These definitions are only useful within the graphical solution of LP models. Thus, a more general definition is required.

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1 BUAD 403 OPERATIONAL RESEARCH I LECTURE IV HAND OUT Binding Constraints & Non-Binding Constraints: Binding Constraints: are the constraints that intersect on the optimum solution point. Non-Binding Constraints: are the constraints that do not intersect on the optimum solution point. These definitions are only useful within the graphical solution of LP models. Thus, a more general definition is required. A constraint in an LP is binding if the LHS and the RHS are equal when the values of the variables in the optimal solution are plugged into the constraint. Vice versa, a constraint in an LP is non-binding if the LHS and the RHS are not equal when the values of the variables in the optimal solution are plugged into the constraint. Sensitivity Analysis: Sensitivity analysis is, the analysis of the effect of parameter changes to the optimal solution. Changes in Objective Function Coefficients Change in RHS Values of the Constraints Other Changes in the Model (Changes in constraint coefficients, additional variables and or constraints) Analyzing Effect of Changes in the Objective Functions Coefficients to the Optimal Solution: Beaver Creek Pottery Company manufactures two products: bowls & mugs. Sale of a bowl yields $40 of profit and sale of a mug yields $50 of profit. In production of bowls & mugs company uses two resources: labor (hours) & clay (libres*). Manufacture of a bowl requires 1 hour of labor & 4 libres of clay. And, manufacture of a mug requires 2 hours of labor and 3 libres of clay. Currently te company as 40 working hours available and 120 lb of clay in storage. Formulate an LP model for the company to aid in profit maximization and suggest a production plan to the company solving your LP model. obj. f/n: ***Graphical method solution: (in class C(24, 8),

2 ***Illustration of changes in the objective function coefficients using graphical method: (in class The sensitivity range for an objective coefficient is, the range of values over whicih the current optimal solution point will remain optimal. Sensitivity range for the objective coefficients are: Reduced Cost of a decision variable is, the required change in the objective function coefficient of the decision variable that would ensure the decision variable s inclusion in the optimal plan (basis). (...that would make the non-basic variable a basic variable.) The LP model and the final table that represents the optimal solution for the Dakota Furniture Company s revenue maximization model was: Z RHS , ,5 1, Z In a final simplex table that represents the optimal basic feasible solution, values in the decision variables segment of the Z row are the reduced costs for the decision variables. Reduced cost of x 1 and x 3 are both zero, which indicates that to ensure their inclusion in the optimal plan no changes in their objective function coefficients are required. (...which indicates that these decision variables are basic variables.) Reduced cost of x 2 is 5, which indicates that tables are underpriced. (...which indicates that x2 is a non-basic variable.) If the company had sold the tables for ( $35 instead of $30, x 2 would have been a basic variable.

3 Analyzing Effect of Changes in the Quantity Values(RHS) of the Constraints to the Optimal Solution: A change in the quantity value of a constraint causes the feasible solution space to change. For the Beaver Creek Pottery Company example: ***Illustration of changes in the quantity values of the constraints using graphical method: (in class The sensitivity range for a RHS value of a constraint is, the range of values over which the quantity values can change without changing the solution variable mix (incuding the additional varaibales). A solution variable mix means the set of basic variables and the set of non-basic variables. Sensitivity range for the constraint quantities are: Shadow Price of a constraint is, the marginal value of one additional unit of resource (quantity of a constraint) to the objective function value. The LP model and the final table that represents the optimal solution for the Dakota Furniture Company s revenue maximization model was: Z RHS ratio , ,5 1, Z In a final simplex table that represents the optimal basic feasible solution, values in the additional variables segment of the Z row are the shadow prices for the constraints.

4 Shadow prices of s 1 and s 4 are both zero, which indicates that an additional unit to the constraint quantity would NOT CHANGE the objective function value. (...which indicates that, these to constraints are non-binding constraints.) (...which indicates that the slack variables associated with these constraints have positive values, thus there is idle capacity for these constraints.) Shadow price of s 2 and s 3 are both 10, which indicates that an additional unit of finishing labor hours or carpentry labor hours would both change the revenue of the company by $10. (...which indicates that constraints associated with these slack variables are binding constraints.) Analyzing Effect of Other Changes in the LP model to the Optimal Solution: Other changes are: An additional variable to the model An additional constraint to the model or Changes in the LHS constraint parameters First and the second type of change requires re-formulation of the LP model. Thus, will be skipped. Yet again the third type of change requires re-formulation of the LP model but can be illustrated (analyzed) using the graphical method. For the Beaver Creek Pottery Company example: Beaver Creek Pottery Company lost one of his master artisans and hired a new less-exprienced artisian. Now manufacture of a bowl requires 1,33 hours of labor instead of 1. ***Illustration of changes in the LHS parameters of the constraint using graphical method: (in class Assignment IV Formulate an LP model, solve the model using the Simplex algorithm and report your answers including sensitivity analysis. (Values of the variables included in the model, objective function value, binding & nonbinding constraints, idle capacities, sensitivity range for objective function coefficients, reduced costs, shadow prices, interpretation of reduced costs & shadow prices.) A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are two brands of fertilizer to choose from: Toros Tarım and TKİ Hümas. Each brand yields a specific amount of nitrogen and phosphate per bag, as follows:

5 Brand Chemical Contribution Nitrogen (kg/bag) Phosphate (kg/bag) Toros Tarım 2 4 TKİ Hümas 4 3 *Chemical Contribution data given above is %100 fictional Farmer s field requires at least 16 kilograms of nitrogen and at least 24 kilograms of phosphate. Toros Tarım costs TL6 per bag, and TKİ Hümas costs TL3 per bag. Farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing. Hint1: Initial BFS must be a corner-point-feasible solution, point(0,0) is NOT FEASIBLE. Your initial simplex tables must represent a basic-feasible solution) Hint2: Assume that all the constraints were less than or equal to constraints and the objective function was a maximization problem. Use the simplex metod to obtain a BFS for the original problem. For visualization, this website might be helpful: Write down your answers in a Word Document, save as STUDENTIDA4LASTNAME (ie. : A4YILDIZ) and send your assignments via . Due Date: November, 5th Announcements! Now you have 5 hours for late submissions. Yet late submissions will be penalized by 10 points. Check out the announcements on website.

Linear Programming: Model Formulation and Graphical Solution

Linear Programming: Model Formulation and Graphical Solution Chapter 2 2-1 Chapter Topics Model Formulation A Maximization Model Example Graphical Solutions of Linear Programming Models A Minimization