Mba or Unit-Ii Notes
Mba or Unit-Ii Notes
Mba or Unit-Ii Notes
The constraints There are always certain limitations (or constraints) on the use of resources, such
as: labour, machine, raw material, space, money, etc., that limit the degree to which an 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.
Assumptions of an LP Model
In all mathematical models, assumptions are made for reducing the complex real-world problems
into a simplified form that can be more readily analyzed. The following are the major assumptions
of an LP model:
1. Certainty: In LP models, it is assumed that all its parameters such as: availability of
resources, profit (or cost) contribution per unit of decision variable and consumption of resources
per unit of decision variable must be known and constant.
2. Additivity: The value of the objective function and the total amount of each resource used
(or supplied), must be equal to the sum of the respective individual contribution (profit or cost) of
the decision variables. For example, the total profit earned from 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 for producing A and B must be equal to the total sum of resources used for A
and B individually.
3. Linearity (or proportionality): The amount of each resource used (or supplied) and its
contribution to the profit (or cost) in objective function must be proportional to the value of each
decision variable. 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 3×5 = 15 hours of that resource.
4. Divisibility (or continuity): The solution values of decision variables are allowed to
assume continuous values. For instance, it is possible to collect 6.254 thousand litres of milk by a
milk dairy and such variables are divisible. But, it is not desirable to produce 2.5 machines and such
variables are not divisible and therefore must be assigned integer values. Hence, if any of the
variable can assume only integer values or are limited to discrete number of values, LP model is no
longer applicable.
The company has a daily order commitment for 20 units of products A and a total of 15 units of
products B and C. Formulate this problem as an LP model so as to maximize the total profit.
Solution
The data of the problem is summarized as follows:
x1, x2, x3 ≥ 0.
Solution
1) Decision variables
Let
x1 = quantity of product A (in ’000 gallons) to be produced in plant 1
x2 = quantity of product A (in ’000 gallons) to be produced in plant 2
x3, = quantity of product B (in quintals) to be produced in plant 1
x4 = quantity of product B (in quintals) to be produced in plant 2
Step 2) Objective function
Practice problems
1) A company sells two different products A and B, making a profit of Rs 40 and Rs 30 per
unit, respectively. They are both produced with the help of a common production process and are
sold in two different markets. The production process has a total capacity of 30,000 man-hours. 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. Subject to these limitations, products can be sold in any
combination. Formulate this problem as an LP model to maximize profit.
2) The manager of an oil refinery must decide on the optimal mix of two possible blending
processes of which the input and output per production run are given as follows:
The maximum available amount of crude A and B are 200 units and 150 units, respectively. Market
requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be
produced. The profit per production run from process 1 and process 2 are Rs 300 and Rs 400,
respectively. Formulate this problem as an LP model to maximize profit.
3) A firm places an order for a particular product at the beginning of each month and that
product is received at the end of the month. The firm sells during the month from the stocks and it
can sell any quantity. The prices at which the firm buys and sells vary every month.
The following table shows the projected buying and selling prices for the next four months:
As on April 1 the firm has no stocks on hand, and does not wish to have any stocks at the
end of July. The firm has a warehouse of limited size, which can hold a maximum of 150 units of
the product. Formulate this problem as an LP model to determine the number of units to buy and sell
each month so as to maximize the profits from its operations.
4) A manufacturer produces three models (I, II and III) of a certain product. He uses two
types of raw materials (A and B) of which 4,000 and 6,000 units, respectively, are available. The
raw material requirements per unit of the three models are as follows:
The labour time of each unit of model I is twice that of model II and three times that of model III.
The entire labour force of the factory can produce equivalent of 2,500 units of model I. A market
survey indicates that the minimum demand of the three models is: 500, 500 and 375 units,
respectively. However, the ratios of the number of units produced must be equal to 3 : 2 : 5. Assume
that the profit per unit of models I, II and III are Rs 60, 40 and 100, respectively. Formulate this
problem as an LP model to determine the number of units of each product which will maximize
profit.
