ORDUDE #3-Model Formulation

Dual Degree – Management UNPLecturerGesit Thabrani

Model Formulation

Dual Degree – Management UNP

Operations Operations ResearchResearch

OR#3


Learning Objectives

1. Understand the basic assumptions and properties of linear programming (LP)

2. Formulate the LP problems

After completing this chapter, students will be abl e to:After completing this chapter, students will be abl e to:


Outline

1. What is Linear Programming?2. Requirements of a Linear

Programming Problem3. Formulating LP Problems


Linear Programming

� Many management decisions involve trying to make the most effective use of limited resources� Machinery, labor, money, time, warehouse space, raw

materials

�� Linear programmingLinear programming (LPLP ) is a widely used mathematical modeling technique designed to help managers in planning and decision making relative to resource allocation

� Belongs to the broader field of mathematical mathematical programmingprogramming

� In this sense, programmingprogramming refers to modeling and solving a problem mathematically


Requirements of a Linear Programming Problem

� LP has been applied in many areas over the past 50 years

� All LP problems have 4 properties in common1. All problems seek to maximizemaximize or minimizeminimize some

quantity (the objective functionobjective function )2. The presence of restrictions or constraintsconstraints that limit the

degree to which we can pursue our objective3. There must be alternative courses of action to choo se

from4. The objective and constraints in problems must be

expressed in terms of linearlinear equations or inequalitiesinequalities


LP Properties and Assumptions

PROPERTIES OF LINEAR PROGRAMS

1. One objective function

2. One or more constraints

3. Alternative courses of action

4. Objective function and constraints are linear

ASSUMPTIONS OF LP

1. Certainty

2. Proportionality

3. Additivity

4. Divisibility

5. Nonnegative variables

Table 7.1


Basic Assumptions of LP

� We assume conditions of certaintycertainty exist and numbers in the objective and constraints are known with certainty and do not change during the period being studied

� We assume proportionalityproportionality exists in the objective and constraints

� We assume additivityadditivity in that the total of all activities equals the sum of the individual activities

� We assume divisibilitydivisibility in that solutions need not be whole numbers

� All answers or variables are nonnegativenonnegative


Formulating LP Problems

� Formulating a linear program involves developing a mathematical model to represent the managerial problem

� The steps in formulating a linear program are1. Completely understand the managerial

problem being faced2. Define the decision variables3. Identify the objective and constraints4. Use the decision variables to write

mathematical expressions for the objective function and the constraints



� Decision variables are mathematical symbols that represent levels of activity by the firm

� For example, an electrical manufacturing firm desires to produce x1 radios, x2 toasters, and x3clocks, where x1, x2, and x3 are symbols representing unknown variable quantities of each item. The final values of x1, x2, and x3, as determined by the firm, constitute a decision (e.g., the equation x 1 = 100 radios is a decision by the firm to produce 100 radios).

� The objective function is a linear mathematical relationship that describes the objective of the firm in terms of the decision variables. The objective function always consists of either maximizing or minimizing some value (e.g., maximize the profit or minimize the cost of producing radios)



� The model constraints are also linear relationships of the decision variables; they represent the restrictions placed on the firm by the operating environment.

� The restrictions can be in the form of limited resources or restrictive guidelines.

� For example, only 40 hours of labor may be available to produce radios during production. The actual numeric values in the objective function and the constraints, such as the 40 hours of available labor, are parameters

� Status Function show that all the variables are nonnegative (nonnegativity constraints)



Identify the problem Determine objective function

Determine the constraints

Define status function

X1 >= 0Number of variable = 2Number of variable >= 2

Maximize

Minimize

inequality “<=”

inequality “>=”

equality “=”

PROBLEMS FORMULATION

Linear Programming Model



� One of the most common LP applications is the product mix problemproduct mix problem

� Two or more products are produced using limited resources such as personnel, machines, and raw materials

� The profit that the firm seeks to maximize is based on the profit contribution per unit of each product

� The company would like to determine how many units of each product it should produce so as to maximize overall profit given its limited resources


Flair Furniture Company

� The Flair Furniture Company produces inexpensive tables and chairs

� Processes are similar in that both require a certai n amount of hours of carpentry work and in the painting and varnishing department

� Each table takes 4 hours of carpentry and 2 hours of painting and varnishing

� Each chair requires 3 of carpentry and 1 hour of painting and varnishing

� There are 240 hours of carpentry time available and 100 hours of painting and varnishing

� Each table yields a profit of $70 and each chair a profit of $50



� The company wants to determine the best combination of tables and chairs to produce to reach the maximum profit

HOURS REQUIRED TO PRODUCE 1 UNIT

DEPARTMENT(T)

TABLES(C)

CHAIRSAVAILABLE HOURS THIS WEEK

Carpentry 4 3 240

Painting and varnishing 2 1 100

Profit per unit $70 $50

Table 7.2



� The decision variables representing the actual decisions we will make areT = number of tables to be produced per weekC = number of chairs to be produced per week

� The objective is toMaximize profit

� The constraints are1. The hours of carpentry time used cannot

exceed 240 hours per week2. The hours of painting and varnishing time

used cannot exceed 100 hours per week



� We create the LP objective function in terms of Tand C

Maximize profit = $70 T + $50C� Develop mathematical relationships for the two

constraints� For carpentry, total time used is

(4 hours per table)(Number of tables produced)+ (3 hours per chair)(Number of chairs produced)

