QUANTITATIVE TECHNIQUES (QUAN 1202)

Lecture 3Linear Programming Modeling

September 2009

You should be able to identify or define: Understand the basic assumptions and properties

of Linear Programming (LP). Graphically solve any LP problem that has only

two variables by both the corner point and isoprofit line methods.

Understand special issues in LP such as infeasibility, unboundedness, redundancy and alternative optimal solutions.

Understand the role of sensitivity analysis.

Learning Objectives

Introduction• In decision-making, model formulation is important

because it represents the essence of business decision problem.

• Linear Programming (LP) is a particular type of technique used for economic allocation of 'scarce' or 'limited' resources (resources that are not unlimited in availability during the planning period), such as labour, material, machine, time, warehouse space, capital, energy, etc. to several competing activities, such as products, services, jobs, new equipment, projects, etc. on the basis of a given criterion of optimality.

Introduction• Linear programming is the most used technique

of decision-making in business.• Its purpose is to assist decision makers in the

allocation of scare resources, i.e., a technique that seeks to solve resource allocation problems using the proportional relationships between two variables.

• Here decisions are made under certainty, i.e., information on available resources and relationship between variables are known.

Allocating Resources• Types of Resources

– The assets of the organisation• Financial: debt, equity, and retained earnings• Physical: buildings, equipment, and raw materials• Human: experiences, skills, knowledge, and

competencies• Intangible: brand names, patents, reputation,

trademarks, copyrights, and databases• Structural/cultural: history, culture, work

systems, working relationships, trust, and policies

Requirements of LP problem Must seek to maximise or minimise some quantity (the

objective function), usually the profit or cost. Presence of restrictions or constraints – that limit the

degree to which we can pursue our objective. There must be alternative courses of action from which

to choose. For ex: if a company produces three different products, management may use LP to decide how to allocate among them its limited production resources (of personnel, machinery and so on). Should it devote all manufacturing capacity to make only the first product, should it produce equal amounts of each product or should it allocate the resources in some ratio?

Objectives and constraints must be expressible as linear equations or inequalities (2A + 5B = 10)

Basic assumptions of LP• Certainty: It is assumed, that all model

parameters such as availability of resources, profit (or cost) contribution of a unit of decision variable and consumption of resources by a unit of decision variable must be known and is constant and do not change during the period being studied.

• Divisibility: The solution values of decision variables and resources are assumed to have either whole numbers (integers) or mixed numbers (integer and fractional).

• Additivity: Meaning that the total of all activities equals the sum of the individual activities. For example, the total profit earned by the sale of two products A and B must be equal to the sum of the profits earned separately from A and B. Similarly, the amount of a resource consumed by A and B must be equal to the sum of resources used for A and B individually.

• Non-negative: Negative values of physical quantities are impossible; you simply cannot produce a negative number of chairs, shirts, lamps or computers.

• Linearity (or proportionality): All relationships in the LP model (i.e. in both objective function and constraints) must be linear. In other words, for any decision variable j, the amount of particular resource say i used and its contribution to the cost one in objective function must be proportional to its amount. For example, if production of one unit of a product uses 5 hours of a particular resource, then making 3 units of that product uses 3 x 5 = 15 hours of that resource.

Formulating LP problems

1. What are the decision variables?2. What is the objective?3. What are the constraints?4. Mathematical formulation.5. Solve the LP problem graphically.

Structure of LPGeneral Structure of LP Model consists of 3 components:i. Decision variables (activities): We need to evaluate

various alternatives (courses of action) for arriving at the optimal value of objective function. The evaluation of various alternatives is guided by the nature of objective function and availability of resources. For this, we pursue certain activities usually denoted by x1, x2, ……xn. For example, in a product-mix manufacturing, the management may use LP to decide how many units of each of the product to manufacture by using its limited resources such as personnel, machinery, money, material, etc. All decision variables are continuous, controllable and non-negative. That is, x1>0, x2>0, ....xn>0.

ii. The objective function: The objective function of each LP problem is a mathematical representation of the objective in terms of a measurable quantity such as profit, cost, revenue, distance, etc. In its general form, it is represented as: Optimise (Maximise or Minimise) Z = c1x1 + c2x2. … cnxn where Z is the measure-of-performance variable, which is a function of x1, x2 ..., xn. Quantities c1, c2…cn are parameters that represent the contribution of a unit of the respective variable x1, x2 ..., xn to the measure-of-performance Z.

iii. The constraints: There are always certain limitations (or constraints) on the use of resources, e.g. labour, machine, raw material, space, money, etc. that limit the degree to which objective can be achieved. Such constraints must be expressed as linear equalities or inequalities in terms of decision variables. The solution of an LP model must satisfy these constraints.

A furniture company produces inexpensive tables and chairs. The production process for each is similar in that both require a certain number of hours of carpentry work and a certain number of labour hours in the painting and varnishing department. Each table takes 4 hours of carpentry and 2 hours in the painting and varnishing. Each chair requires 3 hours in carpentry and 1 hour in painting and varnishing. During the current production period, 240 hours of carpentry time and 6,000 minutes in painting and varnishing time are available. Each table sold yields a profit of Rs 7; each chair produced is sold for a Rs 5 profit.The company’s problem is to determine the best possible combination of tables and chairs to manufacture in order to yield the maximum profit. The company would like this production mix situation be formulated as a LP problem.

Formulation of LP problem

Suppose a farmer has 10 acres of land and a capital of Rs.20,000. It costs Rs.1,400 to sow one acre of wheat and Rs.1,000 to sow one acre of rice. The farmer wants to know the optimal number of acres to use to sow rice or wheat to maximise profits. And suppose that the farmer’s profit is Rs. 800 per acre on wheat and Rs.550 per acre on rice.