5) Consider a small plant which makes two types of automobile parts, say A and B. It buys
castings that are machined, bored and polished. The capacity of machining is 25 per hour for A and
24 per hour for B, capacity of boring is 28 per hour for A and 35 per hour for B, and the capacity of
polishing is 35 per hour for A and 25 per hour for B. Castings for part A cost Rs 2 and sell for Rs 5
each and those for part B cost Rs 3 and sell for Rs 6 each. The three machines have running costs
of Rs 20, Rs 14 and Rs 17.50 per hour. Assuming that any combination of parts A and B can be
sold, formulate this problem as an LP model to determine the product mix which would maximizes
profit.
6) On October 1, a company received a contract to supply 6,000 units of a specialized
product. The terms of contract require that 1,000 units of the product be shipped in October; 3,000
units in November and 2,000 units in December. The company can manufacture 1,500 units per
month on regular time and 750 units per month in overtime. The manufacturing cost per item
produced during regular time is Rs 3 and the cost per item produced during overtime is Rs 5. The
monthly storage cost is Re 1. Formulate this problem as an LP model so as to minimize total costs.
7) A wine maker has a stock of three different wines with the following characteristics:
A good dry table wine should be between 30 and 31 degree proof, it should contain at least 0.25%
acid and should have a specific gravity of at least 1.06. The wine maker wishes to blend the three
types of wine to produce as large a quantity as possible of a satisfactory dry table wine. However,
his stock of wine A must be completely used in the blend because further storage would cause it to
deteriorate. What quantities of wines B and C should be used in the blend. Formulate this problem
as an LP model.
8) ABC foods company is developing a low-calorie high-protein diet supplement called Hi-
Pro. The specifications of Hi-Pro have been established by a panel of medical experts. These
specifications along with the calorie, protein and vitamin content of three basic foods, are given in
the following table:
What quantities of foods 1, 2, and 3 should be used? Formulate this problem as an LP model to
minimize cost of serving.
9) Omega leather goods company manufactures two types of leather soccer balls X and Y.
Each type of ball requires work by two types of employees – semi-skilled and skilled. Basically, the
semi-skilled employees use machines, while the skilled employees stitch the balls. The available
time (per week) for manufacturing each type of employee and the time requirement for each type of
ball are given below:
The cost of an hour of semi-skilled labour is Rs 5.50 and that of an hour of skilled labour is Rs 8.50.
To meet the weekly demand requirements, at least 15 balls of type X and at least 10 balls of type Y
must be manufactured. Formulate this problem as an LP model so as to minimize cost of production.
10) A pharmaceutical company has developed a new pill to be taken by smokers that will
nauseate them if they smoke. This new pill is a combination of four ingredients that are costly and in
limited supply. The available supply and costs are as follows:
IMPORTANT DEFINITIONS
Solution The set of values of decision variables xj ( j = 1, 2, . . ., n) that satisfy the constraints of an
LP problem is said to constitute the solution to that LP problem.
Feasible solution The set of values of decision variables xj ( j = 1, 2, . . ., n) that satisfy all the
constraints and non-negativity conditions of an LP problem simultaneously is said to constitute the
feasible
solution to that LP problem.
Infeasible solution The set of values of decision variables xj ( j = 1, 2, . . ., n) that do not satisfy all
the constraints and non-negativity conditions of an LP problem simultaneously is said to constitute
the infeasible solution to that LP problem.
Basic solution For a set of m simultaneous equations in n variables (n > m) in an LP problem, a
solution obtained by setting (n – m) variables equal to zero and solving for remaining m equations
in m variables is called a basic solution of that LP problem.
The (n – m) variables whose value did not appear in basic solution are called non-basic variables
and the remaining m variables are called basic variables.
Basic feasible solution A feasible solution to an LP problem which is also the basic solution is
called the basic feasible solution. That is, all basic variables assume non-negative values. Basic
feasible solution is of two types:
(a) Degenerate A basic feasible solution is called degenerate if the value of at least one basic
variable is zero.
