Unformatted text preview: – A diﬀerent, but related problem would be to minimize expenditure, subject to a minimum level of proﬁt More Economic InterpretaFon • A furniture company manufactures desk, tables, and chairs. • Each type of furniture requires lumber and two types of skilled labor: ﬁnishing and carpentry. • The amount of each resource needed Resource Desk Table Chair Lumber 8 board ft 6 board ft 1 board ft Finishing hours 4 hours 2 hours 1.5 hours Carpentry hours 2 hours 1.5 hours 0.5 hours • At present, 48 board feet of lumber, 20 ﬁnishing hours, 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $ 20. • Demand for desks and chairs is unlimited. • The company wants to maximize total revenue. 33 Define:
x1 = number of desks produced
x2 = number of tables produced
x3 = number of chairs produced.
The primal is: max ς = 60x1 + 30x2 + 20x3
s.t. 8x1 + 6x2 + x3 ≤ 48 4x1 + 2x2 + 1.5x3 ≤ 20
2x1 + 1.5x2 + 0.5x3 ≤ 8 (Lumber constraint)
(Finishing constraint)
(Carpentry constraint) x1, x2, x3 ≥ 0
The dual is: min ξ = 48y1 + 20y2 + 8y3
s.t. 8y1 + 4y2 + 2y3 ≥ 60 (Desk constraint)
6y1 + 2y2 + 1.5y3 ≥ 30 (Table constraint)
y1 + 1.5y2 + 0.5y3 ≥ 20 (Chair constraint)
y1, y2, y3 ≥ 0 Economic InterpretaFon of the Dual • Decision variable y1 is associated with lumber, y2 with ﬁnishing hours, and y3 with carpentry hours. • Suppose an entrepreneur wants to purchase all of the company’s resources. • The entrepreneur must determine the price he or she is willing to pay for a unit of each of the resources. To determine these prices we deﬁne: y1 = price paid for 1 boards l of lumber y2 = price paid for 1 ﬁnishing hour y3 = price paid for 1 carpentry hour The resource prices y1, y2, and y3 should be determined by solving the dual problem. Economic InterpretaFon of the Dual • The total price that should be paid for these resources is min ξ = 48y1 + 20y2 + 8y3
• In semng resource prices, the prices must be high enough to induce the company to sell. • For example, the entrepreneur must oﬀer the company at least $60 for a combinaFon of resources that includes 8 board feet of lumber, 4 ﬁnishing hours, and 2 carpentry hours because the company could, if it wished, use the resources to produce a desk that could be sold for $60. • Since the entrepreneur is oﬀering 8y1 + 4y2 + 2y3 for the resources used to produce a desk, he or she must chose y1, y2, and y3 to saFsfy: 8y1 + 4y2 + 2y3 ≥ 60 36 Back to Product Mix Example Duality 38 Primal Dual RelaFonship ! x1 ! 0 ! x2 ! 0 ! ! xn ! 0 ! ! !"! y1 ! 0! a11! a12 ! ! a1n ! b1! y2 ! 0 ! a21 ! a22 ! ! a2 n ! !! (((! (((! (((! ! ! ! ym ! 0! am1 ! am 2 ! ! amn ! ! ! !!
c1! !!
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! ! 39 Dual of Dual 40 Gap or No Gap? • Is there a gap between the largest primal objec0ve value and the smallest dual objec0ve value? 41 Simplex Method and Duality 42 Second IteraFon 43 Strong Duality Theorem 44 Duality Gap 45 Under the Hood: LP Optimizers Dual Simplex Optimizers
• a linear programming problem can be stated in primal or dual form, and an optimal
solution (if one exists) of the dual has a direct relationship to an optimal solution of the
primal model.
• CPLEX dual simplex optimizer makes use of this relationship, but still reports the
solution in terms of the primal model.
• The dual simplex method is the first choice for optimizing a linear programming
problem, especially for primaldeg...
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 Winter '09

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