Question 1A company sells two different products A and B. The company makes a profit of Rs.40 and Rs.30 per unit on products A and B respectively. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 30,000 hours of labour. It takes three hours to produce a unit of A and one hour to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 8,000 units and that of B is 12,000 units respectively. Subject to these limitations, products can be sold in any combinations.Formulate the above problem as a LP model and calculate the profit of the company.

1.0 Graphical Solution Method

LP problem can be easily solved graphically when it involves only two decision variables.1. Define the problem mathematically.2. Graph the constraints, treat each inequality as

though it was an equality.3. Identify the feasible region.4. Find the corner points on the feasible solution.5. Evaluate the corner points and select the one that

satisfies the objective function.

Question 2A company is manufacturing two products A and B. Production is limited to 80 units of product A and 60 units of product B due to limited supply of raw materials. Production of each of these products requires 5 units and 6 units of electronic components respectively. The Electronic components are supplied by another manufacturer and the supply is limited to 600 units/day. The company has 160 employees and the production of one unit of product A requires 1 man-day of labour and one unit of product B requires 2 man-day of labour. Each unit of these products is sold in the market at a profit of Rs.50 and Rs.80 respectively. Determine how many units of each product the company should produce to maximise profit.

• Both optimal solutions occurred at "corners" of the feasible region. These vertices are called extreme points.

• The optimal solution to a linear programming problem is always found at an extreme point of the feasible region.

• Therefore, we need only evaluate extreme points and select the one that maximises the objective function.

Advantages of LP• Helps in attaining the optimum use of productive

resources. • Improve the quality of decisions. • Provide possible and practical solutions since there

might be other constraints operating outside the problem which must be taken into account.

• Highlighting of bottlenecks in the production processes is the most significant advantage of this technique.

• Helps in re-evaluation of a basic plan for changing conditions.

Limitations of LP• While solving an LP model, there is no

guarantee that we will get integer valued solutions.

• Linear programming model does not take into consideration the effect of time and uncertainty.

• Sometimes large-scale problems can be solved with linear programming techniques even when assistance of computer is available.

• Parameters appearing in the model are assumed to be constant but in real-life situations, they are frequently neither known nor constant.

2.0 Isoprofit Line Solution Method• The optimal is the point lying in the feasible region that

produces the highest profit.• There are a few approaches that can be taken in solving

for the optimal solution when the feasible region has been established and one of the speediest one to apply is called the isoprofit line method.

• The technique is that you set the profits equal some arbitrary but small amount. For the furniture company, we can choose a profit of Rs 210. This is a profit level that can be obtained easily without violating either of the two constraints. The objective function can be written as Rs 210 = 7X1 + 5X2.

• The expression is just the equation of a line; an isoprofit line.

• Represents all combinations of (X1,X2) that would yield a total profit of Rs 210.

• Plot the profit line (let X1 = 0 and solve for X2 and let X2 = 0 and solve for X1).

• All points on the line represent feasible solutions that produce a profit of Rs 210.

• Draw a series of parallel isoprofit lines until you find the highest isoprofit line, that is, the one with the optimal solution (the one that touches the tip of the feasible region at the corner point (X1 = 30 and X2 = 40) and yields a profit of Rs 410.

3.0 Corner Point Solution Method• This technique is simpler conceptually than the

isoprofit line approach.• Involves looking at the profit at every corner point of

the feasible region.• The mathematical theory behind LP states that an

optimal solution to any problem (that is, the values of X1 and X2 that yield the maximum profit) will lie at a corner point, or extreme point, of the feasible region.

• It is necessary to find the values of the variables at each corner; the maximum profit or optimal solution will lie at one (or more) of them.

Application of LPLinear programming is the most widely used technique of decision-making in business and industry and in various other fields. For example

• Agricultural Applications: Linear programming can be applied in agricultural planning, e.g. allocation of limited resources such as acreage, labour, water supply and working capital, etc. in a way so as to maximise net revenue.

• Military Applications: Military applications include the problem of selecting an air weapon system against enemy so as to keep them pinned down and at the same time minimising the amount of aviation gasoline used.

• Production Management: (Product mix) A company can produce several different products, each of which requires the use of limited production resources. In such cases, it is essential to determine the quantity of each product to be produced knowing its marginal contribution and amount of available resource used by it. The objective is to maximise the total contribution, subject to all constraints.

• Financial Management: (Portfolio selection) This deals with the selection of specific investment activity among several other activities. The objective is to find the allocation which maximises the total expected return or minimises risk under certain limitations.

• Marketing Management: (Media selection) Linear programming technique helps in determining the advertising media mix so as to maximise the effective exposure, subject to limitation of budget, specified exposure rates to different market segments, specified minimum and maximum number of advertisements in various media.

• Personnel Management: (Staffing problem) Linear programming is used to allocate optimum manpower to a particular job so as to minimise the total overtime cost or total manpower.

Question 3A manufacturing firm produces two products, A and B. Each of these products must be processed through two different machines. One machine has 24 hours of available capacity and the second has 16 hours. Each unit of product A requires two hours of time on both machines. Each unit of product B requires three hours of time on the first machine and one hour on the second machine. The incremental profit is Rs. 6 per unit of product A and Rs. 7 per unit of product B, and the firm can sell as many units of each product as it can manufacture. The objective of the firm is to maximise profits. The problem is to determine how many units of product A and product B should be produced within the limits of available machine capacities.