete or is lost or pilfered. Carrying cost can be determined by two different ways: 1. Carrying Cost = (Cost of Carrying one unit of an item in the inventory for a given length of time) * (Average number of units of an item carried in the inventory for a given length of time) 2. Carrying Cost = (Cost to carry one rupee’s worth of inventory per time period) * (Rupee value of units carried)
9
•
GLOBAL
ORDERING (OR SET-UP) COST
• • Ordering cost is associated with the cost of placing orders for procuring items from outside suppliers or producing the items setting up of machinery. Cost per order generally includes ? Requisition cost of handling of purchase orders, invoices, stationery, payments etc, ? Cost of services include cost of mailing, telephone calls and other follow up actions ? Material handling cost incurred in receiving, inspecting and storing the items included in the order. For practical purposes, ordering (set-up) cost is independent of the size of the order, rather it varies with the number of orders placed during a given period of time. Thus if a large number of orders are placed, more money will be required for procuring the items. Ordering cost is calculated as: ? Ordering Cost = (Cost per order or per set-up) * (Number of orders or set-ups in the inventory planning period) When items are produced internally, a set-up cost is incurred and has the same meaning as the ordering cost.
• •
•
•
10
GLOBAL
Total Inventory Cost
• The total inventory cost (TC) is then given by:
? Total Inventory Cost = Purchase cost + Ordering Cost+ Carrying Cost+
Shortage Cost
Total Variable Inventory Cost
• Total Variable Inventory Cost = Ordering Cost+ Carrying Cost+ Shortage Cost
11
GLOBAL
Demand for Inventory Items
• • • • The understanding of the nature of demand (i.e. its rate, size and pattern) for the
inventory items is essential to determine an optimal inventory policy.
Size of demand is referred to the number of items required in each period. The size of demand may be either deterministic or probabilistic. In the deterministic case, the quantities needed over a period of time are known with certainty. This can be fixed (Static) or can vary (dynamic) from period to period. • In the probabilistic case, the quantities needed over a period of time are not known
with certainty but the nature of such requirement can be described by a known
probability distribution.
12
GLOBAL
Order Cycle
• An ordering cycle is the time period between two successive placement of orders.
•
The ordering cycle may be determined in one of the two ways:
? Continuous Review: In this case the level of inventory is updated continuously as current level is reached at which point (also called reorder point) a new order is placed. This is also referred to as the two-bin system, fixed order size system, Q-system or (Q,R) system ? Periodic Review: In this case the orders are placed at equal interval of time, but the size of the order may vary with the variations in demand. This is also
referred to as fixed order interval system, P-system, (S,t) system.
13
GLOBAL
Lead Time or Delivery Lag
• When an order is placed, it may require some time before delivery is reached. The
time between the placement of an order and its receipt is called delivery lag or lead
time. • Lead time may be deterministic or probabilistic
Constraints in the inventory system
• This could be: • Warehouse space constraints
•
• •
Investment constraints
Production capacity constraints Transportation capacity constraints
14
GLOBAL
LIST OF SYMBOLS
• C = Purchase (or manufacturing) cost per unit (Rs / Unit)
•
• • • •
Cp= Ordering (or set-up) cost per order (Rs / Order)
Ch= Cost of carrying one unit in the inventory for a given length of time (Rs/Unit time) r = Cost of carrying one rupee’s worth of inventory per time period (usually expressed in terms of percent of rupee value of inventory) Cs= Shortage cost per unit per time (Rs / Unit-Time) D = Demand Rate, Units per time (Unit / Time). Also denoted by d
•
• • •
Q = Order Quantity i.e. number of units ordered per order (units)
ROL = Reorder Level i.e. the level of inventory at which an order is placed (units) LT = Replenishment Lead Time. n = number of orders per time period (orders / time)
15
GLOBAL
LIST OF SYMBOLS
• • • • • t = reorder cycle time i.e the time interval between successive orders to replenish
(time)
tp= Production period (time) p = Production rate i.e the rate at which quantity Q is added to inventory (quantity / time) TC = Total Inventory Cost (Rs) TVC = Total Variable Inventory Cost (Rs)
16
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
