29 economic interpretafon resource allocafon given

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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....
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