Some Examples on Linear programming

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Some Examples on Linear programming
Operational Research By TMJA Cooray
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Product Mix problem-A Bakery

• A bakery specializes in two pastriesdonuts and
  cakes.profit per dozen donuts is 0.60,and profit
  per cake is 3.25. It takes 1/6 labor hour to
  make a dozen donuts and 2 labor hours to prepare
  a cake.

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• The management employs three people,each working
  40 hours a week.sales of donuts are not expected
  to exceed 500 dozen a week.Construct the LP model
  and determine the optimal product mix for the
  bakery.

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Product Mix problemA kitchenware manufacturer

• Potsnpans manufactures two types of kitchenware
  pans,denoted type A pans and type B pans.
  Each pan is stamped from steel and then sent
  through an oven where a coating of enamel is
  applied.The stamping machine is available 50
  hours a week for production and the oven operates
  45 hours per week.

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• Each type A pan requires 5 minutes on the
  stamping machine and 10 minutes in the oven.type
  B pan requires 7 minutes on the stamping
  machine and 13 minutes in the oven.Profit
  contribution is 2.50 for type A and 4.55 for
  the type B pan. Construct the LP model and
  determine the optimal product mix for Potsnpans.
  

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Diet planning problem-Feed Mix

• Jill owns a small dog ,Snippet,who currently
  enjoys two types of dog food Ruff chow and
  Special treat. Ruffchow is a basic dog food for
  main meals and costs 5 cents per ounce. Ruffchow
  consists of 35 protein 10 crude fiber,and 12
  fat( the rest consists of minor ingredients).

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• Special treat is a cheese flavored cracker that
  costs 3 ½ cents per ounce and consists of 25
  protein,12 crude fiber, and 8 fat.Jill wants to
  be sure that her dog gets at least 7 ounces of
  protein,4 ounces of crude fiber, and 3 ounces of
  fat per week from these two sources.

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• How much of each type of dog food should Jill
  feed snippet per week in order to meet the
  nutritional requirements at the minimum cost?

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• Capital Budgeting problem
• (Which projects should be undertaken?)
• A certain construction company having only Rs(150
  x106) wishes to undertake five different
  projects P1,P2,P3,P4 and P5. .The estimated net
  profit and the total cost for undertaking each
  project are given below.
• ( Because of the different demand and
  requirements for each productman
  power,equipments,transportation to project site
  etc the profit and the cost varies from project
  to project)

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• Some more constraints
• 1.Exactly one project should be selected from
  the set of projects P1,P2,P5
• 1At least one project should be selected from
  the set of projects P1,P2,P5
• 1At most one project should be selected from
  the set of projects P1,P2,P5

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• 2.The company has decided that at most one of the
  two projects P2 and P4 can be selected.
• 3.If P1 and P2 are both selected then P5 should
  be selected
• 4.If P3 is selected then P4 must be selected.

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Assumptions in Linear programming

• Linearity the amount of resources required for
  a give n activity level is directly proportional
  to the activity level
• Divisibility fractional values of the decision
  variables are permitted
• Non negativity Decision variables are permitted
  to have only the values which are equal or
  greater than zero.
• Additivity The total output for a given
  combination of activity levels is the sum of the
  output of each individual process.

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Properties of LP solutions

• Feasible solution If all the constraints of the
  given LP model are satisfied by the solution of
  the model , then the solution is known as a
  feasible solution.
• Optimal solution If there is no other superior
  solution to th solution obtained for a given LP
  model then that solution is treated as the
  optimal solution.

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• Alternate optimal solution For some LP model
  there may be more than one combination of values
  of the decision variables yielding the best
  objective function value.Thy are called alternate
  optimal solutions.
• Unbounded solutions For some LP models ,the
  objective value can be increased/decreased
  without any limitation.such solutions are known
  as unbounded solutions.

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• Infeasible solutions If there is no combination
  of values of the decision variables satisfying
  all the constraints of the LP model then that
  model is said to have infeasible solutions.

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• Degenerate solution In LP problems ,
  intersections of two constraints will define a
  corner point of the feasible region.But if more
  than two constraints pass through any one of the
  corner points of the feasible region excess
  constraint will not serve any purpose. They are
  called redundant constraints and , in such a case
  degeneracy will occur. In the simplex method
  ,some iterations will be carried out without any
  improvement in the objective function if there
  are redundant constraints.

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Geometrical (Graphical) solution

• Step 1 Determine the solution space that
  satisfies all the constraints of the model.
  (The solution space will include all the feasible
  solutions of the model.)
• To determine the solution space , represent all
  the non negativity constraints and all the other
  inequalities on the graph.
• Each inequality divides the (x1,x2 )plane into
  two halves. Only one half will satisfy the
  inequality.Combine all the half planes satisfying
  these inequalities.

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• Step 2 Determine the optimum solution from among
  all the feasible points in the solution
  space.
• There are two methods.
• Method 1 The evaluation method. Evaluate each
  corner point and find the corner point which
  maximizes the objective function.

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• Method 2 The other is plotting the iso-profit or
  iso- cost lines and find the last point of
  contact of the feasible region and the line.
• To draw the iso-profit/cost lines, arbitrarily
  allocate a value for Z and draw the corresponding
  line. Draw several lines by either increasing or
  decreasing this allocated value .

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• After drawing the iso-profit lines move it
  towards the north east direction and find the
  last point of contact of the feasible region and
  the line.
• Exercise-----
• And demonstration------

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