management science

P a g e | 1

KOLHAN UNIVERSITY

Semiester - 2

Management Science

Masters of Business Administration ®

Sandeep Ghatuary

K O L H A N U N I V E R S I T Y

Management Science 2

Management Science Management Science is concerned with developing and applying models and concepts that may prove useful in helping

to illuminate management issues and solve managerial problems. The models used can often be represented

mathematically, but sometimes computer-based, visual or verbal representations are used as well or instead. The range

of problems and issues to which management science has contributed insights and solutions is vast. It includes

scheduling airlines, both planes and crew, deciding the appropriate place to site new facilities such as a warehouse or

factory, managing the flow of water from reservoirs, identifying possible future development paths for parts of the

telecommunications industry, establishing the information needs and appropriate systems to supply them within the

health service, and identifying and understanding the strategies adopted by companies for their information systems.

Their research in management science is concerned with developing novel methods and approaches to well-known

problem areas, and also the use of established approaches to understand new areas of application. Teaching within the

Department utilizes these latest ideas to convey the practical importance of Management Science to organizations.

The Role of Management Science in Decision Making

1. Subduing Emotion - One of the roles of management science in decision making is to subdue human emotion.

Human emotion can get in the way of decision making. For example, a person might be emotionally attached to

a project that logically will not be profitable; management science tools can be used to identify the rational

decision, which is to abandon the project.

2. Evaluating Complex Situations - Often, a decision will involve a complex web of causes and effects. The human

brain simply cannot handle so much data. Management science offers methods for arranging this data in a way

that it can be interpreted easily.

3. Overcoming Biases - Humans are naturally inclined to have biases. Often, people aren't even aware of these

biases. Biases can be very simple, such as a subjective preference for the color yellow rather than the color blue.

Management science removes human biases from the decision making process.

Decision Making Process Decision making can be hard. It is the cognitive process of selecting a course of action from among multiple

alternatives. Almost any decision involves some conflicts or dissatisfaction. The difficult part is to pick one

solution where the positive outcome can outweigh possible losses. Avoiding decisions often seems easier. Yet, making

your own decisions and accepting the consequence is the only way to stay in control of your time, your success, and

your life.

Decision under Certainty- We say that the decision is taken under certainty if each action is known to lead invariably

to a specific outcome (prospect, alternative, etc.).When the decision maker knows with reasonable certainty about what

the available alternatives are, and what conditions are associated with each alternative; then a state of certainty is said

to exist. For example, Air India needs to buy ten jumbo jets. The decision is from whom to buy. Air India has two

choices: McDonnell Douglas, and Airbus. Each of these companies is known for their quality products. Air India can

choose from any of these alternatives. Here, for making the choice, there is less ambiguity and there is a relatively

lower chance of making a bad decision.

Decision under Risk - We say that the decision is taken under risk if each action leads to one of a set of possible

specific outcomes, each outcome occurring with a known probability. In some situations, a manager is able to estimate

the level of probability at which certain variables could occur. The ability to estimate may be due to experience,

incomplete but reliable information or, in some cases, an accurate report. When estimates are made, a degree of risk is

involved. However some amount of information about the situation is available. The situation requires estimating the

probability that one or more known variables might influence the decision being made.


Decision under Uncertainty- We say that the decision is taken under uncertainty if either action has as its

consequence a set of possible specific outcomes, but the probabilities of these outcomes are completely unknown or

are not even meaningful. A condition of uncertainty exists when a manager is faced with reaching a decision with

no historical data concerning the variables and/or unknowns and their probability of occurrence. For instance, the

decision to introduce Kellogg corn flakes in India was made under uncertainty.

Modern Approach to Decision Making Under Uncertainty

Modern approach to decision making under uncertainty helps in improving the quality of decision making. For making

such decisions, there are three approaches: risk analysis, decision trees and preference theory.

� Risk analysis : Risk analysis involves knowledge of the size and the nature of the risk involved, in

choosing a particular course of action. Before the launch of its Versa model, Maruti, conducted risk analysis in

the areas of capital investment, cost of production and pricing.

� Decision trees: A graphical representation of alternative courses of action with the possible outcomes comprises

a decision tree. It depicts the various decision points, chances, events and probabilities involved in various

decision- courses that might be undertaken.

� Preference or utility theory: This theory is based on the notion that individuals' attitudes towards risk will vary.

Some individuals are willing to take risk (gamble), whereas others are not willing to take risk or take only low risk

(risk averters). Managers play both these roles, when they are uncertain about the outcome.

Three Models of Decision Making: In every one of the following models the decision maker knows the (prior)

probabilities of the natural states. However, the models differ in the degree to which the decision maker knows the

actual state or capable of predicting it with some probability of success.

� Model 1: Decision Making under Perfect Information - The decision maker has a Perfect Information of the

(actual) Natural State, prior to choosing the action.

