# 44 example alternative optimal solutions consider the

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44Example:Alternative Optimal SolutionsConsider the following LP problem.Max4x1+ 6x2s.t.x1<62x1+ 3x2<18x1+x2<7x1>0andx2>0
45Boundary constraint 2x1+ 3x2<18and objectivefunction Max4x1+ 6x2are parallel.All points online segment A – B are optimal solutions.x1x27654321123456789102x1+ 3x2<18x1+x2<7x1<6Max 4x1+ 6x2ABExample:Alternative Optimal Solutions
46·InfeasibilityNo solution to the LP problem satisfies all theconstraints, including the non-negativity conditions.Graphically, this means a feasible region does notexist.Causes include:A formulation error has been made.Management’s expectations are too high.Too many restrictions have been placed on theproblem (i.e. the problem is over-constrained).Special Cases
47Example:Infeasible Problem·Solve graphically for the optimal solution:Max2x1+ 6x2s.t.4x1+ 3x2<122x1+x2>8x1,x2>0
48Example:Infeasible Problem·There are no points that satisfy both constraints, sothere is no feasible region (and no feasiblesolution).x2x14x1+ 3x2<122x1+x2>8246810482610
49Infeasible LP – An Example·minimize4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24+8x31+10x32+16x33+5x34Subject to·x11+x12+x13+x14=100·x21+x22+x23+x24=200·x31+x32+x33+x34=150·x11+x21+x31=80·x12+x22+x32=90·x13+x23+x33=120·x14+x24+x34=170·xij>=0, i=1,2,3; j=1,2,3,4Total demand exceeds total supplyHence infeasible
50Example:Unbounded Problem·Solve graphically for the optimal solution:Max3x1+ 4x2s.t.x1+x2>53x1+x2>8x1,x2>0
51·UnboundedThe solution to a maximization LP problem isunbounded if the value of the solution may bemade indefinitely large without violating any of theconstraints.For real problems, this is the result of improperformulation.(Quite likely, a constraint has beeninadvertently omitted.)Special Cases
52Example:Unbounded Solution·Consider the following LPproblem.Max4x1+ 5x2s.t.x1+x2>53x1+x2>8x1,x2>0
53Example:Unbounded Solution·The feasible region is unbounded and the objectivefunction line can be moved outward from the originwithout bound, infinitely increasing the objectivefunction.x2x13x1+x2>8x1+x2>5Max 4x1+ 5x2681024681042
54Product Mix Example (from session 1)Type 1Type 2Profit per unit\$60\$50Assembly time perunit4 hrs10 hrsInspection time perunit2 hrs1 hrStorage space perunit3 cubic ft3 cubic ftResourceAmount availableAssembly time100 hoursInspection time22 hoursStorage space39 cubic feet
55Maximize 60x1+ 50x2subject to4x1+ 10x2<= 1002x1+1x2<= 223x1+3x2<= 39x1,x2>= 0Linear Programming ExampleVariablesObjectivefunctionConstraintsWhat is a Linear Program?A LP is an optimization model that hascontinuous variablesa single linear objective function, and(almost always) several constraints (linear equalities or inequalities)Non-negativity Constraints
56·Decision variables·unknowns, which is what model seeks to determine·for example, amounts of either inputs or outputs·Objective Function·goal, determines value of best (optimum) solution among all feasible (satisfyconstraints) values of the variables

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