Management wants primary consideration given to three factors: long-run profit, stability in the workforce, and an increase in the companys earning next year. In particular, using the units given in the following table, they want to Maximize Z=2P-5C-3D Where P=total( discounted) profit over life of new products C=change(in either direction)in current level of employment, D=decrease(if any)in next years earnings from current year's leve The amount of any increase in earning does not enter into Z, because management is concerned primarily with just achieving some increase to keep the stockholders happy. (It has mixed feelings about a large increase that then would be difficult to surpass in subsequent The impact of each of the new products(per unit rate of production) on each of these factors is shown in the following table Product unit contribution Factor Units Long-run profit Maximize (millions of dollars Employment level (hundreds of employees) Earning next year ≥75 ( Millions of dollars) Assuming that three are no additional constraints on the production rates not described here, use the goal programming technique to formulate a linear programming model for this problem. (10 points) 4. Use the simplex method to solve the following problem (15 points Maximize Z=5x,+4x2 x1+3x2≤90 80 subject to x,+x≤45 x1≥0,x2≥0 5. Consider the following problem (10 points) Maximize Z=6x,+x2+2x3 x3 x3≤3 subject to x1≥0,x2≥0,x3≥0 Let x4, x5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows Basic variable E Coefficient of2 Management wants primary consideration given to three factors: long-run profit, stability in the workforce, and an increase in the company’s earning next year. In particular, using the units given in the following table, they want to Maximize Z=2P-5C-3D Where P=total (discounted) profit over life of new products. C=change (in either direction) in current level of employment, D=decrease (if any) in next year’s earnings from current year’s level. The amount of any increase in earning does not enter into Z, because management is concerned primarily with just achieving some increase to keep the stockholders happy. (It has mixed feelings about a large increase that then would be difficult to surpass in subsequent years.) The impact of each of the new products (per unit rate of production) on each of these factors is shown in the following table: Product unit contribution Factor 1 2 3 Goal Units Long-run profit 20 15 25 Maximize （millions of dollars） Employment level 6 4 11 =50 (hundreds of employees) Earning next year 8 7 5 ≥75 (Millions of dollars) Assuming that three are no additional constraints on the production rates not described here, use the goal programming technique to formulate a linear programming model for this problem. (10 points) 4. Use the simplex method to solve the following problem. (15 points) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ ≥ + ≤ + ≤ + ≤ = + 0, 0 45 2 80 3 90 5 4 1 2 1 2 1 2 1 2 1 2 x x x x x x x x subject to Maximize Z x x 5.Consider the following problem (10 points) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≥ ≥ + + ≤ − − − ≤ + + ≤ = + + 0, 0, 0 1 2 1 2 3 2 3 4 2 2 2 1 2 2 6 2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x x x x subject to Maximize Z x x x Let x4,x5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows: Basic variable Eq. Coefficient of :