Chapter 2Linear Programming: Model Formulation and Graphical SolutionBIT 2406*

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Chapter TopicsModel FormulationA Maximization Model ExampleGraphical Solutions of Linear Programming ModelsA Minimization Model ExampleIrregular Types of Linear Programming ModelsCharacteristics of Linear Programming ProblemsBIT 2406*

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Linear ProgrammingObjectives of business decisions frequently involve maximizing profit or minimizing costs.

Linear Programming is a model that consists of linear relationships representing a firms decision(s), given an objective and resource constraints.

Three Steps in Applying the Linear Programming Technique:Identify the problem as being solvable by linear programmingFormulate the unstructured problem as a mathematical modelSolve the model using established mathematical techniquesBIT 2406*

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Model ComponentsDecision Variables are mathematical symbols that represent levels of activity.Objective Function a linear relationship that reflects the objective of an operation, in terms of the decision variables this function is to be maximized or minimized.Constraint a linear relationship that represents a restriction on decision making.Parameters numerical values that are included in the objective functions and constraints.BIT 2406*

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Model Formulation Steps

Step 1 : Clearly define the decision variables

Step 2 : Construct the objective function

Step 3 : Formulate the constraintsBIT 2406*

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A Maximization Model Example: Product Mix ProblemProblem Statement: Beaver Creek Pottery Company is a small crafts operation run by a Native American tribal council. The company employs skilled artisans to produce clay bowls and mugs with authentic Native American design and colors. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit. Each bowl requires 1 hour of labor and 4 lbs of clay, and sells for $40. Each mug requires 2 hours of labor and 3 lbs of clay, and sells for $50. There are 40 hours of labor and 120 pounds of clay available each day for production.BIT 2406*

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A Maximization Model Example: Product Mix ProblemBIT 2406*

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A Maximization Model Example: Product Mix ProblemResource Requirements for Productionand Profit per Item ProducedBIT 2406*Decision Variables: How many bowls and mugs to produce each day Objective: Maximize profit.Resource: 40 hrs of labor per dayAvailability: 120 lbs of clay

ProductBowlMugProfit ($/Unit)Labor (Hr./Unit)Clay (Lb./Unit)

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A Maximization Model Example: Product Mix ProblemDecision Variables: Objective Function:Resource Constraints:

Non-Negativity Constraints*:

BIT 2406**Non-negativity constraints: restrict the decision variables to zero or positive values.

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A Maximization Model Example: Product Mix ProblemA feasible solution does not violate any of the constraints: Example x1 = 5 bowls x2 = 10 mugs Z = ? Labor constraint check: ? Clay constraint check: ?BIT 2406*

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BIT 2406*A Maximization Model Example: Product Mix ProblemAn infeasible violates at least one of the constraints: Example x1 = 10 bowls x2 = 20 mugs Z = ? Labor constraint check: ? Clay constraint check: ?

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Graphical Solutions of LP ModelsGraphical solutions are limited to linear programming problems with only two decision variables.

Can be used with 3 decision variables but cumbersome.

Graphical methods provide visualization of how a solution for a linear programming problem is obtained.

Graphs provide a clearer understanding of how the computer and mathematical solution approaches work.BIT 2406*

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BIT 2406*Formula of a Line

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BIT 2406*Labor Constraint

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BIT 2406*Clay Constraint

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BIT 2406*Feasible Region

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BIT 2406*Objective Line

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BIT 2406*Optimal Solution

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BIT 2406*Optimal Solution Coordinates

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BIT 2406*Solutions at the Extreme Points

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BIT 2406*Changing the ObjectiveZ = 70 x1 + 20 x2

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Graphical Solutions of LP ModelsIf the objective function coefficients are changed, the solution may change.

If the constraint coefficients are changed, the solution space and solution points may change.

Sensitivity analysis is used to analyze changes in model parameters.BIT 2406*

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Slack VariablesA Slack Variable is added to a constraint to convert it to an equation (=).A Slack Variable represents unused resources.A Slack Variable contributes nothing to the objective function.

BIT 2406*Linear Programming Model in Standard Form (with slack variables)

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Slack Variables in Graphical SolutionBIT 2406*

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Graphical Solution StepsStep 1 : Plot the model constraints as equations on the graph, then considering the inequalities of the constraints, indicate the feasible solution area.Step 2 : Plot the objective function; then move this line out from the origin to locate the optimal solution point. Step 3 : Solve simultaneous equation at the solution point to find the optimal solution values OR solve simultaneous equations at each corner point to find the solution values at each point.Step 4 : Substitute these values into the objective function to find the set of values that results in the maximum Z value.BIT 2406*

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A Minimization Model ExampleProblem Statement: A farmer is preparing to plant a crop in the spring and needs to fertilize a field at a minimum possible cost. There are two brands of fertilizer to choose from, Super-gro and Crop-quick. Super-gro yields 2 lbs of nitrogen and 4 lbs of phosphate per bag, which costs $6. Crop-quick yields 4 lbs of nitrogen and 3 lbs of phosphate and costs $3. The farmers field requires at least 16 lbs of nitrogen and at least 24 lbs of phosphate.BIT 2406*

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A Minimization Model ExampleBIT 2406*

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A Minimization Model ExampleChemical Contribution and CostsBIT 2406*Decision Variables: How many bags of each brand to purchase Objective: Minimize the total cost of fertilizing.Field: 16 lbs of nitrogenRequirements: 24 lbs of phosphate

BrandSuper-groCrop-quickCost ($/Bag)Nitrogen (Lb./Bag)Phosphate (Lb./Bag)

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A Minimization Model ExampleDecision Variables: Objective Function: Model Constraints:

Non-Negativity Constraints*:

BIT 2406**Non-negativity constraints: restrict the decision variables to zero or positive values.

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Graphical Solution Feasible RegionBIT 2406*

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Graphical Solution Objective FunctionBIT 2406*

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Surplus VariablesA Surplus Variable is subtracted from a constraint to convert it to an equation (=).A Surplus Variable represents an excess above a constraint requirement level.A Surplus Variable contributes nothing to the objective function.

BIT 2406*Linear Programming Model in Standard Form

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Graphical Solution Surplus VariablesBIT 2406*

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Irregular Types of LP ProblemsFor some linear programming models, the general rules do not always apply.

Special Types of Problems Include Those With:Multiple Optimal SolutionsInfeasible SolutionsUnbounded SolutionsBIT 2406*

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Multiple Optimal SolutionsBIT 2406*

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Infeasible ProblemBIT 2406*

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Unbounded ProblemBIT 2406*

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Characteristics of LP ProblemsA linear programming problem requires a decision - a choice amongst alternative courses of action.The decision is represented in the model by decision variables.The problem encompasses a goal, expressed as an objective function, that the decision maker wants to achieve.Restrictions (Constraints) exist that limit the extent of achievement of the objective.The objective and constraints must be definable by linear mathematical functional relationships.BIT 2406*

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Properties of Linear Programming ModelsProportionality - The rate of change (slope) of the objective function and constraint equations is constant.Additivity - Terms in the objective function and constraint equations must be additive.Divisibility -Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature.Certainty - Values of all the model parameters are assumed to be known with certainty (non-probabilistic).BIT 2406*

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