Linear Programming. Linear programming provides methods for allocating limited resources among competing activities in an optimal way.

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1 University of Southern California Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research - Deterministic Models Fall 2007: Midterm Review Linear Programming Linear programming provides methods for allocating limited resources among competing activities in an optimal way. Geometric Approach: Linear programming problems may be solved graphically if the model is limited to two variables. Definitions: Objective function, functional constraints, non-negativity constraints, resources, activities, feasible region, feasible solution, infeasible solution, optimal solution, corner point solution, corner point feasible solution (CPF), and adjacent CPF solutions. Assumptions of Linear Programming: Proportionality, additivity, divisibility, and certainty. Theorem: A CPF solution that has no adjacent CPF solutions that are better is optimal. Properties of CPF Solutions Property 1: (a) If there is exactly 1optimal solution, it must be a CPF solution, (b) If there are multiple optimal solutions (and a bounded feasible region), 2 must be adjacent CPF solutions. Property 2: There are a finite number of CPF solutions. Property 3: If a CPF solution has no adjacent CPF solutions that are better, then such a CPF solution is guaranteed to be an optimal solution. Simplex Method Definitions: Slack variable, surplus variable, artificial variable, augmented solution, basic solution, basic feasible (BF) solution, degrees of freedom, basic variable, nonbasic variable, pivot column, and pivot row. Basic Solution Properties Each variable is either a basic or non-basic variable Number of basic variables equals the number of functional constraints Non-basic variables are set equal to zero Values of the basic variables are obtained as the simultaneous solution of the system of equations If the basic variables satisfy the non-negativity constraints, the basic solution is a BF solution. Method Initialize: Introduce slack variables, which are the initial basic variables. Decision variables are the initial non-basic variables. Optimality Test: BFS is optimal iff every coefficient in row 0 is nonnegative. If so, Fall 2007 Page 1

2 we are done, otherwise continue. Iteration: Determine entering basic variable having the most negative coefficient in row 0. Determine leaving basic variable by applying the minimum ratio test. Solve for new BFS using elementary row operations. Solving LP Problems Using the Simplex Method Algebra of the simplex method Tie-breaking in entering basic variable Tie-breaking in leaving basic variable (degeneracy) No leaving basic variable (unbounded Z) Multiple optimal solutions Variations in Model Form (from Standard Form): Equal or greater-than-orequal functional constraint, negative right-hand-side of functional constraints, minimization, and variables allowed to be negative. Solving Problems with Artificial Variables: Big M Method Post-optimality Analysis: shadow prices, binding constraints, sensitivity analysis Revised Simplex Method Initial Basic Coefficients Tableaux Var Z Original Vars. Slack Vars. RHS Z 1 c 0 0 x B 0 A I b Final Basic Coefficients Tableaux Var Z Original Vars. Slack Vars. RHS Z 1 c B B -1 A c c B B -1 c B B -1 b x B 0 B -1 A B -1 B -1 b Book s Basic Coefficients Notation Var Z Original Vars. Slack Vars. RHS Z 1 z* c y* Z* x B 0 A* S* b* Revised Simplex Method 1. Initialize as before (inc. model formulation) 2. Iteration Determine entering basic variable (c B B -1 A c) Determine leaving basic variable (B -1 A col ) Determine new BFS (x B = B -1 b) 3. Optimality test Derive B -1 new from B -1 old x k = entering basic variable a ik = row i from B -1 A col r = row of leaving basic variable Fall 2007 Page 2

3 η i = {-a ik /a rk if i r, 1/a rk if i = r} η = [η 1, η 2,..., η m ] E = identity matrix with rth column replaced with η B -1 new = EB -1 old Duality Theory Definitions: Primal problem, dual problem Primal and Dual: Primal Problem Dual Problem Max Z = cx Min W = yb s.t. Ax b s.t. ya c and x 0 and y 0 Weak Duality Property: If x is a feasible solution for the primal problem and y is a feasible solution for the dual problem, then cx yb. Strong Duality Property: If x* is an optimal solution for the primal problem and y* is an optimal solution for the dual problem, then cx* = y*b. Complementary Solutions Property: At each iteration, the simplex method simultaneously identifies a CPF solution x for the primal problem and a complementary solution y for the dual problem (found in row 0, the coefficients of the slack variables), where cx = yb. If x is not optimal for the primal, what about y? Complementary Optimal Solutions Property: At the final iteration, the simplex method simultaneously identifies an optimal solution x* for the primal problem and a complementary optimal solution y* for the dual problem (found in row 0, the coefficients of the slack variables), where cx* = y*b. In this solution, y* gives the shadow prices for the primal problem. Symmetry Property: The dual of the dual is the primal. Duality Theorem: Possible scenarios Feasible solutions exist and objective function is bounded, then same is true for other problem. Feasible solutions exist and objective function is unbounded, then other problem is infeasible. No feasible solutions exist then other problem is either infeasible or has unbounded objective function. Complementary Basic Solutions Property: Each basic solution in the primal problem has a complementary basic solution in the dual problem, where their respective objective function values (Z and W) are equal. Complementary Optimal Basic Solutions Property: Each optimal basic solution in the primal problem has a complementary optimal basic solution in the dual problem, where their respective objective function values (Z and W) are equal. Complementary Slackness Property: The variables in the primal basic solution and the complementary dual basic solution satisfy the complementary slackness as Fall 2007 Page 3

4 shown: Primal Variable Basic Non-basic Dual Variable Non-basic (m variables) Basic (n variables) Shortcuts for Conversion Between Primal and Dual Primal Dual Maximize Z Minimize W Constraint i: Variable y i : form y i 0 = form unconstrained form y i 0 Variable x j : x j 0 unconstrained x j 0 Constraint j: form = form Sensitivity Analysis Definitions: Sensitive parameter, non-sensitive parameter Sensitivity Analysis deals with the effect on the optimal solution of making changes in the values of the model parameters a ij, b i c j Procedure for Sensitivity Analysis 1. Revision of model 2. Revision of final tableau 3. Conversion to proper form using Gaussian elimination 4. Feasibility test 5. Optimality test 6. Re-optimization Revised Final Tableaux Eq. Coefficients Z Original Vars. Slack Vars. RHS (0) 1 z* - c = y*a - c y* Z* = y*b (1, 2,..., m) 0 A* = S*A S* b* = S*b Revised Coefficients Eq. Final Z Original Vars. Slack Vars. RHS Tableaux (0) 1 (z* - c) = y* A - c y* Z* = y* b (changes) (1, 2,..., m) 0 A* = S* A S* b* = S* b Case 1: Changes in b i Only right-hand-side column changes Can skip conversion to proper form and optimality test steps Fall 2007 Page 4

5 Case 2a: Changes in the Coefficients of a Non-basic Variable Only coefficients of non-basic variable x j change Can skip conversion to proper form and feasibility test steps Shortcut for optimality test: consider if corresponding constraint in the dual problem is satisfied Case 2b: Introduction of a New Variable Treat this case as if it were Case 2a Case 3: Changes in the Coefficients of a Basic Variable Only coefficients of basic variable x j change Case 4: Introduction of a New Constraint First check if optimal solution satisfies new constraint. If so, then the solution is still optimal. If not, move to next step. Introduce new constraint into the final tableau. Fall 2007 Page 5