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					Operations Management (2)

Inventory Models

Prof. Upendra Kachru

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All organizations have inventory Can be a sizable organizational asset Influences sales (revenue generation and customer relations) Influences production/ operations costs Large amounts reduces ROI Costs of having inventory Frequently the largest single expenditure Excesses can result in losses
Prof. Upendra Kachru

Inventory is the stock of any item or resource used in an organization.

WHAT IS INVENTORY?

Prof. Upendra Kachru

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Working Stock Safety Stock Anticipation Stock Pipeline Stock Decoupling Stock Psychic Stock

Prof. Upendra Kachru

Functions To protect against variations (fluctuations) in demand and supply To take advantage of batches and longer production runs To provide flexibility to allow changes in production plans in view of changes in demand etc. To facilitate intermittent production To take advantage of price discounts by bulk purchases To meet anticipated and current demand
Prof. Upendra Kachru

Functions of Inventory
Excess Demand from Inventory Capacity Requirement without Inventory Capacity Requirement with Inventory Excess Production to Inventory

Demand

J F M A M J J A S O N D

 To smooth production requirements from seasonality

Functions of Inventory

Department 1

Department 2

Department 3

Department 4

Interstage Inventory D1

Interstage Inventory D2

Interstage Inventory D3

 To decouple different components of the internal inventory-distribution system.

Supply

(Purchasing)
Safety Stock

Raw Materials & Supplies

(Production)
Decoupling Stock

In-Process Goods

(Production)
Anticipation Stock

INVENTORY FUNCTIONS
 To meet anticipated demand

Finished Goods

(Marketing)
Psychic Stock

Demand
Prof. Upendra Kachru

Functional Area Marketing Production Purchasing Finance

Functional Responsibility Sell the Product Make the Product Buy required materials Provide working capital Design the product

Inventory Goal Maximize customer service Efficient lot sizes Low cost per unit Efficient use of capital Avoid obsolescence

Inventory Inclination High High High Low Low

Engineering

Prof. Upendra Kachru

Area Marketing

Typical Response

Marketing revenue generation customer relations

Production /efficiency I can’t sell without adequate stocks. I can’t Sales cost of operations our customers if we continue to stockout keep cost of operations and there is not sufficient product variety

Production

If I can produce larger lot sizes, I can reduce per unit cost and function efficiency. Finance
liquidity I can reduce our per unit cost if I buy large return on investment

Purchasing
Logistics Handling Capacity Finance Space

quantities in bulk. Where am I going to get the funds to pay for the inventory? The levels should be lower. I am out of space. I can’t fit anything else in the building.
Prof. Upendra Kachru

Warehousing

Balancing Objectives
1. Provide customer service 2. Support plant efficiency 3. Minimize inventory investment
Minimize Inventory Investment

Maximize Customer Service Operating Efficiency

Prof. Upendra Kachru

CONSTRAINTS INPUTS
Mgt. policies Working capital Space Plant capacity

OUTPUTS

OPERATIONS PLANNING

Forecasts Demand rates Production rates Stock-on-hand Backorders Lead times Product structures

INVENTORY PLANNING AND CONTROL

DECISION RULES

1. What to order? 2. When to order? 3. How much? 4. From whom?

COSTS
Purchase Order / Set-up Holding Stockout
Prof. Upendra Kachru

Inventory Costs are additive

Prof. Upendra Kachru

• Holding (or carrying) costs • Ordering Costs/ Setup (or production change) costs • Shortage or Stock-out Costs

Prof. Upendra Kachru

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External Shortage
1. Present Profit Loss (potential sales) 2. Backorder Costs 3. Future Profit Loss (goodwill erosion)

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Internal Shortage
1. 2. 3. 4. Lost Production (idle people / machines) Substitute Cost (alternate) Overtime / Extra Shift Cost Delay Project Completion Date

Prof. Upendra Kachru

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Average Inventory Investment: The rupee value of a company’s average level of inventory is one of the most common measures of inventory. Inventory Turnover Ratio: It is a ratio that measures how many times during a year the inventory turns around.
Inventory turnover = annual cost of goods sold/average inventory investment

Prof. Upendra Kachru

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Days of Inventory: This measure is an indication of approximately how many days of sales can be supplied solely from inventory.
Days of inventory = avg. inventory investment/ (annual cost of gods sold/days per year) Days of inventory = days per year/ inventory turnover rate

Prof. Upendra Kachru

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The inventory of a medium sized business organization would comprise thousands of items, each item with different usage, price, lead time and specifications. There could be different procurement and technical problems associated with different items. In order to escape this quagmire many selective inventory management techniques are used.

