Transcript Linear Programming

Linear Programming Problem
Prepared by
Sayed Mohibul Hossen
Lecturer in Statistics
Linear programming
Linear programming is a method or technique of
determining an optimum program of interdependent activities in a view of available
resources.
Linear programming is a mathematical technique
for determining the optimal allocation of
resources and obtaining a particular objective.
Linear programming
Mathematical programming is used to find the
best or optimal solution to a problem that
requires a decision about how best to use a set of
limited resources to achieve a state goal of
objectives.
Linear programming
Linear programming is a technique for the
optimization of a linear objective function,
subject to linear equality and linear inequality or
constraints.
LP is a mathematical modeling technique used to
determine a level of operational activity in order
to achieve an objective, subject to restrictions
called constraints.
Linear programming
In Linear Programming the term linear implies
that all the mathematical relations used in the
problem are linear and Programming refers to
the method of determining a particular plan of
action from amongst several alternatives.
Examples of LP Problems
A Product Mix Problem
• A manufacturer has fixed amounts of different resources
such as raw material, labor, and equipment.
• These resources can be combined to produce any one of
several different products.
• The decision maker wishes to produce the combination
of products that will maximize total income.
Examples of LP Problems
A Production Scheduling Problem
• A manufacturer knows that he must supply a given number of
items of a certain product each month for the next n months.
• They can be produced either in regular time, subject to a
maximum each month, or in overtime. The cost of producing
an item during overtime is greater than during regular time. A
storage cost is associated with each item not sold at the end of
the month.
• The problem is to determine the production schedule that
minimizes the sum of production and storage costs.
Examples
• Devising of a product schedule that could satisfy future
demands ( seasonal or otherwise) for the firm’s
product and at the same time minimize production
costs.
• Selecting of production–mix to make the best use of
machines, man-hours with a view to maximize profits.
• Selecting the advertising mix that will maximize the
benefit subject to the total advertising budget.
Example
A manufacturer company produces two types of product.
Following table shows the per unit labor, per unit clay and per
unit profit is given.
RESOURCE REQUIREMENTS
PRODUCT
Bowl
Mug
Labor
(hr/unit)
1
2
Clay
(lb/unit)
4
3
profit
($/unit)
40
50
There are 40 hours of labor and 120 pounds of clay available each
day.
How much bowl and how much mug does the manufacturer to
produce for maximizing the profit?
General LPP
Max/min z = c1x1 + c2x2
objective function
Where, x1 and x2 are decision variables.
Subject to
a11x1 + a12x2 (≤, =, ≥) b1
a21x1 + a22x2 (≤, =, ≥) b2
:
x1 ≥ 0, x2 ≥ 0
Constraints
Non-negativity restriction
Decision variables
The decision variables are those quantities whose
values are to be determined. Here x1, x2, x3… xn
are the decision variables which optimize the
objective function and also satisfy the
constraints.
Objective Function
A linear function (z = c1x1 + c2x2 + ... + cnxn)
which reflecting the objective of an operation and
which has to be maximized or minimized is called
a linear objective function. Objective function
represents cost, profit, or some other quantity to
be maximized or minimized subject to the
constraints.
Constraints
The linear inequalities or equations or restrictions
on the available resources (Labor, Capital,
Materials and Machines) of a linear programming
problem are called constraints.
Non-negative constraints
Non-negative constraints included because
variables are usually the number of items
produced and one cannot produce a negative
number of items. The smallest number of items
one could produce is zero.
Example
A manufacturer company produces two types of product.
Following table shows the per unit labor, per unit clay and per
unit revenue is given.
RESOURCE REQUIREMENTS
Labor
Clay
profit
PRODUCT
(hr/unit)
(Kg./unit)
($/unit)
Bowl
1
4
40
Mug
2
3
50
There are 40 hours of labor and 120 Kg. of clay available each day.
How much bowl and how much mug does the manufacturer to
produce for maximizing the profit.
Formulation of Linear Programming
Problem
• Step I: In every LPP, certain decisions are to be made.
These decisions are represented by decision variables.
These decision variables are those quantities whose
values are to be determined. Identify the variables and
denote them by x1, x2 and x3 or x, y and z etc.
• Step II: Identify the objective function and express it as a
linear function of decision variables introduced in step I.
Formulation of Linear Programming
Problem
• Step III: In a LPP, the objective function may be in the form of
maximizing (profits) or minimizing (costs). So identify the type
of optimization i.e., maximization or minimization.
• Step IV: Identify the set of constraints, stated in terms of
decision variables and express them as linear inequalities or
equalities according to the conditions.
