Reference no: EM133034087

A diet planner considers four types of food in his diet: peanut butter, pasta, milk, & steak. The problem was formulated (in LINGO-style) as follows: 

Min = 3*x1 + 6*x2 + 8*x3 + 3*x4;                                Minimize cost

5*x1 + 7*x2 + 2*x3 + 2*x4 >= 20;                                Include at least 20 units of vitamin A

6*x1 + 1*x2 + 4*x3 + 3*x4 >= 20;                                Include at least 20 units of vitamin B

7*x1 + 2*x2 + 1*x3 + 2*x4 <= 40;                                Include no more than 40 units of vitamin F

11*x2 + 8*x3 + 9*x4 >= 28;                                          Include at least 28 units of protein

11*x2 + 8*x3 + 9*x4 < 34;                                            Include at most 34 units of protein

12*x1 + 38*x2 + 80*x3 + 42*x4 >= 70;                         Include at least 70 units of carbohydrates

210*x1 + 900*x2 + 650*x3 + 400*x4 >= 1500;              Include at least 1,500 calories

210*x1 + 900*x2 + 650*x3 + 400*x4 <= 2200;              Include no more than 2,200 calories

A solution to the problem was obtained by LINGO:

  Objective value: 17.31148

 Variable     Value         Reduced Cost

    X1       2.098361         0.000000

    X2       0.7213115        0.000000

    X3       0.000000         5.147541

    X4       2.229508         0.000000

  Row   Slack or Surplus      Dual Price

   1       17.31148           -1.000000

   2       0.000000           -0.5409836

   3       0.000000           -0.4918033E-01

   4       19.40984            0.000000

   5       0.000000           -0.1967213

   6       6.000000            0.000000

   7       76.22951            0.000000

   8       481.6393            0.000000

   9       218.3607            0.000000

Ranges in which the basis is unchanged:

Objective Coefficient Ranges:

                 Current        Allowable        Allowable

 Variable     Coefficient      Increase         Decrease

   X1          3.000000        3.789474         0.4390244

   X2          6.000000        0.4000000        3.666667

   X3          8.000000        INFINITY         5.147541

   X4          3.000000        3.000000         0.3272727

Righthand Side Ranges:

            Current          Allowable        Allowable

  Row        RHS            Increase         Decrease

  2        20.00000         2.933921         4.888889

  3        20.00000         5.866667         8.080890

  4        40.00000         INFINITY         19.40984

  5        28.00000         5.915618         13.04811

  6        34.00000         INFINITY         6.000000

  7        70.00000         76.22951         INFINITY

  8        1500.000         481.6393         INFINITY

  9        2200.000         INFINITY         218.3607

(a) How much of the different types of foods are included in the diet & how much are the (daily) costs? 

(b) Which nutritional requirements are tight, i.e., are satisfied as equations? 

(c) What if the price of pasta increases to $6.50 per serving? Will that change the dietician's plans? 

(d) What is the range of prices of milk within which the solution remains unchanged? 

(e) Suppose that the diet planner has assumed a new job that is physically demanding. The diet should reflect that & so it has determined that the number of calories in the diet should be between 1,900 & 2,400. How will this change influence the solution?