Prof. Upendra Kachru

Vilfredo Pareto’s 80-20 rule.
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The ABC classification is based on focusing efforts where the payoff is highest; i.e. high-value, high-usage items must be tracked carefully and continuously. Typically only 20 percent of all the items account for 80 percent of the total rupee usage, while the remaining 80 percent of the items typically account for remaining 20 percent of the rupee value. The large value items constitute only 20 percent, the ABC analysis makes the task relatively easier.

Prof. Upendra Kachru

PERCENT OF TOTAL DOLLAR USAGE

120 100 80 60

40

A
20

B

C

0
0 20 40 60 80 100
A = HIGH VALUE ITEMS B = MEDIUM VALUE ITEMS C = LOW VALUE ITEMS

PERCENT OF TOTAL ITEMS

Prof. Upendra Kachru

PERCENT OF RUPEE VALUE

80
60 40

A

B
20 0

C

PERCENT OF ITEMS

20
40 60

Prof. Upendra Kachru

Item Degree of Control A B C Tight Moderate Loose

Type of Records

Lot Sizes Frequency of Size of Safety Review Stocks Small Moderate Large

Accurate / Complete Good Simple

Low Medium Large

Continuous Occasional Infrequent

Prof. Upendra Kachru

1. Difficult Procurement Items
2. Short Shelf Life 3. Large Storage Space Requirements 4. Item’s Operational Criticality 5. Likelihood of Theft

6. Difficult Forecast Items

Prof. Upendra Kachru

Title ABC (Level of Usage) HML (High, medium, low usage)

Basis Value of consumption Unit price of the material

Main Uses raw material components and work-in progress inventories Mainly to control purchase. Control obsolescence. Lead time analysis and purchasing strategies Procurement strategies To determine the stocking levels of spare parts. Seasonal items like agriculture products To review the inventories and their use scheduled intervals.
Prof. Upendra Kachru

FSND (Fast, Slow moving, Non moving, Dead )

Consumption pattern of the component

SDE (Scarce, difficult, easy Problems faced in to obtain items) procurement GOLF (Government, Ordinary, Local, Foreign) VED (Vital, Essential, (Desirable) SOS (Seasonal, Offseasonal) XYZ ( Value of Stock) Source of the material Criticality of the component

Nature of suppliers
Value of items in storage

Independent Demand

Inventory systems are predicated on whether demand is derived from an end item or is related to the item itself. There are two types of models that are used in the case of independent demand:
◦ Single Period Models, and ◦ Multiple Period Models.

Finished product

Dependent Demand subassemblies, raw materials, etc)

E( 1)

Component parts

INVENTORY MODELS
Prof. Upendra Kachru

Prof. Upendra Kachru

Single-Period Inventory Models are a special case of periodic inventory systems.
One time purchasing decision (Example: vendor selling food at Siababa temple) Seeks to balance the costs of inventory overstock and under stock It is used for a wide variety of service and manufacturing applications

Prof. Upendra Kachru

F(0.9)= ŷ+1.282σ After prayers at the Siababa temple on Thursdays, people go f(z) to a vendor to eat food. The vendor has collected data over a few months that show, on an average, 100 meals were sold with a standard deviation of 10 meals.
If our vendor wants to be 90 percent sure of not running out of z food each Thursday, how manyz* meals should he prepare?
If we assume that the distribution is normal and the vendor prepared food for exactly 100 persons, the risk of food running out would be 50 percent. The demand would be expected to be less than 100 meals 50 percent of the time, and greater than 100 the other 50 percent. To be 90 percent sure of not falling short, he needs to prepare more food. From the “standard normal distribution“, we can find out that he needs to have additional food to cover 1.282 standard deviations. In order to ensure that he is 90 percent sure having sufficient food: The number extra food required would be 1.282 x 100 = 128.2, or 129 meals.
Prof. Upendra Kachru

