Controlling Air Pollution. A quick review. Reclaiming Solid Wastes. Chapter 4 The Simplex Method. Solving the Bake Sale problem. How to move?

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1 ESE Operations Research 9// Controlling Air Pollution Technology can be use fully () or fractional thereof A quick review ESE Operations Research Reclaiming Solid Wastes Chapter The Simple Method ESE Operations Research Solving the Bake Sale problem How to move?. Consider the edges from current solution Choose one that increase fastest. Move along the edge Until intersecting with a constraint. Solve the constraint. Go to Optimality test ESE Operations Research ESE Operations Research

2 ESE Operations Research 9//. If there is at least one optimal solution There must be one CPF that is optimal Simple only consider CPF. Simple is Iterative ESE Operations Research 7 ESE Operations Research 8. Simple starts at origin, if possible Number of constraints need for CFP? number of decision variable More constraints can t hurt May be redundant. It is easier to move to adacent CPF They different by one constraint Go along an edge ESE Operations Research 9 ESE Operations Research. Simple move in direction of largest gain The rate of improvement along a constraint Is Constant!!! Because the obective/constraint are linear!. Optimal Rates of improvement are all negative Algebraic Simple Method Need to solve a system of equation inequality structural constraints -> equality Add slack variables Augmented Form Maimize Subect to z 8,,,,, ESE Operations Research ESE Operations Research

3 ESE Operations Research 9// Slack variables Slack variable >, within feasible region. Slack variable, on the constraint boundary Slack variable <, with infeasible region Maimize Subect to z 8,,,,, ESE Operations Research Terminology Basic solution CP solution for the augmented problem Basic Feasible (BF) solution CPF solution for the augmented problem Augmented form has nm variables m: # of constraints N: # of decision variables Representing degree of freedom Maimize Subect to Therefore, n variables are nonbasic and set to The rest are basic basis of the solution space for the simultaneous equations z 8,,,,, ESE Operations Research Eample CPF solution (,) Set and as nonbasicvariables Basic Solution (,,,,) Set and as nonbasicvariables Basic Solution (,,,,8) Set and X as nonbasicvariables Basic Solution (,,,,) Set and as nonbasic variables Basic Solution (,9,,-,) Maimize Subect to z 8,,,,, Adacency BF solution are adacent if One basicvariable is swapped with a nonbasicone Set and as nonbasicvariables Basic Solution (,,,,) Set and as nonbasicvariables Basic Solution (,,,,8) Set and X as nonbasicvariables Basic Solution (,,,,) (,,,,) and (,,,,8) are adacent (,,,,) and (,,,,) are not adacent Maimize Subect to z 8,,,,, ESE Operations Research ESE Operations Research Obective function as constraint Applying Simple To the augmented form Maimize Maimize Subect to 8,,,,, Subect to 8,,,,, ESE Operations Research 7 ESE Operations Research 8

4 ESE Operations Research 9// X and are nonbasic variables Initial solution is (,,,,8) Maimize Subect to 8,,,,, Rate of improvement as increase is Rate of improvement as increase is Both are positive t optimal Question: what if rate of improve? Maimize Subect to 8,,,,, ESE Operations Research 9 ESE Operations Research Determine direction of movement Choose entering basic variable Subect to Base on maimum rate of improvement From the list of nonbasic variable It will become non-zero Hence become a basic variable w, something else has to give Maimize ESE Operations Research 8,,,,,,,, Determining where to stop Stop when non-negative constraint is violated Minimum ratio test Choosing leaving basic variable The point is to stay positive (feasible) 8, ESE Operations Research Solve for new BF solution entering (value ) leaving (value ) Maimize Subect to stay nonbasic (value ) Eliminate from obective function 8,,,,, Optimality test t optimal Because coefficient of is positive ESE Operations Research ESE Operations Research

5 ESE Operations Research 9// Determining direction of movement Choose entering basic variable Base on maimum rate of improvement From the list of nonbasic variable Only one place to go ESE Operations Research Determining where to stop ESE Operations Research,,,, Stop when non-negative constraint is violated Minimum ratio test Choosing leaving basic variable The point is to stay positive (feasible) is entering, is leaving Solve for new BF solution entering (value ) leaving (value ) stay nonbasic (value ) Eliminate from obective function ESE Operations Research 7 Optimality test Optimal Because coefficient are not positive ESE Operations Research 8