• Step V: Add the non-negativity restrictions on the decision
variables, as in the physical problems, negative values of
decision variables have no valid interpretation.
Problem 1
• A company is manufacturing two products A and B. The
manufacturing times required to make them, the profit and
capacity available at each work centre are given below.
Work
Matching
Fabrication
Assembly
Centre
Product
A
B
1 hr
2 hrs
5 hrs
4 hrs
3 hrs
1 hr
Total capacity
720 hrs
1800 hrs
900 hrs
Formulate the L.P.P for maximizing the profit.
Profit per
unit
(in Tk.)
80
100
Problem 2
A firm makes two types of furniture: chairs and tables. The
contribution for each product is Tk. 20 per chair and Tk. 30
Per table. Both products are processed on three machines M1,
M2 and M3. The time required in hours by each product and
total time available in hours per week on each machine are as
follows.
Machines
Chair
Table
Available time
M1
3 hrs
3 hrs
36 hours
M2
5 hrs
2 hrs
50 hours
M3
2 hrs
6 hrs
60 hours
• Formulate the L. P. P.
• How should the manufacturer schedule his product in order
to maximize contribution?
Problem 3
Food X contains 6 units of vitamin A per Kg. and 7
units of vitamin B per Kg. and cost Tk. 12 per Kg.
Food Y contains 8 units of vitamin A per kg and 12
units of vitamin B and costs tk. 20 per Kg. The daily
minimum requirements of vitamin A and vitamin B are
100 units and 120 units respectively.
Formulate L. P. P. and find the minimum cost of
product mix.
Problem 4
A company produces two products A and B. There are two
machines M1 and M2 through which the products are
processed. The potential time capacity of machine M1 is
60 hours a week and that of machine M2 is 48 hours a
week. To make one unit of A, it requires 4 hours in M1
and 2 hours in M2. To make one unit of B, it requires 2
hours in M1 and 4 hours in M2. If the profit per unit of A
is Tk. 24 and the profit per unit of B is Tk. 18 can be
expected.
Formulate an LPP in order to maximize the profit.
Problem 5
A dealer wish to buy some numbers of Cycles and
Scooters. He has only Tk.50000 to invest and has a
space for at most 60 items. One Scooter cost him
Tk.2500 and a Cycle cost him Tk.500. His
expectation is that he can sell a Scooter Tk.3000 and
a Cycle Tk.650. Assuming that he can sell all the
items that he can buy.
Formulate L. P. P.
How many Scooters and Cycles can he buy and sell
in order to maximize his profit?
Problem 1
A company produces two products A and B. There are two
machines M1 and M2 through which the products are
processed. The maximum time capacity of machine M1 is
36 hours a week and that of machine M2 is 42 hours a
week. To make one unit of A, it requires 2 hours in M1
and 4 hours in M2. To make one unit of B, it requires 9
hours in M1 and 3 hours in M2. If the profit per unit of A
is Tk. 10 and the profit per unit of B is Tk. 8 can be
expected.
Find out the number of units of A and B to be produced in
order to maximize the profit.
Solution by Graphical method
• Step 1: Formulate the L. P. P.
• Step 2: Plot the constraints graphically.
• Step 3: Locate the feasible solution region. The
feasible solution region is the graphical area which
satisfies all the constraints at the same time.
• Step 4: Identify each of the corner points of the
feasible region from the graph or by solving the
simultaneous equations.
Solution by Graphical method
• Step 5: Calculate the objective function by
putting the co-ordinates (x, y) of each corner
point.
• Step 6: In a maximization problem, the optimal
solution occurs at that corner point which gives
the highest (profit) value of objective function.
• Step 7: In a minimization problem, the optimal
solution occurs at that corner point which gives
the lowest (cost) of the objective function.
Problem 2
Food X contains 6 units of vitamin A per Kg. and 7
units of vitamin B per Kg. and cost Tk. 12 per Kg.
Food Y contains 8 units of vitamin A per kg and 12
units of vitamin B and costs tk. 20 per Kg. The daily
minimum requirements of vitamin A and vitamin B are
100 units and 120 units respectively.
Find the minimum cost of product mix using
graphical method.
Problem 3
A company produces two products P and Q. There are two
machines M1 and M2 through which the products are
processed. The potential time capacity of machine M1 is
60 hours a week and that of machine M2 is 48 hours a
week. To make one unit of P, it requires 4 hours in M1 and
2 hours in M2. To make one unit of Q, it requires 2 hours
in M1 and 4 hours in M2. The profit per unit of P is Tk. 8
and the profit per unit of Q is Tk. 6.
Find out the number of units of P and Q to be produced to
maximize the profit. Also find the maximum profit.