If Co = Cost per unit of demand overage, Cu = Cost per unit of demand underage, The probability that the unit will be sold is ‘P’;

The expected marginal equation can be represented as: P * Co < (1-P) * Cu Solving for P, we obtain P < [Cu / (Co +Cu)]

Here (1-P) is the probability of the service/ product not being sold. cost

Prof. Upendra Kachru

A newspaper vendor is faced with the problem of deciding how many newspapers to order daily so as to maximize the daily profit. Daily demand (d) for newspapers is a random variable. No reordering is possible during a day,
◦ If the newsvendor orders fewer papers than customers demand he or she will lose the opportunity to sell some papers.  If supply exceeds demand, the vendor will be stuck with papers which cannot be sold.

Prof. Upendra Kachru

Based on observations over several weeks, the vendor has established the following probability distribution of daily demand:
Demand d Probability P(d) Cumulative Prob. F(d) = P(D  d)

35 or less 36 37 38 39 40 41 42 43 44 45 46 or more

0.00 0.05 0.07 0.08 0.15 0.15 0.20 0.15 0.10 0.03 0.02 0.00

0.00 0.05 0.12 0.20 0.35 0.50 0.70 0.85 0.95 0.98 1.00 1.00

The vendor purchases daily papers at Rs.2 and sells them at Rs. 5 apiece. Leftover papers are valueless and are discarded (i.e. no salvage value).
Prof. Upendra Kachru

The vendor identifies two penalty costs which he/she will incur, regardless of his/her decision: Cost of Overage
CO = Purchase Price - Salvage Value = c - s

For each paper overstocked the newsvendor incurs a penalty cost of:
CO = Rs. 2.00 – Rs.0.00 = Rs. 2.00

Cost of Underage
CU = Selling Price - Purchase Price = p - c

For each paper understocked the newsvendor incurs a penalty (opportunity) cost of:
CU = Rs. 5.00 – Rs. 2.00 = Rs. 3.00

Prof. Upendra Kachru

Assume that there is already a policy in place to order a certain number of papers daily, say 38. Consider the decisions: D1 : Continue the present policy: Stock 38 papers. D2 : Order one more paper: Stock 39 papers. The possible events are: E1 : The 39th paper sells (i.e. demand  39 = demand > 38). E2 : The 39th paper does not sell (i.e. demand  39 = demand  38).

Prof. Upendra Kachru

Item 39 will not sell on a given day only if demand on that day is for 38 or fewer items: P(D  38) = F(38) = 0.20. The probability that an item will not sell is the cumulative probability associated with the previous item. Item 39 will sell on a given day only if demand on that day is for 39 or more items: P(D  39) = 1 - P(D  38) = 1 - F(38) = 1 - 0.2 = 0.80.

To Stock or Not to Stock!
This implies an increase in profit of Rs. 2.00 as compared to the alternative decision which has a payoff of Rs. 0.00. He should stock the 39th paper.

The expected payoff is:
Rs. 3(0.8) + (- Rs. 2)(0.2) = Rs. 2.

