OPERATIONS RESEARCH

OR

MANAGEMENT SCIENCE

Operations Research (OR) is the study of

mathematical models for complex

organizational systems.

COMPLEXITY IN AN ORGANIZATION Aviation Industry:

Revenue management- cost &fuel efficiency, market share

Flight Scheduling -(fleet Planning, Schedule development, Capacity Planning, routing)

Operations (Manpower planning, Maintenance planning, Crew scheduling)

Irregular operations Retail Industry:

Distribution System-(number of warehouses, trucks, routes)

Retail site selection Optimal mix of merchandise to maximize sales Production optimization-What organization, machines,

processes, and work flows will maximize quality, minimize costs, and maximize output

EMERGENCE OF OPERATIONS RESEARCH

Increased complexity and specialization Limited available resources

“OR is applied the problem that concern how to conduct and co-ordinate the operations within an organization”

EMERGENCE OF OPERATIONS RESEARCH Activities of Operations Research were

initiated in England during the World War II.

Urgent need to allocate scare resources the various activities within each operations (activities) in an effective manner

“Scientist were asked to research on (military) operations”

APPLICATION OF OR

Extensively applied in such diverse area as:ManufacturingTransportationConstructionFinancial planningLabor schedulingSelection of advertising mediaRouting of delivery vehicleHospital staffing

PROBLEMS DISCUSSED IN OR

Optimization Problem: Seeks to maximize or minimize a numerical function

Minimize cost of the food purchased subjected to the constraints that purchased foods provide at least a certain total amount of each nutrient.

Programming Problem: Deals with determining optimal allocation of limited resources to meet given objective. or they deal with situations where number of resources such as men, material, machines and land, are available (with certain restrictions on amount and quality) and are to be combined to yield one or more product.

Out of all permissible resources, it is desired to find the one or ones which maximize or minimize some numerical quantity, such as profit or cost.

Ex. We may be interested in finding the cheapest way of transporting a product from a number of origin to number of destinations.

“Linear programming deals with that class of programming problems for which all relations among the variables are linear.”

EXAMPLE Let us consider a shop with three types of machines,

A, B, and C, Which can turn out four products, 1, 2, 3, 4. Any one of the products has to undergo some operation on each of the three types of machines (lathes, drill and milling machines). We assume that the production is continuous, and that each product must first go on machine type A, then B, and finally C. Furthermore, we shall assume that the time requires for adjusting the setup of each machine to a different operation, when production shifts from one product to another is negligible. Table shows:

The hours required on each machine type per unit of each product

The total available machine hours per week The profit realized on the sale of one unit of any one

products.

DATA FOR EXAMPLE:

Machine Type

Products Total time available per week

1 2 3 4

A 1.5 1 2.4 1 2000

B 1 5 1 3.5 8000

C 1.5 3 3.5 1 5000

Unit profit

5.24 7.30 8.34 4.18

It is assumed that profit is directly proportional to the number of units sold. We wish to determine the weekly output for each product in order to maximize profits.

OBSERVATIONS: Maximum profit will not be achieved by

restricting production to a single item. Available machine time is limited so we

can not increase input arbitrarily of any product.

Production must be allocated among the product 1, 2, 3, 4 so that profit will be maximized without exceeding the maximum number of machine hours available.

LINEAR PROGRAMMING:“Given a set of m linear inequalities or equations in r variables, we wish to find non negative values of these variables which will satisfy the constraints and maximize or minimize some linear function of variables.”

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LINEARITY: Additivity Multiplicativity

Assumptions of the linear programming model The parameter values are known with certainty. The objective function and constraints exhibit constant returns to scale (proportionality /Multiplicativity assumption). There are no interactions between the decision variables (the Additivity assumption). The Continuity assumption: Variables can take on any value within a given feasible range.

APPROACH USED MANAGEMENT SCIENCE

Define Problem Identify Alternative Determine Criteria Evaluate criteria Choose an alternative

MODEL“Models are representation of reality”.

Ex. Equations, An outline, A diagram, A map

Symbolic Model Mathematical Model

Deterministic versus Probabilistic Models

Models are based on some assumption: Technical AssumptionOperational Assumption

FORMULATION OF LP MODEL

Define the decision variable Determine the objective function Identify the constraints Determine appropriate value of

parameters Determine the upper limit, lower limit or

equality called for Build a model

THE SERVER PROBLEM General Description- A firm that assembles computers and

computer equipment is about to start production of two new Web server models. Each type of model will require assembly time, inspection time and storage space. The amount of each of these resources that can be devoted to the production of the server is limited. The manager of the firm would like to determine the quantity of each model to produce in order to maximize the profit generated by sales of these servers.

Additional Information- In order to develop a suitable model of the problem, the manager has met with design and manufacturing personnel. As a result of those meeting, the manager has obtained the following information: Type 1 Type 2

Profit per unit $60 $50

Assembly time per unit 4 hours 10 hours

Inspection time per unit 2 hours 1 hours

Storage space per unit 3 cubic feet 3 cubic feet

The manager also has acquired information on the availability of company resources. These amount are:

Resources Amount availableAssembly time 100 hours

Inspection time 22 hours

Storage space 39 cubic feet

The manager also met with the firm’s marketing manager and learned that demand for the server was such that whatever combination of these two models of servers is produced, all of the output can be sold.

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EXCEPTIONAL CASES:

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TOY MANUFACTURER A toy manufacturer makes three version of a toy

robot. The first version requires 10 minutes each for fabrication and packaging and 2 pounds of plastic, the second version requires 12 minutes for fabrication and packaging and 3 pounds of plastic and the third version requires 15 minutes of fabrication and packaging and 4 pounds of plastic. There are 8 hours of fabrication and packaging time available and 200 pounds of plastic available for the next production cycle. The unit profits are $1 for each version 1, $5 for each version 2, and $6 for each version3. A minimum of 10 units of each version of robot must be made to fill previous orders.

Formulate an LP model that will determine the optimal production quantities for profit maximization.