Duality in Linear Programming

A model consisting of linear relationshipsrepresenting a firm’s objective and resource constraintsLinear ProgrammingLP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

LP Model FormulationDecision variablesmathematical symbols representing levels of activity of an operationObjective functiona linear relationship reflecting the objective of an operationmost frequent objective of business firms is to maximize profitmost frequent objective of individual operational units (such as a production or packaging department) is to minimize costConstrainta linear relationship representing a restriction on decision making

LP Model Formulation (cont.)Max/min z = c1x1 + c2x2 + ... + cnxnsubject to: a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1 a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2 : am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bmxj = decision variables bi = constraint levelscj= objective function coefficientsaij = constraint coefficients

Geometry of the Prototype ExampleMax 3 P1 + 5 P2s.t. P1 + < 4 (Plant 1) 2 P2 < 12 (Plant 2) 3 P1 + 2 P2 < 18 (Plant 3) P1, P2 > 0 (nonnegativity)P2Every point in this nonnegative quadrant isassociated with a specific production alternative.( point = decision = solution )P10

Geometry of the Prototype ExampleMax 3 P1 + 5 P2s.t. P1 + < 4 (Plant 1) 2 P2 < 12 (Plant 2) 3 P1 + 2 P2 < 18 (Plant 3) P1, P2 > 0 (nonnegativity)P2P1(4,0)(0,0)

Geometry of the Prototype ExampleMax 3 P1 + 5 P2s.t. P1 + < 4 (Plant 1)2 P2 < 12 (Plant 2) 3 P1 + 2 P2 < 18 (Plant 3) P1, P2 > 0 (nonnegativity)P2(0,6)P1(4,0)(0,0)

Geometry of the Prototype ExampleMax 3 P1 + 5 P2s.t. P1 + < 4 (Plant 1)2 P2 < 12 (Plant 2)3 P1 + 2 P2 < 18 (Plant 3) P1, P2 > 0 (nonnegativity)P2(0,6)(2,6)(4,3)(9,0)P1(4,0)(0,0)

Max 3 P1 + 5 P2In Feasible RegionP2(0,6)(2,6)Feasible region is the set of points (solutions) that simultaneously satisfy all the constraints. There are infinitely many feasible points (solutions).(4,3)(9,0)P1(4,0)(0,0)

Geometry of the Prototype ExampleMax 3 P1 + 5 P2P2(0,6)(2,6)Objective function contour(iso-profit line)(4,3)(9,0)P1(4,0)3 P1 + 5 P2 = 12(0,0)

Geometry of the Prototype ExampleMax 3 P1 + 5 P2s.t. P1 + < 4 (Plant 1)2 P2 < 12 (Plant 2)3 P1 + 2 P2 < 18 (Plant 3) P1, P2 > 0 (nonnegativity)3 P1 + 5 P2 = 36P2(0,6)(2,6)Optimal Solution: the solution for the simultaneous boundary equations of two active constraints(4,3)(9,0)P1(4,0)(0,0)

LP Terminologysolution (decision, point): any specification of values for all decision variables, regardless of whether it is a desirable or even allowable choicefeasible solution: a solution for which all the constraints are satisfied.feasible region (constraint set, feasible set): the collection of all feasible solutionoptimal solution (optimum): a feasible solution that has the most favorable value of the objective functionoptimal (objective) value: the value of the objective function evaluated at an optimal solution

Unbounded or Infeasible CaseOn the left, the objective function is unboundedOn the right, the feasible set is empty

LP Model: ExampleRESOURCE REQUIREMENTSLabor Clay RevenuePRODUCT (hr/unit) (lb/unit) ($/unit)Bowl 1 4 40 Mug 2 3 50 There are 40 hours of labor and 120 pounds of clay available each dayDecision variablesx1 = number of bowls to producex2 = number of mugs to produce

LP Formulation: ExampleMaximize Z = $40 x1 + 50 x2Subject to x1 + 2x2 40 hr (labor constraint) 4x1 + 3x2 120 lb (clay constraint) x1 , x2 0Solution is x1 = 24 bowls x2 = 8 mugs Revenue = $1,360

Graphical Solution MethodPlot model constraint on a set of coordinates in a planeIdentify the feasible solution space on the graph where all constraints are satisfied simultaneouslyPlot objective function to find the point on boundary of this space that maximizes (or minimizes) value of objective function

50 –40 –30 –20 –10 –0 –x24 x1 + 3 x2 120 lbArea common toboth constraintsx1 + 2 x2 40 hr|10|60|50|20|30|40x1Graphical Solution: Example

40 –30 –20 –10 –0 –x2x1 + 2x2 = 40 4x1 + 3x2 = 120 4x1 + 8x2 = 160 -4x1 - 3x2 = -120 5x2 = 40 x2 = 8 x1 + 2(8) = 40x1 = 244 x1 + 3 x2 120 lbx1 + 2 x2 40 hrx1|10|20|30|40Z = $50(24) + $50(8) = $1,360Computing Optimal Values824