• The inventory control system in this case can be described in terms of the following
assumption:
? Demand (D) rate is constant and known throughout the reorder cycle time. ? Production rate (rp) is infinite (the entire order quantity Q is received at one time as soon as the order is placed) ? Lead Time (LT) is constant and known with certainty (No shortage of inventory) ? Purchase price (or Cost, C) per unit of the given item is constant i.e quantity
discount is not available
? Unit costs of carrying inventory (Ch) and ordering (Cp) are known and constant
17
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
1. Q (EOQ)
=
2 D Cp
Ch
=
2 X Annual Demand X Ordering Cost Carrying Cost
2. Optimal length of inventory replenishment cycle time (t) i.e optimal interval between 2 successive orders • Q = Annual Demand x Reorder Cycle time = D X t • t = Q /D • t = 1/D x 2 D Cp = 2 Cp 2 x Ordering Cost = Ch D Ch Annual Demand x Carrying Cost
18
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND 3. Optimal number of orders (N) to be placed in the given time period (assumed as one year) • N = Annual Demand /Optimal Order Quantity = D/Q = D x Ch 2 x Cp = Annual Demand x Carrying Cost 2 x Ordering Cost
19
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND 4. Optimal (minimum ) Total Variable Cost (TVC) = 2 D C pC h
=
2 x Annual Demand x Ordering Cost x Carrying Cost
20
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Problem: 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 Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. The purchase manager wishes to determine an ordering policy for raw material. Calculate : 1. Optimal Lot Size 2. Optimal Order Cycle Time 3. Minimum yearly variable Inventory Cost
4. Minimum yearly total cost or Total Inventory Cost
21
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Given Data: • D = 3,600 kg per year
•
•
Cp= Rs. 36 per order
Ch= 25 percent of the investment in inventories = Rs. 10 x 0.25 = Rs. 2.5 per kg /year
Calculation of Optimal Lot Size:
EOQ, Q
=
2 X Annual Demand X Ordering Cost Carrying Cost
22
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Calculation of Optimal Lot Size:
Q
=
2 X 3,600 X 36
2.5 = 322 kg per order
2. Optimal Order Cycle Time:
• t = Q/D = 322/3600 = 0.0894 year = 32.6 days
23
GLOBAL
MODEL I (a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND
Calculation of Optimal Lot Size:
3. Minimum Yearly Variable Inventory Cost: • TVC = 2DCpCh
=
2 x 3,600 x 36 x 2.5 = Rs. 805 per year
4. Minimum Yearly Variable Inventory Cost: • TC = TVC + (D*C) = Rs. 805 + (3,600 kg) (Rs. 10 / kg) = Rs. 36,805 per year
24
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
This modes is also based on the assumptions in model I (a) except that of instantaneous replenishment. This is because of the fact that in many situations, the amount ordered is not delivered all at once, but available at a finite rate per unit of time.
Economic Order Quantity = 2DCp p Q (EOQ) P-d Ch Total Minimum Inventory Variable Cost TVC
=
2DCpCh 1 – d/p
25
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Optimal Order Cycle Time
t = Q/D
=
2Cp DCh
p P-d
Optimal number of production runs per year
N = D/Q = DCh 2 Cp
26
p-d p
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Problem
A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He
finds that, when he starts production run, he can product 25,000 bearings per day. The cost
of holding a bearing in stock for a year is Rs. 2 and the set up cost of a production run is Rs. 1800. Assuming 300 working days in a year, calculate • • Economic Order Quantity Order Cycle Time
27
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution
Given Data: Cp = Rs. 1800 per production run Ch=Rs. 2 per year per bearing p = 25,000 bearings per day d = 10,000 bearings per day D = 10,000 * 300 = 30,00,000 units / year
28
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution Q (EOQ) = =
2DCp p P-d Ch
2 X30,00,000 x 1800
2
X 25,000 25,000 – 10,000
= 94,868 bearings
29
GLOBAL
MODEL I (b) – ECONOMIC LOT SIZE MODEL FINITE REPLENISHMENT (SUPPLY RATE)
Solution
Frequency of Production runs is given by
t = Q/D = 94, 868 /30,00,000 = 0.031 year = 9.486 days
30
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH SHORTAGES (BACKORDERING)
This model is based on all the assumptions of model I (a), except that the inventory
system run out of stock for a certain period of time, i.e. shortages are allowed. Usually,
two types of situations occur when an inventory system runs out of stock. •Customers are not ready to wait and therefore any sale that would have resulted is lost.