� Model 2: Decision Making with Sampling (External) Information. The decision maker has no Perfect

Information on the Natural State, but possesses External Information that Forecasts the Natural State, prior to

choosing the action. This external information is also commonly referred to as sampling information, because it

is often based on statistical sampling. For example, by sampling the earth one may predict if there is oil in the

ground or not, prior to deciding whether to dig for oil or not. The external (sampling) information provides

prediction of the natural state.

� Model 3: No Perfect Information and No Sampling (External) Information. The decision maker has neither

Perfect Information nor even External (Sampling) Information on the Natural State. The decision maker has no

choice but to choose his act without external information and then face the natural state with its consequences.

Types of Managerial Decisions 1. A decision is a choice made from available alternatives.

2. Managers often are referred to as decision makers

3. Decision-making is the process of identifying problems and opportunities and selecting a course of action to deal

with a specific problem or take advantage of an opportunity.

4. Managerial decision-making differs from personal decision making in the systematic, specialized attention that

managers give to decision-making.

5. Decision-making is not easy. It must be done amid ever-changing factors, unclear information, and conflicting

points of view.

6. Good decision-making is a vital part of good management, because decisions determine how the organization

solves its problems, allocates resources, and accomplishes its goals.

7. Although many of their important decisions are strategic, managers also make decisions about every other

aspect of an organization, including structure, control systems, responses to the environment, and human

resources.

8. Plans and strategies are arrived at through decision making; the better the decision making, the better the

strategic planning.

9. Managers scout for problems and opportunities, make decisions for solving or taking advantage of them, and

monitor the consequences to see whether additional decisions are required.


Linear Programming Linear Programming is a technique for making decisions under certainty i.e.; when all the courses of options available to

an organisation are known & the objective of the firm along with its constraints are quantified. That course of action is

chosen out of all possible alternatives which yield the optimal results. Linear Programming can also be used as a

verification and checking mechanism to ascertain the accuracy and the reliability of the decisions which are taken solely

on the basis of manager's experience without the aid of a mathematical model. Linear Programming is the analysis of

problems in which a linear function of a number of variables is to be optimized (maximized or minimized) when whose

variables are subject to a number of constraints in the mathematical near inequalities.

The characteristics or the basic assumptions of linear programming are as follows: 1. Decision or Activity Variables & Their Inter-Relationship. The decision or activity variables refer to any activities

which are in competition with other variables for limited resources.

2. Finite Objective Functions. Such objectives can be: cost-minimization, sales, profits or revenue maximization &

the idle-time minimization etc

3. Limited Factors/Constraints. These are the different kinds of limitations on the available resources e.g.

important resources like availability of machines, number of man hour’s available, production capacity and

number of available markets.

4. Presence of Different Alternatives. Different courses of action or alternatives should be available to a decision

maker, who is required to make the decision which is the most effective.

5. Non-Negative Restrictions. Since the negative values of (any) physical quantity has no meaning, therefore all the

variables must assume non-negative values.

6. Linearity Criterion. The relationship among the various decision variables must be directly proportional Le.; Both

the objective and the constraint,$ must be expressed in terms of linear equations or inequalities

7. Additively. It is assumed that the total profitability and the total amount of each resource utilized would be

exactly equal to the sum of the respective individual amounts

8. Mutually Exclusive Criterion. All decision parameters and the variables are assumed to be mutually exclusive In

other words, the occurrence of anyone variable rules out the simultaneous occurrence of other such variables

9. Divisibility. Variables may be assigned fractional values. i.e.; they need not necessarily always be in whole

numbers. If a fraction of a product cannot be produced, an integer programming problem exists. Not hold good

at all times.

Advantages of Linear Programming 1. Scientific Approach to Problem Solving. Linear Programming is the application of scientific approach to problem

solving. Hence it results in a better and true picture of the problems-which can then be minutely analyzed and

solutions ascertained.

2. Evaluation of All Possible Alternatives. Most of the problems faced by the present organizations are highly

complicated - which cannot be solved by the traditional approach to decision making. The technique of Linear

Programming ensures that’ll possible solutions are generated - out of which the optimal solution can be

selected.

3. Helps in Re-Evaluation. Linear Programming can also be used in .reevaluation of a basic plan for changing

conditions. Should the conditions change while the plan is carried out only partially, these conditions can be

accurately determined with the help of Linear Programming so as to adjust the remainder of the plan for best

results?

4. Quality of Decision. Linear Programming provides practical and better quality of decisions’ that reflect very

precisely the limitations of the system i.e.; the various restrictions under which the system must operate for the

solution to be optimal. If it becomes necessary to deviate from the optimal path, Linear Programming can quite

easily evaluate the associated costs or penalty.

