Linear Programming Optimal Solutions and Models Without Unique Optimal Solutions

Models With No Solutions - Infeasibility Max 8X 1 + 5X 2 s.t.2X 1 + 1X 2 ≤ X 1 + 4X 2 ≤ 2400 X 1 - X 2 ≤ 350 X 1 ≥ 800 X 1, X 2 ≥ 0

Solver - Infeasible

Solver Results

Infeasibility SolveExcel – When Solve is clicked: infeasibleA problem is infeasible when there are no solutions that satisfy all the constraints. Infeasibility can occur from –Input Error –Mis-formulation –Simply an inconsistent set of constraints

Models With An “Unbounded” Solution Max 8X 1 + 5X 2 s.t. X 1 - X 2 ≤ 350 X 1 ≥ 200 X 2 ≥ 200

Solver - Unbounded

Solver Results - Unbounded

Unboundedness An unbounded solution means you left out some constraints – you cannot make an “infinite” profit. SolveExcel – When Solve is clicked Means the problem isunbounded

Multiple Optimal Solutions Max 8X 1 + 4X 2 s.t.2X 1 + 1X 2 ≤ X 1 + 4X 2 ≤ 2400 X 1 - X 2 ≤ 350 X 1, X 2 ≥ 0

Multiple Optimal Solutions canWhen slope of objective function line equals slope of binding constraint the problem can have multiple optimal solutions. slopeThe slope of a line written as: aX 1 + bX 2 = d is: Object function: 8X 1 + 4X 2 Slope is -8/4 = -2 Plastic constraint: 2X 1 + 1X 2 ≤ 1000 Slope is -2/1 = -2 -a/b

Multiple Optimal Solutions –The constraint must not be a redundant constraint but must be a boundary constraint. –The objective function must move in the direction of the constraint— MINIn the previous example if the objective function had been MIN 8X 1 + 4X 2, then it is moved in the opposite direction of the constraint and (0,0) would be the optimal solution. Multiple optimal solutions allow the decision maker to use secondary criteria to select one of the optimal solutions that has another desirable characteristic (e.g. Max X 1 or X 1 = 3X 2, etc.)

Alternate Optimal Solution To find the second optimal solution: 1.Observe that an Allowable Increase or Allowable Decrease for the objective function coefficient of some variable X j is 0 2.Add a constraint that sets the value function cell to the optimal value from the first optimal solution. 3.Change objective function to If the Allowable Increase = 0, change objective to maximize X j If the Allowable Decrease = 0, change objective to minimize X j

Original Optimal Solution in Excel

Multiple Optimal Solutions in Excel Excel – Identification of multiple solutions Sensitivity Report If an Allowable Decrease or an Allowable Increase of an Objective Function Coefficient is 0.

Alternate Optimal Solution in Excel

Generating the Multiple Optimal Solutions Any weighted average of optimal solutions is also optimal. –In the previous example it can be shown that the two optimal extreme points are (320,360) and (450, 100). Thus.5(320,360) +.5(450,100) = (385,230) is also an optimal point that is half-way between these two points..8(320,360) +.2(450,100) = (346,308) is also an optimal point that is 8/10 of the way up the line toward (320,360).

Review When a linear programming model is solved it: –Has a unique optimal solution –Has multiple optimal solutions –Is infeasible –Is unbounded Identification of each –By Excel Find alternative solutions for multiple optimal solutions problem