linear programming (maximisation and minimisation)

8/13/2019 linear programming (maximisation and minimisation)

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Dr. Md Nusrate Aziz

Graduate School of Management

Multimedia University

October 20131


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Quantitative Analysis for Management, by Barry Render andRalph M. Stair, JR. - Pearson International Edition.

Fundamental Methods of Mathematical Economics by AlphaC. Chiang (3rdEdition), McGraw-Hill International Editions.

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Linear Programming (LP)is a widely used mathematical technique

designed to help managers in planning and decision making relativeto resource allocation.

LP Properties:

First: Problems seek to maximize or minimize an objective (profit,production or cost).

Second: Restriction or constraints limit the degree to which theobjective can be obtained. We want to maximize or minimize theobjective function subject to limited resources (the constraints).

Third: There must be alternatives (combinations of resources)available.

Fourth: The objective and constraints in linear programmingproblems must be expressed in terms of linear equations orinequalities.

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Problem: The Flair Furniture Company produces inexpensive

tables and chairs. The production process for each is similar in thatboth require a certain number of hours of carpentry work and acertain number of labour hours in the painting and varnishing. Eachtable takes 4 hours of carpentry and 2 hours in the painting andvarnishing. Each chair requires 3 hours in carpentry and 1 hours in

painting and varnishing. During the current production period, 240hours of carpentry time and 100 hours in painting and varnishingtime are available. Each table sold yields a profit of $7; each chairproduced is sold for a $5 profit.

Flair Furnitures problem is to determine the best possiblecombination of tables and chairs to manufacture in order to reachthe maximum profit.

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Formulating a linear program involves developing a mathematical

model to represent the managerial problem The steps in formulating a linear program are

1. Completely understand the managerial problem being faced

2. Identify the objective and constraints

3. Define the decision variables

4. Use the decision variables to write mathematical expressionsfor the objective function and the constraints

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We can covert the above information to formulate the problem whichis as follows:

Suppose, X1= number of tables to be produced

X2= number of chairs to be produced

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The objective is to

Maximize profit The constraints are

1. The hours of carpentry time used cannot exceed 240 hours perweek

2. The hours of painting and varnishing time used cannot exceed

100 hours per week The decision variables representing the actual decisions we will

make are

X1= number of tables to be produced per week

X2= number of chairs to be produced per week

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The easiest way to solve a small LP problem such as that of

Flair Furniture Company is with the graphical solutionapproach. The graphical procedure is useful only when thereare two decision variables (X1 and X2 in this case) in theproblem. When there are more than two variables it is notpossible to plot and solve the problem in two dimensionalgraph.

We look at the constraints and plotthem. Feasible Region: In an LP problem we need to find a set of

solutions that satisfies all of the constraints simultaneously. Itis known by the term area of feasible solution or simplyfeasible solution

. So, the feasible region is the overlappingarea of constraints that satisfies all of the restrictions on

resources.

Now we solve the problem in the white board -

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We suppose profit equal to some arbitrary but small dollar

amount. For the Flair Furniture problem we may choose aprofit of $210.

$210 = 7X1+5X2We then find (X1, X2) ={(0, 42); (30, 0)}

$280=7X1+5X2

We find (X1, X2) = {(0, 56); (40,0)}

Following the same way we draw a series of parallel isoprofitlines (iso-profit map) until we find the highest isoprofit line,that is, the one with the optimal solution.

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Alternatively, we can solve the LP problem by using cornerpoint method. It involves looking at every corner point of thefeasible region. Optimal solution will lie at one (or more) ofthem.

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In point (4), we need to find the value in the intersection point

of two constraints.4X1+ 3X2= 240 [Carpentry equation]

2X1+ 1X2= 100 [Painting equation]

Solving them gives the values of X1and X2, which is -

(X1, X2) = (30, 40)

So, the total profit is, = 7 (30) + 5 (40) = $ 410

Because point (4) produces the highest profit of any cornerpoint, the product mix of X

1=30 tables and X

2=40 chairs is

the optimal solution to Flair Furnituresproblem.

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ISOPROFIT METHOD

1. Graph all constraints and find the feasible region.2. Select a specific profit (or cost) line and graph it to find the slope.

3. Move the objective function line in the direction of increasingprofit (or decreasing cost) while maintaining the slope. The lastpoint it touches in the feasible region is the optimal solution.

4. Find the values of the decision variables at this last point andcompute the profit (or cost).

CORNER POINT METHOD

1. Graph all constraints and find the feasible region.

2. Find the corner points of the feasible reason.

3. Compute the profit (or cost) at each of the feasible corner points.

4. select the corner point with the best value of the objectivefunction found in Step 3. This is the optimal solution.


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See Excel Sheet [Computer Exercise].

We solve the Flair Furniture Problem using xls tool. Thefindings are as follows:

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Suppose, 240 acres of land available.

Profit: $40/acres corn; $30/acre oats. Have 320 hours of labour available.