Prof. Upendra Kachru

Product Burger Pizza Patties Samosa Sandwich Pastry

Price 18.00 23.00 8.00 6.00 10.00 7.00

Sell 7.00 12.00 4.00 2.00 5.00 3.00

Don’t Sell (Overage) 9.00 23.00 4.00 6.00 5.00 3.50

Stockout (Underage) 11.00 17.00 6.00 3.00 7.00 5.00

Hotdog

11.00

4.00

11.00

6.00

Prof. Upendra Kachru

Prof. Upendra Kachru

◦ Fixed-Order Quantity Models: Event triggered (Example: running out of stock) ◦ Fixed-Time Period Models: Time triggered (Example: Monthly sales call by sales representative)

Inventory Level ‘Q’

Time

T1

T2

Prof. Upendra Kachru

Prof. Upendra Kachru

Feature
Order quantity When to place order

Fixed-order quantity Model

Fixed-Time Period Model

The same amount ordered each Quantity varies each time order time is placed Reorder point when inventory Reorder when the review period position dips to a predetermined arrives level Each time a withdrawal or addition Counted only at review period. is made Less than fixed-time period model Higher due to perpetual record keeping Higher-priced, critical, or important items. Larger than fixed-order quantity model

Record keeping Size of inventory Time to maintain Type of items

Prof. Upendra Kachru

Economic Order Quantity (EOQ) models, due to simplicity and versatility, are fixed order quantity models used for material planning. When independent demand is the most important issue, the EOQ model provides a solution to the problem.

Prof. Upendra Kachru

The inventory cycle determines when an order should be placed and how much should be ordered so as to minimize average annual variable costs.

Q
Quantity on hand

Profile of Inventory Level Over Time

Usage rate

Reorder point

Receive order

Place Receive order order

Place order

Receive order

Time

Lead time
Prof. Upendra Kachru

The EOQ Model
The basic assumptions in the EOQ Model are as follows:  The rate of demand for the item is deterministic and is a constant ‘D’ units per annum independent of time.  Lead time is zero or constant and it is independent of both demand as well as the quantity ordered.  Price per unit of product is constant  Inventory holding cost is based on average inventory  Ordering or setup costs are constant

Prof. Upendra Kachru

Prof. Upendra Kachru

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By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs
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average inventory level:

Q  2

C O S T

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Q vr The holding cost per unit: HQ 2   D 2D
The setup cost per unit:

Total Cost Holding Costs Annual Cost of Items (DC) Ordering Costs

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A  Q

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The production cost per unit:
QOPT

P

Order Quantity (Q)
Prof. Upendra Kachru

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Total Annual = Cost

Annual Purchase Cost

Annual Annual + Ordering + Holding Cost Cost

TC =

P*D + D*A / Q + Q*v*r / 2

TC=Total annual cost D =Demand P =Cost per unit Q =Order quantity A =Cost of placing an order or setup cost R =Reorder point L =Lead time H = v*r =Annual holding and storage cost per unit of inventory

Prof. Upendra Kachru

QOPT =

2DA = rv

2DA  H

2(Annual Demand)(Or or Setup Cost) der Annual Holding Cost
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We also need a reorder point to tell us when to place an order

R eo rd er p o in t, R = d L
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d = average daily demand (constant) L = Lead time (constant)

Prof. Upendra Kachru

EOQ Model Problem
A company, for one of its class ‘A’ items, placed 8 orders each for a lot of 150 numbers, in a year. Given that the ordering cost is Rs. 5,400.00, the inventory holding cost is 40 percent, and the cost per unit is Rs. 40.00. Find out if the company is making a loss in not using the EOQ Model for order quantity policies. What are your recommendations for ordering the item in the future? And what should be the reorder level, if the lead time to deliver the item is 6 months? ‘D’ = Annual demand = 8*150 = 1200 units ‘v’ = Unit purchase cost = Rs. 40.00 ‘A’ = Ordering Cost = Rs. 5400.00 ‘r’ = Holding Cost = 40%

Prof. Upendra Kachru

QEOQ = √ (2*5400*1200)/(0.40*40) Using the Economic Order Equation:

TC= = √ 2*5400*1200*0.40*40

QEOQ = √ (2*A*D /r*v) = 900 units. Minimum Total Annual Cost (TC) = √ 2*A*D*r*v = Rs. 14,400.00 The Total annual Cost under the present system = Rs. 45,000.00 The loss to the company = Rs. 45,000 – Rs. 14,400 = Rs. 30,600.00 Reorder Level = Ro = L*D = (6/12)* 1200 = 600 units
The company should place orders for economic lot sizes of 900 units in each order. It should have a reorder level at 600 units. = Rs. (1200*5400/150 + 0.40*40*150/2) = Rs. (43,800 + 1200)
Prof. Upendra Kachru