� We know thatCarpentry time used ≤ Carpentry time available

4T + 3C ≤ 240 (hours of carpentry time )



� SimilarlyPainting and varnishing time used ≤ Painting and varnishing time available

2 T + 1C ≤ 100 (hours of painting and varnishing time)

This means that each table produced requires two hours of painting and varnishing time

� Both of these constraints restrict production capacity and affect total profit



� The values for T and C must be nonnegative

T ≥ 0 (number of tables produced is greater than or equal to 0)

C ≥ 0 (number of chairs produced is greater than or equal to 0)

� The complete problem stated mathematically

Maximize profit = $70 T + $50C

subject to4T + 3C ≤ 240 (carpentry constraint)

2T + 1C ≤ 100 (painting and varnishing constraint)

T, C ≥ 0 (nonnegativity constraint)


Shader ElectronicsShader Electronics

The productThe product--mix problem at Shader mix problem at Shader ElectronicsElectronics�� Two productsTwo products

1.1. Shader Walkman, a portable CD/DVD Shader Walkman, a portable CD/DVD playerplayer

2.2. Shader WatchShader Watch--TV, a wristwatchTV, a wristwatch--size size InternetInternet--connected color TVconnected color TV

�� Determine the mix of products that will Determine the mix of products that will produce the maximum profitproduce the maximum profit



WalkmanWalkman WatchWatch--TVsTVs Available HoursAvailable HoursDepartmentDepartment ((XX11)) ((XX22)) This WeekThis Week

Hours Required Hours Required to Produce 1 Unitto Produce 1 Unit

ElectronicElectronic 44 33 240240

AssemblyAssembly 22 11 100100

Profit per unitProfit per unit $7$7 $5$5

Decision Variables:Decision Variables:XX11 = number of Walkmans to be produced= number of Walkmans to be producedXX22 = number of Watch= number of Watch--TVs to be producedTVs to be produced



�� Objective Function:Objective Function:Maximize Profit = Maximize Profit = $7$7XX11 + + $5$5XX22

Or, usually we can state it as:Or, usually we can state it as:Max Z = 7XMax Z = 7X11 + 5X+ 5X22



First Constraint:First Constraint:

ElectronicElectronictime availabletime available

ElectronicElectronictime usedtime used is ≤is ≤

44XX11 + + 33XX22 ≤ 240 ≤ 240 (hours of electronic time)(hours of electronic time)

Second Constraint:Second Constraint:

AssemblyAssemblytime availabletime available

AssemblyAssemblytime usedtime used is ≤is ≤

22XX11 + + 11XX22 ≤ 100 ≤ 100 (hours of assembly time)(hours of assembly time)



� The complete problem stated mathematically

subject to44XX11 + + 33XX22 ≤ 240 ≤ 240 (hours of electronic time)(hours of electronic time)22XX11 + + 11XX22 ≤ 100 ≤ 100 (hours of assembly time)(hours of assembly time)

XX11, X, X22 ≥≥ 00 (nonnegativity constraint)(nonnegativity constraint)

Max Z = 7XMax Z = 7X11 + 5X+ 5X22


Minimization Case (Fertilizer)

� A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are two brands of fertilizer to choose from, Super-gro and Crop-quick. Each brand y ields a specific amount of nitrogen and phosphate per bag , as follows:

� The farmer's field requires at least 16 pounds of n itrogen and 24 pounds of phosphate. Super-gro costs $6 per bag, and Crop-quick costs $3. The farmer wants to know h ow many bags of each brand to purchase in order to min imize the total cost of fertilizing.

BRANDCHEMICAL CONTRIBUTION

NITROGEN (LB./BAG)

PHOSPHATE

(LB./BAG

Super-gro 2 4

Crop-quick 4 3



Summary of LP Model Formulation Steps� Step 1. Define the decision variables

� How many bags of Super-gro and Crop-quick to buy

� Step 2. Define the objective function� Minimize cost

� Step 3. Define the constraints� The field requirements for nitrogen and

phosphate



� Decision Variables� This problem contains two decision

variables, representing the number of bags of each brand of fertilizer to purchase:

� x1 = bags of Super-gro� x2 = bags of Crop-quick



� The Objective Function� The farmer's objective is to minimize the total

cost of fertilizing. � The total cost is the sum of the individual

costs of each type of fertilizer purchased. � The objective function that represents

total cost is expressed asminimize Z = $6x 1 + $3x2

where� $6x1 = cost of bags of Super-gro� $3x2 = cost of bags of Crop-quick



� Model ConstraintsEach bag of fertilizer contributes a number of pounds of nitrogen and phosphate to the field

� The constraint for nitrogen is2x1 + 4x2 ≥≥ 16 lb.

� The constraint for phosphate is constructed like the constraint for nitrogen:

4x1 + 3x2 ≤≤ 24 lb.

� Nonnegativity constraints in this problem to indicate that negative bags of fertilizer cannot be purchased:

x1, x2 ≥≥ 0



� The complete model formulation for this minimization problem is

Min Min Z = 6x1 + 3x2

subject to2x1 + 4x2 ≥ 16 lb, of nitrogen4x1 + 3x2 ≥ 24 lb, of phosphate

x1, x2 ≥ 0 (nonnegativity constraint)

ORDUDE #3-Model Formulation

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