•Customers leave order (s) with the supplier and this backorder is filled on stock
availability Thus all of the costs -Costs of keeping backlog orders, cost of shipping the items to the customers, loss or goodwill etc are costs of back order. The backorder is expressed in
rupees per unit of time. This is denoted by Cs
31
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Q = 2 DCp Ch + Cs Ch Cs
Optimal Maximum Stock Level,
M =
2 DCp Ch
Cs Cs + Ch
32
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME Optimal Backorder Quantity
R =
Q
Ch Cs + Ch
Total Variable Cost
TVC = 2 DCpCh
Cs Cs + Ch
33
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Problem
A dealer supplies you the following information with regard to a product dealt in by him: • Annual Demand – 10,000 units • Ordering Cost – Rs. 10 per order • Price : Rs. 20 per unit • Inventory Carrying Cost – 20% of the value of inventory per year. The dealer is considering the possibility of allowing some backorder (stock-out) to occur. He has estimated that the annual cost of backordering will be 25% of the value of inventory. 1. What are the economic order quantities with backorder and without backorder? 2. What quantity of the product should be allowed to be back-ordered, if any? 3. Would you recommend to allow back-ordering? If so, what would be the annual cost saving in Total Variable Cost by adopting the policy of back-ordering?
34
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Given Data: D = 10,000 units / year Cp = Rs. 10 per order C = Rs. 20 per unit Ch = 20% of Rs. 20 = Rs. 4 per unit per year Cs = 25% of Rs. 20 = Rs. 5 per unit per year.
35
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Economic Order Quantity without back-ordering Q
=
2 X Annual Demand X Ordering Cost Carrying Cost
2 X 10,000 x 10
=
4
=
223.6 units
36
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Economic Order Quantity with back-ordering
Q = 2 DCp Ch + Cs Ch Cs
=
2 X 10,000 x 10
4+5
4
=
5
300 units
37
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Optimal Quantity to be back-ordered R =
Q
335
Ch Cs + Ch
4
=
4+5
= 149 units
38
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Minimum Total Variable Cost without backordering
TVC (223.6) =
2 x Annual Demand x Ordering Cost x Carrying Cost
=
2 x 10,000 x 10 x 4
= Rs. 894.272
39
GLOBAL
MODEL I I(a) – ECONOMIC LOT SIZE MODEL WITH CONSTANT DEMAND AND VARIABLE ORDER CYCLE TIME
Solution
Minimum Total Variable Cost with backordering
TVC (335)
=
=
2 DCpCh
Cs Ch+ Cs
5/9
2 x 10,000 x 10 x 4
= Rs. 666.67
Since TVC (223.6) > TVC (666.67), dealer should accept the proposal for back ordering as this will save him Rs. 227 every year (894-666)
40
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Suppose the following price discount schedule is quoted by a supplier in which a price break (Quantity discount) occurs at quantity b1. This means: Quantity Price per unit
0 < Q1 < b1
C1
b1<=Q2 < b2
Step 1
C2 (< C1)
• First, calculate EOQ, Q2 using both C2, the lowest price.
• If Q2 lies in the range b1<=Q2 < b2, the Q2 is the Economic Order Quantity. •
41
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Step 2
• If, Q2 <b1, then taking quantity discount is not viable. Calculate EOQ with price C1 and
the corresponding minimum total cost, TC1.
42
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Problem The annual demand of a product is 10,000 units. Each unit costs Rs. 100 if orders placed
in quantities below 200 units but for orders of 200 or above the price is Rs. 95. The annual
inventory holding costs is 10% of the value of the item and the ordering cost is Rs. 5 per order. Find the economic lot size.
43
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Solution Given Data
• D = 10,000 units / year
•Cp = Rs. 5 per order
•R = 10 percent of price of an item = Rs. 0.10
•The unit cost for the range of quantities is given by: Quantity Price per unit
0 < Q1 < 200
100
200 <=Q2
44
95
GLOBAL
MODEL I V(a) – EOQ Model with All Units Discounts Available
Solution Given Data
• The optimum order quantity based on price C2 = Rs 95 is given by: Q = 2 X 10,000 X 5 95 x 0.10 = 103 units
Since 103 < 200, the given discount will not minimize the Total Variable Inventory Cost.
• The correct optimum order quantity based on price C2 = Rs 95 is given by:
Q = 2 X 10,000 X 5
100 x 0.10
45
= 100 units
doc_627255137.pptx