Operations Research Project 2



Operations Research Project 2Kade EnsleyBlake FulbrightZachary Turkowski12/11/2013Original Problem:For this project our team will be using the information for Nori & Leets Co. to develop a model in order to support the company’s decisions regarding pollution abatement methods. Our original model is as follows.Table SEQ Table \* ARABIC 1. Nori & Leets Co. Original ModelMinimize: Z=8x1+10x2+7x3+6x4+11x5+9x6Subject To: 12 x1 +9 x2 +25 x3 +20 x4 +17 x5 +13 x6 >= 60 35 x1 +42 x2 +18 x3 +31 x4 +56 x5 +49 x6 >= 150 37 x1 +53 x 2+28 x3 +24 x4 +29 x5 +20 x6 >= 125 Xj <= 1 (j=1…6) & Xj >= 0 (j=1…6)The objective function is a minimization of Z, which makes sense as Nori & Leets Co. is just trying to meet the required pollution reductions. The next three constraints involve what abatement method (xj) is used on the left of the constraint and the required reduction for each pollutant is on the right. The last two constraints are a technological constraint which says each abatement method cannot perform above its maximum level of one, and lastly there is a non-negativity constraint.Part A:In part A of this project we were tasked with creating a standard form model, i.e. a maximization model, of the original problem. This standard form model is the primal model our team will be using throughout the rest of this project, the model we created is in table two below. Table SEQ Table \* ARABIC 2. Nori & Leets Co. Standard Model Maximize: Z=-8x1-10x2-7x3-6x4-11x5-9x6Subject To: -12 x1 -9 x2 -25 x3 -20 x4 -17 x5 -13 x6 <= -60 -35 x1 -42 x2 -18 x3 -31 x4 -56 x5 -49 x6 <= -150 -37 x1 -53 x 2-28 x3 -24 x4 -29 x5 -20 x6 <= -125 Xj <= 1 (j=1…6) & Xj >= 0 (j=1…6)The objective function for this problem is now a maximization of Z; however if one inspects the coefficients of these equations with the originals they would see that this model was obtained by taking the original equations and multiplying through by -1. Part B:After creating a standard form of the original model we were tasked with creating the dual problem. The dual is obtained by switching the right hand side of the constraints with the objective function, and then redefining each constraint to fit the new system of variables.Table SEQ Table \* ARABIC 3. Nori & Leets Co. Dual ModelMinimize: W=-60y1-150y2-125y3Subject To: -12y1 - 35y2 - 37y3 >= -8 -9y1 - 42y2 - 53y3 >= -10 -25y1 - 18y2 - 28y3 >= -7 -20y1 - 31y2 - 24y3 >= -6 -17y1 - 56y2 - 29y3 >= -11 -13y1 - 49y2 - 20y3 >= -9 yj <= 1 (j=1…6) & yj >= 0 (j=1…6)Part C:The next part of this project involved taking the model that was created in part a, and translating it into code OPL could use to solve for the objective function (Z). After running the software OPL returned an optimal Z value of 32.16. This means that in order for Nori & Leets Co. to meet the required pollutant reductions, they will have to spend a minimum of 32.16 million dollars annually on various abatement methods. The code used to find this optimal value is located in the appendix.Part D:Table 4. Nori & Leets Co. B Inverse (B-1)0001000000.0270.007-0.029-0.504000.06300.103-0.0460.017-0.006-0.16700-0.5300.116000000100-0.006-0.0280.023-0.19300-0.430-0.989000000001-0.027-0.0070.0290.50410-0.0630-0.1030.046-0.0170.0060.167010.530-0.1160.0060.028-0.0230.193000.4310.989B-1 can be obtained by taking the inverse of the basis matrix, B. The B matrix includes all of the values that correspond to the basic variables in the original objective function. Once B-1 is found, it can be used to update the rest of the final tableau. Multiplying B-1 by b, the values of the original right hand side, yields b*, the final values of the right hand side. Also, multiplying by A, the coefficients of the decision variables for the constraint equations, gives A*, the final coefficients for the constraint equations.Part E:The optimal values of the dual variables are given by y*.Table 5. Nori & Leets Co. Optimal Dual Variable Values0.11110.12680.069800.37730.656500.95240The total optimal cost rises as required reduction of sulfur oxide rises and vice-versa.Part F:-28892566230500The following table was constructed using the fundamental insight of the simplex method.Table 6. Nori & Leets Co. Tableau for the Optimal Primal SolutionPart G:When changing the costs ten percent either way it is just a change to the basic variables in the objective function. When changing a basic variable in the objective function it can either change the feasibility or the optimal solution. For example increasing the cost of having taller smokestacks for the blast furnaces by 10% increased the total cost of reducing emissions. This is the same for increasing or decreasing any of the other individual costs in the objective function. Making these changes to the coefficient of the objective function affects the values of the current basic variables and doesn’t change which variables are basic. To find the new optimal solution use the fundamental insight with your new objective function values to obtain the new optimal solution. We take our new C coefficients which correspond to our Cb coefficients because all the variables are basic and multiply by B-1 then multiply by b(the right-hand side of the constraints) to obtain the new optimal solution. Equation SEQ Equation \* ARABIC 1 Z*newCb * B-1 *b = ZnewPart H:Nori & Leets Co. set limits on the amount of filters they could use as an abatement method. They recently understood that they can only implement half as many filters as they previously thought. This represents a change in the right-hand side of the constraints corresponding to filter method. These changes affect the feasibility of the problem and after making the changes to the constraints it reveals a negative in the right-hand side indicating infeasibility. Since the original value of x4 is greater than one-half it is the only basic variable that becomes infeasible when changing the right-hand side of these constraints. Part I:If changes were to be made to the emission rates associated to the different abatement methods then it would indicate changes to the coefficients of the constraint equations. This can affect either the feasibility or the optimality of our final solution; to obtain the new optimal solution we used the fundamental insight to update our final tableau. The equations we used can be seen below. Equation SEQ Equation \* ARABIC 2 z*new & A*newz*=cb*B-1*AA*=B-1*AConclusion:For this project we worked as OR analysts for Nori & Leets Co. in order to help the company make decisions regarding pollution abatement methods. After we constructed the primal problem, we were able to use it to construct the corresponding dual problem. Next, we used OPL to solve the primal problem and find the optimal solution. In addition, we found B-1, and we were able to use the fundamental insight of the simplex method to create the final tableau and conduct sensitivity analysis. Appendix OPL Code:/*Ranges*/range Abatement=1..6;range Pollutant=1..3;/*Parameter Definitions*/float Emission[Abatement]=[-8,-10,-7,-6,-11,-9];float A[Pollutant][Abatement]=[[-12,-9,-25,-20,-17,-13],[-35,-42,-18,-31,-56,-49],[-37,-53,-28,-24,-29,-20]];int Required[Pollutant]=[-60,-150,-125];/*Decision Var.*/dvar float+ Reduction[Abatement];/*Objective Function*/maximize sum(i in Abatement) Emission[i]*Reduction[i];/*Constraints*/subject to{?forall(i in Pollutant)??ctReduce:??sum(j in Abatement) A[i,j]*Reduction[j]<= Required[i];?forall(i in Abatement)?ctConstraint:??Reduction[i]<=1; ?} ................
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