(b) Non-degenerate A basic feasible solution is called non-degenerate if value of all m basic
variables is non-zero and positive.
Optimum basic feasible solution A basic feasible solution that optimizes (maximizes or
minimizes) the objective function value of the given LP problem is called an optimum basic feasible
solution.
Unbounded solution A solution that can increase or decrease infinitely the value of the objective
function of the LP problem is called an unbounded solution.
Since all constraints have been graphed, the area which is bounded by all the constraints
lines including all the boundary points is called the feasible region (or solution space).
The feasible region is shown in by the shaded area OABCD.
Since the optimal value of the objective function occurs at one of the extreme points of the
feasible region, it is necessary to determine their coordinates. The coordinates of extreme points of
the
feasible region are:
O = (0, 0), A = (60, 0), B = (60, 20), C = (30, 40), D = (0, 40).
1) – x1 + x2 = 1 2) – 0.5x1 + x2 = 2
x1 x2 x1 x2
0 1 0 2
-1 0 -4 0
Since the maximum value of the objective function Z = 1,160 occurs at the extreme point (9, 10),
the optimum solution to the given LP problem is: x1 = 9, x2 = 10 and
Max. Z = Rs 1,160.
5) Use the graphical method to solve the following LP problem.
Minimize Z = 3x1 + 2x2
subject to the constraints
5x1 + x2 ≥10, x1 + x2 ≥ 6, x1 + 4x2 ≥12
and x1, x2 ≥0.
Solution
1) 5x1 + x2 = 10 2) x1 + x2 = 6 3) x1 + 4x2 = 12
x1 x2 x1 x2 x1 x2
0 10 0 6 0 3
2 0 6 0 12 0
The minimum (optimal) value of the objective function Z = 13 occurs at the extreme point C (1, 5).
Hence, the optimal solution to the given LP problem is: x1 = 1, x2 = 5, and
Min Z = 13.
6) A firm plans to purchase at least 200 quintals of scrap containing high quality metal X and
low quality metal Y. It decides that the scrap to be purchased must contain at least 100 quintals of
metal X and not more than 35 quintals of metal Y. The firm can purchase the scrap from two
suppliers (A and B) in unlimited quantities. The percentage of X and Y metals in terms of weight in
the scrap supplied by A and B is given below.
The price of A’s scrap is Rs 200 per quintal and that of B is Rs 400 per quintal. The firm wants to
determine the quantities that it should buy from the two suppliers so that the total cost is minimized.
Solution:
Let us define the following decision variables:
x1 and x2 = quantity (in quintals) of scrap to be purchased from suppliers A and B, respectively.
Then LP model of the given problem is:
Minimize Z = 200x1 + 400x2
subject to the constraints
x1 + x2 ≥200; x1 + 3x2 ≥400 ; x1 + 2x2 ≤350
and x1, x2 ≥0.
Since minimum (optimal) value of objective function, Z = Rs 60,000 occurs at the extreme point
A (100, 100), firm should buy x1 = x2 = 100 quintals of scrap each from suppliers A and B.
Practice problems:
Solve the following LP problems graphically and state what your solution indicates.
(iv) Max Z = x1 + x2
subject to
x1 – x2 ≥ 0
3x1 – x2 ≤– 3
and x1, x2 ≥ 0
1) A furniture manufacturer makes two products: chairs and tables. These products are
processed using two machines – A and B. A chair requires 2 hours on machine A and 6
hours on machine B. A table requires 5 hours on machine A and no time on machine B.
There are 16 hours per day available on machine A and 30 hours on machine B. The profit
gained by the manufacturer from a chair is Rs 2 and from a table is Rs 10. Solve this
problem to find the daily production of each of the two products.
2) A company produces two types of leather belts, A and B. Belt A is of a superior quality and
B is of an inferior quality. The profit from the two are 40 and 30 paise per belt, respectively.