Extreme Corner Pointsx1 = 0 bowlsx2 =20 mugsZ = $1,000x2x1 = 224 bowlsx2 =8 mugsZ = $1,36040 –30 –20 –10 –0 –x1 = 30 bowlsx2 =0 mugsZ = $1,200ABC|20|30|40|10x1

Theory of Linear ProgrammingAn LP problem falls in one of three cases:Problem is infeasible: Feasible region is empty.Problem is unbounded: Feasible region is unbounded towards the optimizing direction.Problem is feasible and bounded: then there exists an optimal point; an optimal point is on the boundary of the feasible region; and there is always at least one optimal corner point (if the feasible region has a corner point).When the problem is feasible and bounded,There may be a unique optimal point or multiple optima (alternative optima).If a corner point is not “worse” than all its neighbor corners, then it is optimal.

Duality Theory The theory of duality is a very elegant and important concept within the field of operations research. This theory was first developed in relation to linear programming, but it has many applications, and perhaps even a more natural and intuitive interpretation, in several related areas such as nonlinear programming, networks and game theory.

Duality TheoryThe notion of duality within linear programming asserts that every linear program has associated with it a related linear program called its dual. The original problem in relation to its dual is termed the primal.

it is the relationship between the primal and its dual, both on a mathematical and economic level, that is truly the essence of duality theory. ExamplesThere is a small company in Melbourne which has recently become engaged in the production of office furniture. The company manufactures tables, desks and chairs. The production of a table requires 8 kgs of wood and 5 kgs of metal and is sold for $80; a desk uses 6 kgs of wood and 4 kgs of metal and is sold for $60; and a chair requires 4 kgs of both metal and wood and is sold for $50. We would like to determine the revenue maximizing strategy for this company, given that their resources are limited to 100 kgs of wood and 60 kgs of metal. Problem P1

Now consider that there is a much bigger company in Melbourne which has been the lone producer of this type of furniture for many years. They don't appreciate the competition from this new company; so they have decided to tender an offer to buy all of their competitor's resources and therefore put them out of business. The challenge for this large company then is to develop a linear program which will determine the appropriate amount of money that should be offered for a unit of each type of resource, such that the offer will be acceptable to the smaller company while minimizing the expenditures of the larger company. Problem D1

Standard form of the Primal Problem

Standard form of the Dual Problem

DefinitionDual ProblemPrimal Problemb is not assumed to be non-negative

Primal-Dual relationship Primal ProblemDual Problemopt=maxopt=minVariable i :yi >= 0yiursConstraint j: >= form = formConstraint i : <= form = formVariable j:xj >= 0xjurs

♣ Duality in LPIn LP models, scarce resources are allocated, so they should be, valuedWhenever we solve an LP problem, we solve two problems: the primal resource allocation problem, and the dual resource valuation problemIf the primal problem has nvariables and m constraints, the dual problem will have mvariables and nconstraints

Primal and Dual AlgebraPrimalDual

General Rules for Constructing Dual1. The number of variables in the dual problem is equal to the number of constraints in the original (primal) problem. The number of constraints in the dual problem is equal to the number of variables in the original problem.2. Coefficient of the objective function in the dual problem come from the right-hand side of the original problem.3. If the original problem is amax model, the dual is aminmodel; if the original problem is amin model, the dual problem is themaxproblem.4. The coefficient of the first constraint function for the dual problem are the coefficients of the first variable in the constraints for the original problem, and the similarly for other constraints.5. The right-hand sides of the dual constraints come from the objective function coefficients in the original problem.

Relations between Primal and Dual1. The dual of the dual problem is again the primal problem.2. Either of the two problems has an optimal solution if and only if the other does; if one problem is feasible but unbounded, then the other is infeasible; if one is infeasible, then the other is either infeasible or feasible/unbounded.3.Weak Duality Theorem: The objective function value of the primal (dual) to be maximized evaluated at any primal (dual) feasible solution cannot exceed the dual (primal) objective function value evaluated at a dual (primal) feasible solution.cTx >= bTy (in the standard equality form)

Relations between Primal and Dual (continued)4.Strong Duality Theorem: When there is an optimal solution, the optimal objective value of the primal is the same as the optimal objective value of the dual.cTx* = bTy*

Weak Duality DLP provides upper bound (in the case of maximization) to the solution of the PLP.Ex) maximum flow vs. minimum cutWeak duality : any feasible solution to the primal linear program has a value no greater than that of any feasible solution to the dual linear program. Lemma : Let x and y be any feasible solution to the PLP and DLP respectively. Then cTx≤ yTb.

Strong Duality Strong duality : if PLP is feasible and has a finite optimum then DLP is feasible and has a finite optimum. Furthermore, if x* and y* are optimal solutions for PLP and DLP then cTx*= y*Tb

Four Possible Primal Dual Problems

Jyothimon CM.Tech Technology ManagementUniversity of KeralaSend your feedbacks and queries tojyothimonc@yahoo.com

Duality in Linear Programming

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