Supplementary Slides for LP

8/8/2019 Supplementary Slides for LP

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Linear Programming(LP)


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Linear programming (LP)

Linear programming (LP) is a widely used

mathematical technique designed to help

operations managers plan and make the

decisions necessary to allocate resources.

LP problems seek to minimise or

maximise some quantity (e.g. cost or

profit)


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Examples ofLP in Practice

Minimising total distance travelled by a

school bus when carrying students.

Minimising the cost of feeding chicken ina farm, while making sure the nutritional

values are right.

Planning the size and layout of hotelrooms to maximise revenues from

customers staying.


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LP Terminology

What we try to minimise or maximise in an

LP problem is called the objective function.

As in real life, there are restrictions orconstraints limiting the degree we can achieve

out objective (e.g. we have limited resources

to maximise profit!)

That means, we try to minimise or maximisethe objective function subject to the

constraints.


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Formulating LP Problems

First define the decision variables.

Next express the objective function

and constraints as equalities orinequalities involving the decision

variables.


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Example

A company produces pans and saucers. Theymake a profit of $25 for selling a pan and $10for selling a saucer. Steel and plastic are

needed to make pans and saucers. Making apan requires 4 kgs of steel and 0.5 kgs of plastic; whereas making a saucer requires 2kgs of steel and 0.2 kgs of plastic. They have4000 kgs of steel and 900 kgs of plastic. Howmany pans and saucers should they produceto maximise profit?


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Formulating LP Problems


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Formulation of the Example

Our constraints for the saucer/penproblem are:

4X

1 + 2X

2 40000.5X1 + 0.2X2 900

X1 0

X2

0

And the objective function is:

Max 25X1 + 10X2


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Graphical Representation ofConstraints

We start by drawing the lines:

4X1 + 2X2 = 4000

5X1 + 2X2 = 9000

In the graph, the region satisfying the conditions iscalled the feasible region. It is showed as a shadedregion.

Any point in the region will satisfy the conditionsbut only a few (or even sometimes one) point(s) willoptimise the objective function.

Once we have the graph, we can continue withsolving the problem.


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Iso-profit line solution method

(A graphical way of solving an LP problem)We can start by assigning a small value to our

objective function (for example, a profit or cost

value)

25X1 + 10X2 = 500

We can draw this line on the graph and assign

a bigger value so that our next line will still be

passing through the feasible region.25X1 + 10X2 = 20000


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Iso-profit line solution method contd.

We continue drawing parallel lines.

The further we move from the 0 origin, the

bigger our profit will be.

By carefully moving our ruler and drawing

parallel lines further away from the origin but

still passing through the feasible region, we

approach the optimum solution. The highest profit line that still touches some

points on the feasible region will be our

optimum solution.


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Corner-point solution method(note that this method fails in special cases of LP problems)

Optimum solution should be one or more of the

corner (or extreme) points of the feasible region OR one

of the edges of the feasible region.

In a bounded and feasible problem, we can find theoptimum solution by testing each corner.

Decision variable values leading to the optimum

solution may or may not be integers.

As the number of constraint increase, the cornerpoints to examine will increase.

It is very important to know that this method

fails to solve the LP problem in certain special

cases!


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LP Example 1

Each sports bike requires 6 hours work in the

component manufacture workshop and 3 hours in

the assembly workshop. Each shopper requires 4

hours in the component manufacture workshopand 10 hours in the assembly workshop. The

workshops are available for 120 and 180 hours per

week, respectively. The profit contribution per bike

is 30 for each sports bike and 25 per shopper.Assuming that the manufacturer wishes to

maximize profits, how many bikes of each type

should be produced per week?


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LP Example 2

You need to buy some filing cabinets. You know

that Cabinet X costs $10 per unit, requires six

square feet of floor space, and holds eight cubic

feet of files. Cabinet Y costs $20 per unit, requireseight square feet of floor space, and holds twelve

cubic feet of files. You have been given $140 for this

purchase, though you don't have to spend that

much. The office has room for no more than 72square feet of cabinets. How many of which model

should you buy, in order to maximize storage

volume?


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LP Example 3

max 12X1 + 9X2

s.t. X1 1000

X2 1500

X1 + X2 1750

4X1 + 2X2 4800

X1 0

X2 0


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LP Example 4

A carpenter makes tables and chairs. Each table can be

sold for a profit of 30 and each chair for a profit of 10.

The carpenter can afford to spend up to 40 hours per

week working and takes six hours to make a table andthree hours to make a chair. Customer demand requires

that he makes at least three times as many chairs as

tables. Tables take up four times as much storage space

as chairs and there is room for at most four tables each

week.

Formulate this problem as a linear programming

problem and solve it graphically.


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LP Example 5

A company makes two products (X and Y) using two machines (A

and B). Each unit of X that is produced requires 50 minutes

processing time on machine A and 30 minutes processing time on

machine B. Each unit of Y that is produced requires 24 minutes

processing time on machine A and 33 minutes processing time on

machine B.At the start of the current week there are 30 units of X and 90 units of

Y in stock. Available processing time on machine A is forecast to be

40 hours and on machine B is forecast to be 35 hours.

The demand for X in the current week is forecast to be 75 units and

for Y is forecast to be 95 units. Company policy is to maximise thecombined sum of the units of X and the units of Y in stock at the end

of the week.

Formulate the problem of deciding how much of each product to

make in the current week as a linear program. Solve this linear

program graphically.

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Supplementary Slides for LP