Cost

Adding Purchasing cost TC with doesn’t change EOQ Purchasing Cost TC without Purchasing Cost

Purchasing Cost

0

EOQ

Quantity
Prof. Upendra Kachru

TCa
Total Cost

TCb TCc

Decreasing Price

CC a,b,c
OC

EOQ

Quantity
Prof. Upendra Kachru

Novelty Ltd carries a wide assortment of items for its customers. One item, Gaylook, is very popular. Desirous of keeping its inventory under control, a decision is taken to order only the optimal economic quantity, for this item, each time. You have the following information. Make your recommendations:
◦ ◦ ◦ ◦ Annual demand Price per unit Carrying cost Cost per order : : : : 1,60,000 units Rs.20 Re.1 per unit or 5 per cent Rs. 50

Determine the optimal economic quantity.

Prof. Upendra Kachru

Order per year 1 10 40 80 100

Size

Average Carrying Ordering inventory cost (Re.1) cost (Rs.50 per order) 80,000 8,000 4,000 2,000 2,000 800 80,000 8,000 4,000 2,000 1,000 800 50 500 1,000 2,000 4,000 5,000

Total cost per year 80,000 8,500 5,000 4,000 5,000 5,800

1,60,000 16,000 8,000 4,000 2,000 1,600

The optimum economic quantity (lot size) for this item is 4,000 numbers.

Prof. Upendra Kachru

Show that changing the order quantity by a small amount has very little effect on the cost.
Prof. Upendra Kachru

Quantity Discounts
 Quantity discounts, which are price incentives to purchase large quantities, create pressure to maintain a large inventory.  For any per-unit price level, P, the total cost is:

Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials
D Q C= (H) + (A) + PD Q 2

Prof. Upendra Kachru

Quantity Discounts
C for P = Rs.4.00 C for P = Rs.3.50 C for P = Rs.3.00
Total cost (Rupees)

EOQ 4.00 EOQ 3.50 EOQ 3.00

PD for P = Rs.4.00

PD for P = Rs.3.50

PD for P = Rs.3.00

Total cost (Rupees)

First price break
0

Second price break
300 0

First price break

Second price break
300

100 200 Purchase quantity (Q)

100 200 Purchase quantity (Q)

Total cost curves with purchased materials added

EOQs and price break quantities
Prof. Upendra Kachru

Finding Q with Quantity Discounts
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Step 1. Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found.
 It is feasible if it lies in the range corresponding to its price.

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Step 2. If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size.
 Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.

Prof. Upendra Kachru

Problem
A supplier for Apollo Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is:
Order Quantity 0 – 299 300 – 499 500 or more Price per Unit Rs. 60.00 Rs. 58.80 Rs. 57.00

Annual demand (D) = 936 units Ordering cost (A) = Rs. 45 Holding cost (H) = rv = 25% of unit price

Prof. Upendra Kachru

Step 1: Start with lowest price level: 2DS H 2DS H 2DS H 2(936)(45) 0.25(57.00) 2(936)(45) 0.25(58.80) 2(936)(45) 0.25(60.00)

EOQ 57.00 =

=

= 77 units
Not feasible

EOQ 58.80 =

=

= 76 units
Not feasible

EOQ 60.00 =

=

= 75 units

Feasible

This quantity is feasible because it lies in the range corresponding to its price.