5. Focus on Grey-Areas. Highlighting of grey areas or bottlenecks in the production process is the most significant

merit of Linear Programming. During the periods of bottlenecks, imbalances occur in the production

department. Some of the machines remain idle for long periods of time, while the other machines are unable

toffee the demand even at the peak performance level.


6. Flexibility. Linear Programming is an adaptive & flexible mathematical technique and hence can be utilized in

analyzing a variety of multi-dimensional problems quite successfully.

7. Creation of Information Base. By evaluating the various possible alternatives in the light of the prevailing

constraints, Linear Programming models provide an important database from which the allocation of precious

resources can be don rationally and judiciously.

8. Maximum optimal Utilization of Factors of Production. Linear Programming helps in optimal utilization of

various existing factors of production such as installed capacity, Labour and raw materials etc.

Limitations of Linear Programming

1. Linear Relationship. Linear Programming models can be successfully applied only in those situations where a

given problem can clearly be represented in the form of linear relationship between different decision variables.

2. Constant Value of objective & Constraint Equations. Before a Linear Programming technique could be applied

to a given situation, the values or the coefficients of the objective function as well as the constraint equations

must be completely known.

3. No Scope for Fractional Value Solutions. There is absolutely no certainty that the solution to a LP problem can

always be quantified as an integer quite often, Linear Programming may give fractional-varied answers,

4. Degree Complexity. Many large-scale real life practical problems cannot be solved by employing Linear

Programming techniques even with the help of a computer due to highly complex and Lengthy calculations.

5. Multiplicity of Goals. The long-term objectives of an organisation are not confined to a single goal. An

organisation, at any point of time in its operations has a multiplicity of goals or the goals hierarchy - all of which

must be attained on a priority wise basis for its long term growth.

6. Flexibility. Once a problem has been properly quantified in terms of objective function and the constraint

equations and the tools of Linear Programming are applied to it, it becomes very difficult to incorporate any

changes in the system arising on account of any change in the decision parameter.

Sensitivity analysis A technique used to determine how different values of an independent variable will impact a particular dependent

variable under a given set of assumptions. It is a technique for systematically changing variables in a model to determine

the effects of such changes. This technique is used within specific boundaries that will depend on one or more input

variables, such as the effect that changes in interest rates will have on a bond's price. In any budgeting process there are

always variables that are uncertain. Future tax rates, interest rates, inflation rates, headcount, operating expenses and

other variables may not be known with great precision. Sensitivity analysis answers the question, "if these variables

deviate from expectations, what will the effect be (on the business, model, system, or whatever is being analyzed)?"

Sensitivity analysis is very useful when attempting to determine the impact the actual outcome of a particular variable

will have if it differs from what was previously assumed. By creating a given set of scenarios, the analyst can determine

how changes in one variable(s) will impact the target variable. For example, an analyst might create a financial model

that will value a company's equity (the dependent variable) given the amount of earnings per share (an independent

variable) the company reports at the end of the year and the company's price-to-earnings multiple (another

independent variable) at that time. The analyst can create a table of predicted price-to-earnings multiples and a

corresponding value of the company's equity based on different values for each of the independent variables.

Microsoft Excel can generate a sensitivity report in two parts - a changing cells report and a constraints report.

1. Cell 2. Name

3. Final

Value

4. Reduced

Cost

5. Objective

Coefficient

6. Allowable

Increase

7. Allowable

Decrease

1. Changing Cells (Adjustable Cells) - For the Changing Cells report, the allowable increase and decrease refers to

how much the objective function decision variable coefficient can change without changing the values of any of

the decision variables. However, the objective function value will have to change if a coefficient changes and the

corresponding decision variable does not change. Note though, that multiplying each term in the objective

function by a constant does not change the values of the decision variables. The 100% rule can be used to

determine if a change in multiple objective function coefficients will change the values of the decision variables.


Under this rule, any combination of changes can occur without a change in the solution as long as the total

percentage deviation from the coordinate extremes does not exceed 100%. However, the objective value would

change since the objective coefficients are changing. For the purpose of this analysis, the decision variable

coefficient is the effective number that is multiplied by the decision variable when the objective function is

simplified so that each decision variable appears once. Reduced Cost is how much more attractive the variable's

coefficient in the objective function must be before the variable is worth using. Ignore the sign reported by

Excel.

2. Constraints - The shadow price is the amount that the objective function value would change if the named

constraint changed by one unit. The shadow price is valid up to the allowable increase or decrease in the

constraint. The shadow price after the constraint is changed by the entire allowable amount is unknown, but is

always less favorable than the reported value due to the law of diminishing returns. To determine if a constraint

is binding, compare the Final Value with the Constraint R.H. Side. If a constraint is non-binding, its shadow price

is zero.

Linearity from Non-Linear Problems - Many problems that initially may be non-linear may be made linear by careful

formulation. For example, one can avoid using the inequality ≠. For binary integer variables, X + Y ≠ 1 is the same as

saying X = Y.