Corn takes 2 hours of labour per acre of land; however,

Oats requires 1 hour of labour per acre of land.

Problem: How many acres of each should be planted tomaximize profit?(1) Formulate the problem.

(2) Solve the problem using graphical method (cornersolution).

[Watch the following video:

http://www.youtube.com/watch?v=M4K6HYLHREQ]

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Many LP problems involve minim izing as object ive such as cost.

For example:(a) A manufacturer may seek to distribute its products from several

factories to its many regional warehouses in such a way as to

minimize total shipping cost.

(b) A hospital may want to provide a daily meal plan for its patients

that meets certain nutritional standards while minimizing foodpurchase costs.

Minimization problems can be solved graphically by first setting upthe feasible so lut ionregion and then using either the corner pointmethodor an iso-cost l ine app roach (which is analogous to the

iso-profit approach in maximization problems) to find the values ofthe decision variables (e.g.,X1andX2) that yield the minimum cost.

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The Holiday Meal Turkey Ranch is considering buying two different

brands of turkey feed and blending them to provide a good, low-costdiet for its turkeys. Each feed contains, in varying proportions, someor all of the three nutritional ingredients essential for fatteningturkeys. Each pound of brand 1purchased, for example, contains 5ounces of ingredient A, 4 ounces of ingredient B, and ounces ofingredient C. Each pound of brand 2 contains 10 ounces of

ingredient A, 3 ounces of ingredient B, but no ingredient C. Thebrand 1 feed costs the ranch 2 cents a pound, while the brand 2feed costs 3 cents a pound. The minimum monthly requirement perturkey are 90 ounce of A, 48 ounce of B and 1 of C. The owner ofthe ranch would like to use LP to determine the lowest-cost diet that

meets the minimum monthly intake requirement for each nutritionalingredient.

We summarize the problem as follows -

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Now, formulate the problem and solve it using graphicalmethod.

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INGREDIENT

COMPOSITION OF EACH POUND

OF FEED (OZ.) MINIMUM MONTHLYREQUIREMENT PER

TURKEY (OZ.)BRAND 1 FEED BRAND 2 FEED

A 5 10 90

B 4 3 48

C 0.5 0 1.5

Cost per pound 2 cents 3 cents


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The Holiday Meal Turkey Ranch is considering buying two different

brands of turkey feed and blending them to provide a good, low-costdiet for its turkeys.

Let, X1= number of pounds of brand 1 feed purchased.

X2= number of pounds of brand 2 feed purchased.

Minimize cost (in cents) = 2X1+ 3X2subject to:

5X1+ 10X2 90 ounces (ingredient constraint A)

4X1+ 3X2 48 ounces (ingredient constraint B)

0.5X

1

1.5 ounces (ingredient constraint C)X1 0 (non-negativity constraint)

X2 0 (non-negativity constraint)

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Using the cornerpoint method.

First we constructthe feasible

solution region.

The optimalsolution will lie aton of the corners asit would in a

maximizationproblem.

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We solve for the values of the three corner points (a, b and c).

Point ais the intersection of ingredient constraints C and B4X1+ 3X2= 48

X1= 3

Substituting 3 in the first equation, we findX2= 12

Solving for point bwith basic algebra we findX1= 8.4 andX2= 4.8

Solving for point cwe find X1= 18 andX2= 0

Substituting these value back into the objective function wefind:

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The lowest cost solution is to purchase 8.4 poundsof brand 1 feed and 4.8 pounds of brand 2 feedfor a total cost of 31.2 cents per turkey.


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Using the isocostapproach.

Choosing an initialcost of 54 cents, it isclear improvement ispossible.

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Four special cases and difficulties arise at times when

using the graphical approach to solving LP problems: Infeasibility

Unboundedness

Redundancy

Alternate Optimal Solutions.

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No feasible solution Exists when there is no solution to the

problem that satisfies all the constraints.

No feasible solution region exists.

This is a common occurrence in the real

world. Generally one or more constraints are

relaxed until a solution is found.

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The Problem with no feasible solution

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Unboundedness

Sometimes a linear program will not have a finitesolution.

In a maximization problem, one or more solutionvariables, and the profit, can be made infinitely

large without violating any constraints. In a graphical solution, the feasible region will be

open ended.

This usually means the problem has been

formulated improperly.

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A solution region unbounded to the right

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Redundancy A redundant constraint is one that does not affect

the feasible solution region.

One or more constraints may be more binding.

This is a very common occurrence in the realworld.

It causes no particular problems, but eliminatingredundant constraints simplifies the model.

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A problem with Redundant constraint

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Alternate Optimal Solutions Occasionally two or more optimal solutions may

exist.

Graphically this occurs when the objectivefunctions iso-profit or iso-cost line runs perfectlyparallel to one of the constraints.

This actually allows management great flexibility indeciding which combination to select as the profitis the same at each alternate solution.

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A problem with un-unique solution

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LP solutions by using

Simplex Method

linear programming (maximisation and minimisation)

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