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modified is a binding constraint (that is, has zero slack or excess).
• Apply the 100% rule. Case 1 • The calorie requirement is decreased to 400 calories • The fat requirement is increased to 15 oz.  ∞ = 500 ∞ ≤ calorie requirement(400) ≤ 500+250= 750  ∞ = 8 ∞ ≤ fat requirement(15) ≤ 8+5= 13 Case 2
• The chocolate requirement is increased to 10 oz.
• The sugar requirement is decreased to 5 oz. ∑r j ≤ 1? j rj =
rj = Δb j
Ij
−Δb j
Dj if Δb j ≥ 0 where Δb j = change in RHS if Δb j ≤ 0 where I j , D j = max allowable increase/decrease The Dual Problem
• Duality theory provides the
economic interpretation of
shadow prices. It also
provides the straightforward
calculation of rhs ranging,
obj. function coefficients
ranging, and sensitivity
analysis. • Duality theory exists for
convex programming, not
just linear programming,
where the dual variables are
called Lagrange multipliers. Duality Theory • The theory of duality is a very elegant and important concept within the ﬁeld of opFmizaFon. • The noFon 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 relaFon to its dual is termed the primal. • The relaXonship between the primal and its dual lies on both a mathema0cal and economic level – Finding upper bounds – Economic InterpretaFon 28 Finding Upper Bounds max s.t. 4x1 + x2 + 3x3 x1 + 4x2 3x1  x2 + x3 x1, x2, x3 ≥ 0 ≤ 1 ≤ 3 • Every feasible soluFon provides a lower bound on the opFmal objecFve funcFon value, ς*. Pick a feasible solu0on. (x1, x2, x3)=(1,0,0)? (x1, x2, x3)=(0,0,3)? What is the lower bound each feasible soluFon provides? How good is our lower bound? • At every pivot step, the dual simplex tableau is the negaFve transpose of the primal simplex tableau. 29 Economic InterpretaFon: Resource AllocaFon • Given a cache of raw materials and a factory for turning these raw materials in a variety of ﬁnished products, – how many of each product type should we make so as to maximize proﬁt? 30 Outsourcing 31 Economic InterpretaFon • Need a pracFcal way to link proﬁt maximizaFon to what we observe in markets. • Duality provides link between ordinary (uncompensated) and compensated demand • Primal = Proﬁt MaximizaFon – We have already seen how we can derive ordinary demand funcFons from proﬁt maximizaFon. • Dual = Expenditure MinimizaFon....
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 Winter '09

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