Binary Variables - When relating continuous decision variables to binary switch variables, the following form often is

useful: continuous variable expression < (some large number) (binary variable)

Simulation - Linear programming techniques assume certainty and by themselves do not deal well with significant

randomness. The following is one possible procedure for maximizing or minimizing some objective function that

contains random variables.

1. Express the objective function in terms of the decision variable.

2. Define a search range and incremental search value for the decision variable, possibly using problem

information to reduce the search range.

3. Run a simulation for each incremental value of the decision variable using a Monte Carlo simulator such as

Crystal Ball.

4. Compare the mean expected values of the objective function and their confidence intervals, possibly using

statistical hypothesis testing to identify the best solution.

Confidence Intervals ---- 95% confidence intervals: +/- 1.96 sigma, 90% confidence intervals: +/- 1.645 sigma

Useful Functions

MIN(x, y) or MAX(x, y) = x, y can be any variable (possibly a random variable), expression, or number.

Applications include revenue calculations that might be limited by either demand or production quantity.

~N (μ = x, σ = y) = Normal distribution with mean x, std dev. y; directly to right of random variable

~U[x, y] = Uniform distribution with min x, max y; directly to right of random variable


Transshipment Model or Transportation Model The transportation problem is one of the subclasses of linear programming problem where the objective is to transport

various quantities of a single homogeneous product that are initially stored at various origins, to different destinations in

such a way that the total transportation is minimum. Transportation models or problems are primarily concerned with

the optimal (best possible) way in which a product produced at different factories or plants (called supply origins) can be

transported to a number of warehouses (called demand destinations). The objective in a transportation problem is to

fully satisfy the destination requirements within the operating production capacity constraints at the minimum possible

cost. Whenever there is a physical movement of goods from the point of manufacture to the final consumers through a

variety of channels of distribution (wholesalers, retailers, distributors etc.), there is a need to minimize the cost of

transportation so as to increase the profit on sales. It is a transportation model with intermediate destination between

the source and the destination. For example – goods are often transported for manufacturing plants to distribution

centers or warehouse, then finally to stores constraint involving source & destination are similar –

1. Everything leaving source must not exceed supplies.

2. Everything entering destination must not exceed demand.

New constraint – Everything entering an intermediate point must equal everything leaving.

Initial solution for a transportation problem 1. North – west corner method - The North West corner rule is a method for computing a basic feasible solution of

a transportation problem where the basic variables are selected from the North – West corner (i.e., top left

corner). Steps

� Select the north west (upper left-hand) corner cell of the transportation table and allocate as many units

as possible equal to the minimum between available supply and demand requirements, i.e., min (s1, d1).

� Adjust the supply and demand numbers in the respective rows and columns allocation.

� If the supply for the first row is exhausted then move down to the first cell in the second row.

� If the demand for the first cell is satisfied then move horizontally to the next cell in the second column.

� If for any cell supply equals demand then the next allocation can be made in cell either in the next row or

column.

� Continue the procedure until the total available quantity is fully allocated to the cells as required.

2. Minimum Matrix Method (MMM) - Matrix minimum method is a method for computing a basic feasible

solution of a transportation problem where the basic variables are chosen according to the unit cost of

transportation. Steps

� Identify the box having minimum unit transportation cost (cij).

� If there are two or more minimum costs, select the row and the column corresponding to the lower

numbered row.

� If they appear in the same row, select the lower numbered column.

� Choose the value of the corresponding xij as much as possible subject to the capacity and requirement

constraints.

� If demand is satisfied, delete the column.

� If supply is exhausted, delete the row.

� Repeat steps 1-6 until all restrictions are satisfied.

3. Vogel’s Approximation Method (VAM) - The Vogel approximation method is an iterative procedure for

computing a basic feasible solution of the transportation problem. Steps

� Identify the boxes having minimum and next to minimum transportation cost in each row and write the

difference (penalty) along the side of the table against the corresponding row.

� Identify the boxes having minimum and next to minimum transportation cost in each column and write

the difference (penalty) against the corresponding column

� Identify the maximum penalty. If it is along the side of the table, make maximum allotment to the box

having minimum cost of transportation in that row. If it is below the table, make maximum allotment to

the box having minimum cost of transportation in that column.

� If the penalties corresponding to two or more rows or columns are equal, select the top most rows and

the extreme left column


Assignment Model Assignment model are like transportation model except you decide whether or to assign a source to estimation (or

employee to a task). Decision variable are binary suppose you have three employees and three tasks. How many

different possible assignments are there? Assignment problem is one of the special cases of the transportation problem.

It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while

minimizing the total assignment costs. One of the important characteristics of assignment problem is that only one job

(or worker) is assigned to one machine (or project). Hence the number of sources are equal the number of destinations

and each requirement and capacity value is exactly one unit. Although assignment problem can be solved using the

techniques of Linear Programming or the transportation method, the assignment method is much faster and efficient.

