maxz=420(x1+x12+x13)+360x21+x2+x23)+300x31+x2+x3) 20x1+15x1,+12x12≤1300 20x2+15x2+12x2≤1200 20x12+15x+12x2≤5000 x1+x21+x31≤750 x2+x2+x2≤900 st x1+x12+x13≤900 +x22+x23≤1200 x+x2+x3≤750 33 900 450 0 3. The research and development division of a certain company has developed three new products. The problem is to decide which mix of these products should be produced 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, ease(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 man 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 Fact Goal Units Long-run profit 20 25 Maximize (millions of dollars Employment level 6 4 (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) Solution: we set produce production 1, 2, 3 are x1, x2, X3, the goal programming model is2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ + + = + + = + + + + ≤ + + ≤ + + ≤ + + ≤ + + ≤ + + ≤ + + ≤ + + ≤ + + ≤ = + + + + + + + + 0 750 900 450 750 1200 900 450 900 750 20 15 12 5000 20 15 12 1200 20 15 12 1300 . . max 420( ) 360( ) 300( ) 11 21 31 12 22 32 13 23 33 31 32 33 21 22 23 11 12 13 13 23 33 12 22 32 11 21 31 13 23 33 12 22 32 11 12 13 11 12 13 21 22 23 31 32 33 ij x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x st Z x x x x x x x x x 3. The research and development division of a certain company has developed three new products. The problem is to decide which mix of these products should be produced. 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) Solution: we set produce production 1, 2, 3 are x1, x2, x3, the goal programming model is