29 economic interpretafon resource allocafon given

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: g 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....
View Full Document

This document was uploaded on 04/30/2013.

Ask a homework question - tutors are online