This method was developed by D.Konig, a Hungarian mathematician and is therefore known as the Hungarian method of

assignment problem. In order to use this method, one needs to know only the cost of making all the possible

assignments. Each assignment problem has a matrix (table) associated with it. Normally, the objects (or people) one

wishes to assign are expressed in rows, whereas the columns represent the tasks (or things) assigned to them. The

number in the table would then be the costs associated with each particular assignment.

It may be noted that the assignment problem is a variation of transportation problem with two characteristics.

1. The cost matrix is a square matrix

2. The optimum solution for the problem would be such that there would be only one assignment in a row or

column of the cost matrix.

Constraints –

1. Each assignment must get at most I assignee.

2. Each assignee must get at most I assignment.

3. Non negativity constraint

4. Integer constraint (use integer programming)

Application Areas of Assignment Problem - 1. In assigning machines to factory orders.

2. In assigning sales/marketing people to sales territories.

3. In assigning contracts to bidders by systematic bid evaluation.

4. In assigning teachers to classes.

5. In assigning accountants to accounts of the clients

The assignment problem can be solved by the following four methods 1. Enumeration method - In this method, a list of all possible assignments among the given resources and activities

is prepared. Then an assignment involving the minimum cost, time or distance or maximum profits is selected. If

two or more assignments have the same minimum cost, time or distance, the problem has multiple optimal

solutions. This method can be used only if the number of assignments is less. It becomes unsuitable for manual

calculations if number of assignments is large.

2. Simplex method - The graphical method is capable of solving problems having a maximum of two variables.

Hence, this method is used which can solve LP problems with any no. of variable or constraints it is geared

towards solving optimization problems which have constraints of less than or equal to type.

3. Transportation method –

4. Hungarian method - There are various ways to solve assignment problems. Certainly it can be formulated as a

linear program (as we saw above), and the simplex method can be used to solve it. In addition, since it can be

formulated as a network problem, the network simplex method may solve it quickly. However, sometimes the

simplex method is inefficient for assignment problems (particularly problems with a high degree of degeneracy).

The Hungarian Algorithm developed by Kuhn has been used with a good deal of success on these problems and

is summarized as follows.


Queuing theory

Queuing theory is generally considered a branch of operations research because the results are often used when making

business decisions about the resources needed to provide service. Queuing theory is the mathematical study of waiting

lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the (back of

the) queue, waiting in the queue (essentially a storage process), and being served at the front of the queue. The theory

permits the derivation and calculation of several performance measures including the average waiting time in the queue

or the system, the expected number waiting or receiving service, and the probability of encountering the system in

certain states, such as empty, full, having an available server or having to wait a certain time to be served. Queuing

theory has applications in diverse fields, including telecommunications, traffic engineering, computing and the design of

factories, shops, offices and hospitals. Applications are frequently encountered in customer service situations as well as

transport and telecommunication. Queueing theory is directly applicable to intelligent transportation systems, call

centers, PABXs, networks, telecommunications, and server queueing, mainframe computer of telecommunications

terminals, advanced telecommunications systems, and traffic flow. For example, "G/D/1" would indicate a General (may

be anything) arrival process, a Deterministic (constant time) service process and a single server. More details on this

notation are given in the article about Queueing models.

Arrivals �� Queue �� Service ��Departures

Here are details of four queuing disciplines: 1. First in first out - This principle states that customers are served one at a time and that the customer that has

been waiting the longest is served first.

2. Last in first out - This principle also serves customers one at a time, however the customer with the shortest

waiting time will be served first it is also known as a stack.

3. Processor sharing - Customers are served equally. Network capacity is shared between customers and they all

effectively experience the same delay.

4. Priority - Customers with high priority are served first.

Inventory management A proper planning of purchase of raw material, handling, storing and recording is to be considered as a part of inventory

manager. T means management of raw material and related items. Inventory management considers what to purchase,

how to purchase, how much to purchase, from where to purchase, were to store and when to use for production.

Methods of Inventory Management –

• FIFO (first in first out)

• LIFO

• HIFO.

Approaches of inventory management - *ABC Analysis *Economic Order Quantity (EOQ)

1. ABC Analysis - The ABC analysis is a business term used to define an inventory categorization technique often

used in materials management. It is also known as Selective Inventory Control. It provides a mechanism for

identifying items that will have a significant impact on overall inventory cost, while also providing a mechanism

for identifying different categories of stock that will require different management and controls. The ABC

analysis suggests that inventories of an organization are not of equal value. Thus, the inventory is grouped

into three categories (A, B, and C) in order of their estimated importance.

� 'A' items are very important for an organization. Because of the high value of these ‘A’ items, frequently

value analysis are required. In addition to that, an organization needs to choose an appropriate order

pattern (e.g. ‘Just- in- time’) to avoid excess capacity.