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Step 2: The first feasible EOQ of 75 does not correspond to the lowest price
level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels (Rs.58.80 and Rs.57.00):

D Q C= (rv) + (A) + PD Q 2 C75 = 936 75 [(0.25)(Rs. 60.00)] + (Rs. 45) + Rs. 60.00(936) 75 2

C75 = Rs. 57,284 936 300 C300 = [(0.25)(Rs. 58.80)] + (Rs. 45) + Rs. 58.80(936) 300 2 = Rs. 57,382 C500 =

936 500 [(0.25)(Rs.57.00)] + (Rs.45) + Rs. 57.00(936) 500 2
The best purchase quantity is 500 units, which qualifies for the deepest discount. = Rs. 56,999
Prof. Upendra Kachru

Decision Point:
If the price per unit for the range of 300 to 499 units is reduced to Rs. 58.00, the best decision is to order 300 catheters.
Annual Demand Ordering Cost Holding Cost 936 Rs. 45.00 25%

Price Rs. 60.00 Rs. 58.00 Rs. 57.00

EOQ 75 300 500

Inventory Cost Rs. 562.50 Rs. 2175 Rs. 3563

Order Cost Rs. 562.50 Rs. 140.40 Rs. 84.24

Purchase Cost Rs. 56,160 Rs. 54,288 Rs. 53,352

Total Cost Rs. 57,284 Rs. 56,603 Rs. 56,999

This shows that the decision is sensitive to the price schedule. A reduction of slightly more than 1 percent is enough to make the difference in this example.
© 2007 Pearson Education
Prof. Upendra Kachru

QUANTITY DISCOUNTS
Advantages
Lower unit cost

Disadvantages
Higher holding costs

Lower ordering costs
Fewer stockouts Price increase hedge

Larger inventory investment
Older stock Slow inventory turnover

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In many retail merchandising systems, a fixedtime period system is used. Sales people make routine visits to customers and take orders. Inventory, therefore, is counted only at particular times. Fixed-time period models generate order quantities that vary from period to period, depending on the usage rates.

Prof. Upendra Kachru

Prof. Upendra Kachru

T = Time between orders

d  Average period usage Average Order Quantity  d  T  A
L = Lead Time I = Existing Inventory
Q = Order Size

Q  d T  d L I Q  d( T  L)  I

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Total Annual Cost = Purchase Cost + Ordering Cost + Holding Cost

T  C  P  D  (D  A/Q)  (H  Q/2)

Prof. Upendra Kachru

Order Quantity = Average demand over the vulnerable period + safety stock - Inventory currently on hand

Accounting for Safety Stock:

Q  d( T  L)  I  SS
Q  d(T  L )  I  z  T  L
Where z = Number of standard deviations for a specified service probability σT + L= Standard deviation of demand over the review and lead time

Prof. Upendra Kachru

Fixed Time Period System - Advantages
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Fewer orders are placed Purchase discounts more likely Lower shipping and freight costs

Prof. Upendra Kachru

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Consumes capital Requires storage space Incurs taxes Requires insurance Can become lost, stolen, damaged, outdated, or obsolete Must be counted, sorted, verified, stored, retrieved, moved, issued, and protected

Prof. Upendra Kachru

Classical Inventory Prblems
Ever - increasing storage space needs  Slow-moving materials  Disposition of scrap, obsolete, & surplus materials  Transaction recording errors  Misplaced materials


Prof. Upendra Kachru

Inputs

Surplus / Idle Excess Stock Safety Stock Working Stock

Nonproductive

Productive

Outputs
Prof. Upendra Kachru

Inputs

Unreleased Orders

Backlog

Orders in Transit Orders in Temporary Storage
Orders Waiting to be Worked Orders Being Inspected Orders Being Worked
Outputs Finished Goods Productive

In-Process Inventory

Nonproductive

Prof. Upendra Kachru

Inventory System Improvement
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Standardize Stock Items Reduce Lead Times Reduce Cycle Times Use Fewer Suppliers Inform Suppliers of Expected Demand Contract for Minimum Annual Purchases Buy on Consignment Consider Transportation Costs Order Economical Quantities Control Access to Storage Areas Obtain Better Forecasts

Prof. Upendra Kachru

Inventory System Improvement
12. Dispose of Excess Stock 13. Improve Record Accuracy (cycle count) 14. Improve Capacity Planning 15. Minimize Setup Times 16. Simplify Product Structures 17. Multishift operations 18. Continuous Improvement

Prof. Upendra Kachru

Operations Management (2)

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