Each belt of type A requires twice as much time as required by a belt of type B. If all the
belts were of type B, the company could produce 1,000 belts per day. But the supply of
leather is sufficient only for 800 belts per day. Belt A requires a fancy buckle and only 400
of them are available per day. For belt B only 700 buckles are available per day. Solve this
problem to determine how many units of the two types of belts the company should
manufacture in order to have the maximum overall profit?
3) Consider a small plant which makes two types of automobile parts, say A and B. It buys
castings that are machined, bored
and polished. The capacity of machining is 25 per hour for A and 40 per hour for B, the
capacity of boring is 28 per hour for A and 35 per hour for B, and the capacity of polishing is
35 per hour for A and 25 per hour for B. Casting for part A costs Rs 2 each and for part B it
costs Rs 3 each. The company sells these for Rs 5 and Rs 6, respectively. The three
machines have running costs of Rs 20, Rs 14 and Rs 17.50 per hour. Assuming that any
combination of parts A and B can be sold, what product mix would maximize profit? Solve
this problem using the graphical method.
Theory of Games
INTRODUCTION
In general, the term ‘game’ refers to a situation of conflict and competition in which two or more
competitors (or participants) are involved in the decision-making process in anticipation of certain
outcomes over a period of time. The competitors are referred to as players. A player may be an
individual, individuals, or an organization. A few examples of competitive and conflicting decision
environment, that involve the interaction between two or more competitors are:
• Pricing of products, where sale of any product is determined not only by its price but also by the
price set by competitors for a similar product
• The success of any TV channel programme largely depends on what the competitors presence in
the same time slot and the programme they are telecasting.
• The success of a business strategy depends on the policy of internal revenue service regarding the
expenses that may be disallowed,
• The success of an advertising/marketing campaign depends on various types of services offered to
the customers.
The models in the theory of games can be classified based on the following factors:
Number of players
If a game involves only two players (competitors), then it is called a two-person game. However, if
the number of players are more, the game is referred to as n-person game.
Strategy
The strategy for a player is the list of all possible actions (moves, decision alternatives or courses of
action) that are likely to be adopted by him for every payoff (outcome). It is assumed that the
players are aware of the rules of the game governing their decision alternatives (or strategies). The
outcome resulting from a particular strategy is also known to the players in advance and is
expressed in terms of numerical values (e.g. money, per cent of market share or utility). The
particular strategy that optimizes a player’s gains or losses, without knowing the competitor’s
strategies, is called optimal strategy. The expected outcome, when players use their optimal strategy,
is called value of the game.
Generally, the following two types of strategies are followed by players in a game:
(a) Pure Strategy
A particular strategy that a player chooses to play again and again regardless of other
player’s strategy, is referred as pure strategy. The objective of the players is to maximize
their gains or minimize their losses.
(b) Mixed Strategy
A set of strategies that a player chooses on a particular move of the game with some fixed
probability are called mixed strategies. Thus, there is a probabilistic situation and objective
of the each player is to maximize expected gain or to minimize expected loss by making the
choice among pure strategies with fixed probabilities.
Payoff matrix
The payoffs (a quantitative measure of satisfaction that a player gets at the end of the play) in terms
of gains or losses, when players select their particular strategies (courses of action), can be
represented in the form of a matrix, called the payoff matrix. Since the game is zero-sum, the gain of
one player is equal to the loss of other and vice versa. In other words, one player’s payoff table
would contain the same amounts in payoff table of other player, with the sign changed. Thus, it is
sufficient to construct a payoff table only for one of the players.
If player A has m strategies represented by the letters: A1, A2, . . ., Am and player B has n strategies
represented by the letters: B1, B2, . . ., Bn. The numbers m and n need not be equal. The total
number of possible outcomes is therefore m × n. It is assumed that each player not only knows his
own list of possible strategies but also of his competitor. For convenience, it is assumed that player
A is always a gainer whereas player B a loser. Let aij be the payoff that player A gains from player
B if player A chooses strategy i and player B chooses strategy j. Then the payoff matrix is shown in
the Table