� 'B' items are important, but of course less important, than ‘A’ items and more important than ‘C’ items.

Therefore ‘B’ items are intergroup items.


� 'C' items are marginally important.

The process of ABC analysis is given below:

� Step 1: Obtain the list of the items along with information on their unit cost and the periodic consumption

(usually annual)

� Step 2: Determine the annual usage value for each of the items by multiplying the unit cost with number of units

and rank them in descending order on the basis of their respective usage values

� Step 3: Express the value for each item as a percentage of the aggregate usage value. Then cumulate the

percent of annual usage value

� Step 4: Obtain the percentage value for each of the items. For n items, each item should represent 100/n

percent. For example, if there are 20 items involved in the classification, then each item would represent 100/20

= 5% of the materials. Cumulate these percentage values as well.

� Step 5: Using the data on cumulated values of items and the cumulated percentage usage values, plot the curve

by showing these, respectively in X and Y axes.

� Step 6: Determine the appropriate divisions for the A, B and C categories. The curve would rise steeply up to a

point. This point is marked and the items up to that point constitute the A- type items. The curve would be

moderately sloped towards upright. The point beyond which the slope of the curve is negligible is marked. The

items covered beyond that point are classified as category C type. The items falling between point A and C will

be items under category B.

2. Economic order quantity (EOQ) – EOQ is the level of inventory order that minimize the total cost associated

with inventory management. The objective is to find out and maintain optimum level of investment in inventory

to minimize the total cost associated with it. The total cost include –

� Carrying cost – are associated with the maintenance / holding of inventory.

� Ordering cost – are associated with acquisition of planning order of inventory.

The quantity to be ordered is fixed under this method. Reorder are made once the stock reaches a certain pre-

determined level called Reorder level. This typically fixed based on the average consumption deriving the lead

time plus some buffer stock. The best way to determine the fixed quantity to be ordered would be using the

concept of EOQ. This concept is designed in a manner to ensure that the overall inventory cost is lowest. In

other words EOQ is the quantity level to be ordered each time so as to keep the inventory cost minimum. There

are three alternate methods to determine the EOQ namely.

1. Algebraic method

2. Tabular method

3. Graphical method.

EOQ = √2CO/S Where, C = Annual consumption of material, O = ordering cost per order, S=Annual storage cost per unit.

Critical path method or Critical path analysis IN 1957, DuPont developed a project management method designed to address the challenge of shutting down

chemical plants for maintenance and then restarting the plants once the maintenance had been completed. Given the

complexity of the process, they developed the Critical Path Method (CPM) for managing such projects. It is a step-by-

step technique for process planning that defines critical and non-critical tasks with the goal of preventing time-frame

problems and process bottlenecks. The CPM is ideally suited to projects consisting of numerous activities that interact in

a complex manner. It helps you to plan all tasks that must be completed as part of a project. They act as the basis both

for preparation of a schedule, and of resource planning. During management of a project, they allow you to monitor

achievement of project goals. They help you to see where remedial action needs to be taken to get a project back on

course. Critical Path Analysis formally identifies tasks which must be completed on time for the whole project to be

completed on time. It also identifies which tasks can be delayed if resource needs to be reallocated to catch up on

missed or overrunning tasks. CPM is commonly used with all forms of projects, including construction, aerospace and

defense, software development, research projects, product development, engineering, and plant maintenance, among

others. Any project with interdependent activities can apply this method of mathematical analysis. Although the original


CPM program and approach is no longer used, the term is generally applied to any approach used to analyze a project

network logic diagram.

The essential technique for using CPM is to construct a model of the project that includes the following: 1. A list of all activities required to complete the project (typically categorized within a work breakdown structure),

2. The time (duration) that each activity will take to completion, and

3. The dependencies between the activities.

CPM provides the following benefits - 1. Provides a graphical view of the project.

2. Predicts the time required to complete the project.

3. Shows which activities are critical to maintaining the schedule and which are not.

Steps in CPM Project Planning 1. Specify the individual activities.

2. Determine the sequence of those activities.

3. Draw a network diagram.

4. Estimate the completion time for each activity.

5. Identify the critical path (longest path through the network)

6. Update the CPM diagram as the project progresses.

CPM can help you to figure out- 1. how long your complex project will take to complete

2. Which activities are ‘critical’ meanings that they have to be done on time or else the whole project will take

longer.

3. if you put in information about the cost of each activity and how much it cost to speed up each activity

4. Its help to figure out whether you should try to speed up the project and if so what is the least costly way to

speed up project.

CPM Limitations -CPM was developed for complex but fairly routine projects with minimal uncertainty in the project

completion times. For less routine projects there is more uncertainty in the completion times, and this uncertainty limits

the usefulness of the deterministic CPM model. An alternative to CPM is the PERT project planning model, which allows

a range of durations to be specified for each activity.

