*,*,Click to edit Master title style,Chapter 3-Linear Programming:Computer Solution and Sensitivity Analysis,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,200,7,/0,8,Sami Fethi,EMU,All Right Reserved,.,Operations Research,Ch,3,:,Computer Solution and Sensitivity analysis,Department of Business Administration,FALL 200,7,-0,8,Management Science,by,Asst.Prof.Sami Fethi,2007 Pearson Education,1,Chapter Topics,Standard form,Sensitivity Analysis,Dual Problem,Example Problems,2,Linear Programming Problem:Standard Form,Standard form requires all variables in the constraint equations to appear on the left of the inequality(or equality)and all numeric values to be on the right-hand side.,Examples:,(Equation),x,3,x,1,+x,2,must be converted to x,3,-x,1,-x,2,0,x,1,/(x,2,+x,3,),2 becomes x,1,2(x,2,+x,3,),and then x,1,-2x,2,-2x,3,0,3,Linear Programming Problem:Standard Form,Models are also transformed into,Standard form.,Examples:,(model),Having defined the profit or cost function as well as constraints functions within the system,eqn,standard form can be formulated as follows,:,Z=12,x,1,+16,x,2,Subject to:3x,1,+2,x,2,500,4x,1,+5,x,2,800,x,1,x,2,0,Standard form,:,4,Beaver Creek Pottery Example,Sensitivity Analysis,(1 of 4),Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations.,Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model.,5,Maximize Z=$40 x,1,+$50 x,2,subject to:1x,1,+2x,2,40,4x,2,+3x,2,120,x,1,x,2,0,Figure 3.1,Optimal Solution Point,Beaver Creek Pottery Example,Sensitivity Analysis,(2 of 4),6,Maximize Z=$100 x,1,+$50 x,2,subject to:1x,1,+2x,2,40,4x,2,+3x,2,120,x,1,x,2,0,Figure 3.2,Changing the x,1,Objective Function Coefficient,Beaver Creek Pottery Example,Change x,1,Objective Function Coefficient(3 of 4),7,Maximize Z=$40 x,1,+$100 x,2,subject to:1x,1,+2x,2,40,4x,2,+3x,2,120,x,1,x,2,0,Figure 3.3,Changing the x,2,Objective Function Coefficient,Beaver Creek Pottery Example,Change x,2,Objective Function Coefficient(4 of 4),8,The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal.,The sensitivity range for the x,i,coefficient is designated as c,i.,Objective Function Coefficient,Sensitivity Range(1 of 3),9,objective function Z=$40 x,1,+$50 x,2,sensitivity range for:,x,1,:25,c,1,66.67 x,2,:30,c,2,80,Figure 3.4,Determining the Sensitivity Range for c,1,Objective Function Coefficient,Sensitivity Range for c,1,and c,2,(2 of 3),10,Minimize Z=$6x,1,+$3x,2,subject to:,2x,1,+4x,2,16,4x,1,+3x,2,24,x,1,x,2,0,sensitivity ranges:,4,c,1,0,c,2,4.5,Objective Function Coefficient,Fertilizer Cost Minimization Example(3 of 3),Figure 3.5,Fertilizer Cost Minimization Example,11,Changes in Constraint Quantity Values,Sensitivity Range(1 of 4),The sensitivity range for a right-hand-side value is the range of values over which the quantitys value can change without changing the solution variable mix,including the slack variables.,12,Changes in Constraint Quantity Values,Increasing the Labor Constraint(2 of 4),Maximize Z=$40 x,1,+$50 x,2,subject to:1x,1,+2x,2,40,4x,2,+3x,2,120,x,1,x,2,0,Figure 3.6,Increasing the Labor Constraint Quantity,13,Changes in Constraint Quantity Values,Sensitivity Range for Labor Constraint(3 of 4),Sensitivity range for:,30,q,1,80 hr,Figure 3.7,Determining the Sensitivity Range for Labor Quantity,14,Changes in Constraint Quantity Values,Sensitivity Range for Clay Constraint(4 of 4),Sensitivity range for:,60,q,2,160 lb,Figure 3.8,Determining the Sensitivity Range for Clay Quantity,15,Changing individual constraint parameters,Adding new constraints,Adding new variables,Other Forms of Sensitivity Analysis,Topics(1 of 4),16,Other Forms of Sensitivity Analysis,Changing a Constraint Parameter(2 of 4),Maximize Z=$40 x,1,+$50 x,2,subject to:1x,1,+2x,2,40,4x,2,+3x,2,120,x,1,x,2,0,Figure 3.9,Changing the x,1,Coefficient in the Labor Constraint,17,Adding a new constraint to Beaver Creek Model:0.20 x,1,+0.10 x,2,5 hours for packaging Original solution:24 bowls,8 mugs,$1,360 profit,Exhibit 3.17,Other Forms of Sensitivity Analysis,Adding a New Constraint(3 of 4),To find out Optimal solution coordinates,we,u,se,t,he both constraints.,x,1,=,40,-,2x,2,4,(,40,-,2x,2,),+3x,2,=,120,160-8,x,2,+3x,2,=,120,-5,x,2,=-40,X,2,=,8,x,1,=,2,4,Z=$40,(24),+$50,(8),Z,=,$,1360 max daily profit possible,18,Adding a new variable to the Beaver Creek model,x,3,a third product,cups,Maximize Z=$40 x,1,+50 x,2,+30 x,3,subject to:,x,1,+2x,2,+1.2x,3,40 hr of labor,4x,1,+3x,2,+2x,3,120 lb of clay,x,1,x,2,x,3,0,Solving model shows that change has no effect on the original solution(i.e.,the model is not sensitive to this change).,Other Forms of Sensitivity Analysis,Adding a New Variable(4 of 4),19,Defined as the marginal value of one additional unit of resource.,The se