Solutions to Assignment 7

  • 3-10

    • Let xi be the the number of units of product i (i=1,2) 
      Max 4x1 + 3x2 
   st
  3x1 + 2x2 <= 10  (Machine 1 capacity)
  x1 + 4x2  <= 16  (Machine 2 capacity)
  5x1 +3x2  <= 12  (Machine 3 capacity)
  x1,x2 > =0.
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  • 3-11
    •     Let S, W and C be the number of different models
    Min 12S +   15W +    24C
    St
      6000S + 8000W + 11000C >= 12,000,000
          S                  >= 100
                  W          >= 200 
                           C >= 300
    Note:
    Although the problem does state that "...she wants to make sure that total contribution margin equals total cost ..." writing the constraint as a greater-than-or-equal-to makes more sense. Had the demand for the models been much higher, with that constraint written as a strict equality, it would have been impossible to satisfy all demand and not make a profit (namely total contribution margin = fixed costs).
    Solver sheet

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  • 3-13. Let T, C, M, and B be the millions of dollars invested in Treasury Bonds, Common Stock, Money Market and Municipal Bonds respectively.
    • Max .08T + .06C + .12M +.09B
S.t.
     T + C + M + B = 10 ( all 10 million to be invested)

         T   <= 5            (Max on T)
         C   <= 7            (Max on C)
         M   <= 2            (Max on M)
         B   <= 4            (Max on B)
         T + C >= 3          (T + C no more that 30%, since total investment will be 10 30% of the total is 3)
         M + B <= 4          (M + B no less than 40%, again 40% of total investment is 4 million) 
     M, C, T, B >= 0     (non-negativity)
    Solver sheet

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  • 3-15.
    •      Let x ij be the gallons of vintage i (i=1,2,. .4)   mixed in blend j (j=A,B,C)
     Max 80(x1A + x2A + x3A + x4A)+
         50(x1B + x2B + x3B + x4B)+
         35(x1C + x2C + x3C + x4C)      
     st
          x1A + x1B + x1C <= 130
          x2A + x2B + x2C <= 200
          x3A + x3B + x3C <= 150
          x4A + x4B + x4C <= 350

          x2A + x3A >= .75(x1A + x2A + x3A + x4A)
          x4A       >= .08(x1A + x2A + x3A + x4A)
          x2B       >= .10(x1B + x2B + x3B + x4B)
          x4B       <= .35(x1B + x2B + x3B + x4B)
          x2C + x3C >= .35(x1C + x2C + x3C + x4C) 
               xij   >= 0 for all i and j.
    Note:
    Total of blend is the sum of the vintages used in that blend. For instance, total of blend A (in gallons) is given by x1A+x2A+x3A+x4A etc. Likewise, total of any vintage used is the sum of that vintage mixed in all the blends. For instance, total vintage 2 used is given by x2A+x2B+x2C etc.
    Solver sheet

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  • 3-19
    • Let Xij be the number of sailors starting their 
four day shift on ith day (i=1,2,3,4,5)  (1 is monday etc.)

and jth period (j= a, (for AM), j=p (for PM). Assume the scheduling is done as a pattern to be used week after week.

Min X1a +  X2a + X3a +  X4a + X5a + X1p +  X2p +  X3p +  X4p + X5p
s. t 
    X3a + X4a + X5a + X1a >=  900
    X4a + X5a + X1a + X2a >= 1000
    X5a + X1a + X2a + X3a >=  450
    X1a + X2a + X3a + X4a >=  800
    X2a + X3a + X4a + X5a >=  700

    X3p + X4p + X5p + X1p >=  800
    X4p + X5p + X1p + X2p >=  500
    X5p + X1p + X2p + X3p >= 1000
    X1p + X2p + X3p + X4p >=  300
    X2p + X3p + X4p + X5p >=  750 
    Xij >=0 i=1,2,3,4,5;   j= a,p


    Solver sheet

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  • 3-20.
    •   Let x1 and x2 be the hours processes 1 and 2 used.

     Max 450 x1 + 390 x2
st
     15x1 +  9x2 >=600
      6x1 + 24x2 >=225
      3x1 + 12x2 <=300
      9x1 +  6x2 <=450
          x1, x2 >=0
    Note:
    Each output and each input can be written solely in terms of hours of operation of the two processes


    Solver sheet

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