Program Evaluation Review Technique (PERT) The Program (or Project) Evaluation and Review Technique, commonly abbreviated PERT, PERT is a method to analyze

the involved tasks in completing a given project, especially the time needed to complete each task, and identifying the

minimum time needed to complete the total project. It is commonly used in conjunction with the critical path method or

CPM. A PERT chart is a project management tool used to schedule, organize, and coordinate tasks within a project. It

was developed primarily to simplify the planning and scheduling of large and complex projects. It was developed for the

U.S. Navy Special Projects Office in 1957 to support the U.S. Navy's Polaris nuclear submarine project. It was able to

incorporate uncertainty by making it possible to schedule a project while not knowing precisely the details and durations

of all the activities. It is more of an event-oriented technique rather than start- and completion-oriented, and is used

more in projects where time, rather than cost, is the major factor. It is applied to very large-scale, one-time, complex,

non-routine infrastructure and Research and Development projects. This project model was the first of its kind, a revival

for scientific management, founded by Frederick Taylor (Taylorism) and later refined by Henry Ford (Fordism). DuPont

Corporation’s critical path method was invented at roughly the same time as PERT.


Advantage of PERT 1. PERT chart explicitly defines and makes visible dependencies between the WBS elements.

2. PERT facilitates identification of the critical path and makes this visible.

3. PERT facilitates identification of early start, late start, and stock for each activity.

4. It provides for potentially reduced project duration due to better understanding of dependencies leading to

improved overlapping of activities and tasks where feasible.

5. The large amount of project data can be organized & presented in diagram for use in decision making.

Disadvantages of PERT 1. These can be potentially hundreds, thousands of activities and individual dependency relationships.

2. The network chart to be large and unwisely requiring several pages to print and requiring special size paper.

3. The lack of a time frame on most PERT/CPM charts makes it harder to show status although colors can help (e.g.

specific color for completed notes)

4. When the PERT/CPM charts become unwieldy they are no longer used to manage the project.


GAME THEORY Game theory is the study of how optimal strategies are formulated in conflict. It is concerned with the requirement of

decision making in situations where two or more rational opponents are involved under conditions of competition and

conflicting interests in anticipation of certain outcomes over a period of time. You are surely aware of the fact that in a

competitive environment the strategies taken by the opponent organizations or individuals can dramatically affect the

outcome of a particular decision by an organization. In the automobile industry, for example, the strategies of

competitors to introduce certain models with certain features can dramatically affect the profitability of other

carmakers. So in order to make important decisions in business, it is necessary to consider what other organizations or

individuals are doing or might do. Game theory is a way to consider the impact of the strategies of one, on the strategies

and outcomes of the other. In this you will determine the rules of rational behavior in the game situations, in which the

outcomes are dependent on the actions of the interdependent players.

A GAME refers to a situation in which two or more players are competing. It involves the players (decision makers) who

have different goals or objectives. They are in a situation in which there may be a number of possible outcomes with

different values to them. Although they might have some control that would influence the outcome, they do not have

complete control over others. Unions striking against the company management, players in a chess game, firm striving

for larger share of market are a few illustrations that can be viewed as games.

Game Theory Strategy may be of two types: 1. Pure strategy - If the players select the same strategy each time, then it is referred as pure – strategy. In this

case each player knows exactly what the opponent is going to do and the objective of the players is to maximize

gains or to minimize losses.

2. Mixed Strategy - When the players use a combination of strategies with some fixed probabilities and each

player kept guessing as to which course of action is to be selected by the other player at a particular occasion

then this is known as mixed strategy. Thus, there is probabilistic situation and objective of the player is to

maximize expected gains or to minimize losses strategies. Mixed strategy is a selection among pure strategies

with fixed probabilities.

SIMULATION What is Simulation?

Simulation means imitation of reality. The purpose of simulation in the business world is to understand the behavior of a

system. Before making many important decisions, we simulate the result to insure that we are doing the right thing.

Simulation is used under two conditions.

1. First, when experimentation is not possible. Note that if we can do a real experiment, the results would

obviously be better than simulation.

2. Second condition for using simulation is when the analytical solution procedure is not known. If analytical

formulas are known then we can find the actual expected value of the results quickly by using the formulas. In

simulation we can hope to get the same results after simulating thousands of times.

