If the nlp is a maximization problem then any point x in the feasible region for which f(x)> f(x holds true for all points x in the feasible region is an optimal solution to the NLp NLPs can be solved with lingo Even if the feasible region for an nlp is a convex set he optimal solution need not be a extreme point of the NLps feasible region For any NLP (maximization, a feasible point X (X1, X2,)is a local maximum if fo,,n) sufficiently small e, any feasible point X (X1,X2Xn) having|×1-×l∈(=1,2,…, satisfies f(×)≥f(x)
6 ◼ If the NLP is a maximization problem then any point in the feasible region for which f( ) ≥ f(x) holds true for all points x in the feasible region is an optimal solution to the NLP. ◼ NLPs can be solved with LINGO. ◼ Even if the feasible region for an NLP is a convex set, he optimal solution need not be a extreme point of the NLP’s feasible region. ◼ For any NLP (maximization), a feasible point x = (x1 ,x2 ,…,xn) is a local maximum if for sufficiently small , any feasible point x’ = (x’1 ,x’2 ,…,x’n) having | x1–x’1|< (i = 1,2,…,n) satisfies f(x)≥f(x’). x x
Example 11: Tire Production Firerock produces rubber used for tires by combining three ingredients: rubber, oil, and carbon black, the costs for each are given The rubber used in automobile tires must have d a hardness of between 25 and 35 d an elasticity of at 16 aa tensile strength of at least 12 To manufacture a set of four automobile tires 100 pounds of product is needed the rubber to make a set of tires must contain between 25 and 60 pounds of rubber and at least 50 pounds of carbon black
7 Example 11: Tire Production ◼ Firerock produces rubber used for tires by combining three ingredients: rubber, oil, and carbon black. The costs for each are given. ◼ The rubber used in automobile tires must have a hardness of between 25 and 35 an elasticity of at 16 a tensile strength of at least 12 ◼ To manufacture a set of four automobile tires, 100 pounds of product is needed. ◼ The rubber to make a set of tires must contain between 25 and 60 pounds of rubber and at least 50 pounds of carbon black
Ex 11-continued ■ Define R= pounds of rubber in mixture used to produce four tires 0= pounds of oil in mixture used to produce four tires C= pounds of carbon black used to produce four tires Statistical analysis has shown that the hardness elasticity and tensile stregth of 100-pound mixture of rubber, oil, and carbon black is Tensile strength= 12.5-.10(0)-.001(0)2 Elasticity=17-+.35R.04(O)-.002(O)2 Hardness=34+.10R+.06(O)-3(C+.001(R)(O)+.005(0)2+.001C2 formulate the nlp whose solution will tell firerock how to minimize the cost of producing the rubber product needed to manufacture a set of automobile tires
8 Ex. 11 - continued ◼ Define: R = pounds of rubber in mixture used to produce four tires O = pounds of oil in mixture used to produce four tires C = pounds of carbon black used to produce four tires ◼ Statistical analysis has shown that the hardness, elasticity, and tensile strnegth of a 100-pound mixture of rubber, oil, and carbon black is Tensile strength = 12.5 - .10(O) - .001(O) 2 Elasticity = 17 - + .35R .04(O) - .002(O) 2 Hardness = 34 + .10R + .06(O) -.3(C) + .001(R)(O) +.005(O) 2+.001C2 ◼ Formulate the NLP whose solution will tell Firerock how to minimize the cost of producing the rubber product needed to manufacture a set of automobile tires
Example 11: Solution ■ After defining ts= Tensile strength E= Elasticity H= Hardness of mixture the Lingo program gives the correct formulation
9 Example 11: Solution ◼ After defining TS = Tensile Strength E = Elasticity H = Hardness of mixture the LINGO program gives the correct formulation
It is easy to use the Excel solver to solve NLPs The process is similar to a linear model For NLPs having multiple local optimal solutions the solver may fail to find the optimal solution because it may pick a local extremum that is not a global extremum. 10
10 ◼ It is easy to use the Excel Solver to solve NLPs. ◼ The process is similar to a linear model. ◼ For NLPs having multiple local optimal solutions, the Solver may fail to find the optimal solution because it may pick a local extremum that is not a global extremum