Why Simulation? This is a fundamental and quantitative way to understand complex systems/phenomena which is complementary to the

traditional approaches of theory and experiment. Simulation is concerned with powerful methods of analysis designed

to exploit high performance computing. This approach is becoming increasingly widespread in basic research and

advanced technological applications, cross cutting the fields of physics, chemistry, mechanics, engineering, and biology

Advantage of Simulation 1. relative straight forward

2. can solve large, complex problems

3. allows "what if" questions


4. Does not interfere with real world systems

5. allows study of interactive variables

6. allows time compression

7. allows inclusion of real world complication

Disadvantage of simulation 1. require generation of all condition and constraints of real world problem

2. each model is unique

3. often require long expensive process

4. does not generate optimal solutions

Monte Carlo Method of Simulation The Monte Carlo method owes its development to the two mathematicians, John Von Neumann and Stanislaw Ulam,

during World War II. The principle behind this method of simulation is representative of the given system under analysis

by a system described by some known probability distribution and then drawing random samples for probability

distribution by means of random number. In case it is not possible to describe a system in terms of Standard probability

distribution such as normal, Poisson, exponential, gamma, etc., an empirical probability distribution can be constructed.

The deterministic method of simulation cannot always be applied to complex real life situations due to inherently high

cost and time values required so as to obtain any meaningful results from the simulated model. Since there are a large

number of interactions between numerous variables, the system becomes too complicated to offer an effective

simulation approach. In such cases where it is not feasible to use an expectation approach for simulating systems,

Monte Carlo method of simulation is used. It can be usefully applied in cases where the system to be simulated has a

large number of elements that exhibit chance (probability) in their behaviour. As already mentioned, the various types

of probability distributions are used to represent the uncertainty of real-life situations in the model. Simulation is

normally undertaken only with the help of a very high-speed data processing machine such as computer. The user of

simulation technique must always bear in mind that the actual frequency or probability would approximate the

theoretical value of probability only when the number of trials are very large i.e. when the simulation is repeated a large

no. of times. This can easily be achieved with the help of a computer by generating random numbers

Following are the steps involved in Monte-Carlo simulation:- � Step I. - Obtain the frequency or probability of all the important variables from the historical sources.

� Step II. - Convert the respective probabilities of the various variables into cumulative problems.

� Step III. - Generate random numbers for each such variable.

� Step IV. - Based on the cumulative probability distribution table obtained in Step II, obtain the interval (i.e.; the

range) of the assigned random numbers.

� Step V. - Simulate a series of experiments or trails.

Remarks. Which random number to use?


Random Number Generation A random number generator (often abbreviated as RNG) is a computational or physical device designed to generate a

sequence of numbers or symbols that lack any pattern, i.e. appear random. The many applications of randomness have

led to the development of several different methods for generating random data. Many of these have existed since

ancient times, including dice, coin flipping, the shuffling of playing cards, the use of yarrow stalks (by divination) in the I

Ching, and many other techniques. Because of the mechanical nature of these techniques, generating large amounts of

sufficiently random numbers (important in statistics) required a lot of work and/or time. Thus, results would sometimes

be collected and distributed as random number tables. Nowadays, after the advent of computational random number

generators, a growing number of government-run lotteries, and lottery games, are using RNGs instead of more

traditional drawing methods. RNGs are also used today to determine the odds of modern slot machines. Random

number generators have applications in gambling, statistical sampling, computer simulation, cryptography, completely

randomized design, and other areas where producing an unpredictable result is desirable. Random number generators

are very useful in developing Monte Carlo method simulations as debugging is facilitated by the ability to run the same

sequence of random numbers again by starting from the same random seed. They are also used in cryptography so long

as the seed is secret. Sender and receiver can generate the same set of numbers automatically to use as keys. Some

simple examples might be presenting a user with a "Random Quote of the Day", or determining which way a computer-

controlled adversary might move in a computer game. Weaker forms of randomness are also closely associated with

hash algorithms and in creating amortized searching and sorting algorithms. Some applications which appear at first

sight to be suitable for randomization are in fact not quite so simple. For instance, a system that "randomly" selects

music tracks for a background music system must only appear to be random, and may even have ways to control the

selection of music; a true random system would have no restriction on the same item appearing two or three times in

succession.

Inventory Simulation ---- While Inventory Optimization selects the mathematically optimal inventory levels,

Inventory Simulation demonstrates “How” the inventory levels and policies will perform in the “real world” given real

demand and supply variability. In business oriented simulations, especially those involving quantitative techniques, it is

often difficult to isolate the decision making process from the mechanics of the simulation. This is particularly true with

inventory simulations because of the amount of bookkeeping required for inventory management. The Inventory

Management Simulation (IMS), a computer based simulation, is designed to alleviate this problem. It acts both as

customer and bookkeeper for the student. It generates demands, and maintains all the necessary records. The student is

then free to focus on individual decisions and the decision making process. The interactive nature of the software makes

the use of the simulation relatively unstructured, so the student can control the time and frequency of its use. The

flexibility of the software allows the instructor to adapt it to a wide variety of learning situations

Common business challenges and business problem 1. Optimized inventory levels not generating expected service performance

2. Demand or supply distribution not reflective of real variability

3. Highly volatile demand or seasonality

4. Extended and variable supplier lead times

5. Erratic fluctuations in inventory levels

6. Inconsistent or low fill rates and increasing stock-outs

management science

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