Andrew, who is ill, takes vitamin pills. Each day, he must take at least 16 units of myra X, at least 5 units of myra Y, and at least 20 units of myra Z. He can choose between tablets and capsules. Each tablet contains 8 units of myra X, 1 unit of myra Y, and 2 units of myra Z, while each capsule contains 2 units of myra X, 1 unit of myra Y and 7 units of myra Z. The cost per tablet is Php15, and Php30 per capsule. How many tablets and capsules should he buy in order to minimize his cost?

z (min) = 105

x₁ = 3

x₂ = 2

Step-by-step explanation:

Let´s call  x₁   (number of tables) and  x₂ ( number of capsules)

assuming it is only possible to get a whole table or capsule we have:

Table  x₁         8 u myra X      1 u of myra Y  and  2 u of myra Z

Capsule x₂     2       "              1         "             and  7 u       "

Objective function

z  =  15* x₁  +  30* x₂    to minimize

First constraint  ( Myra X )

8*x₁ + 2*x₂  ≥  16

Second constraint ( Myra Y )

x₁   +  x₂    ≥  5

Third constraint  ( Myra Z )

2*x₁  +  7*x₂  ≥  20

General constraints  x₁ ≥ 0     x₂  ≥ 0  both integers

Then the model is:

z  =  15 * x₁   +  30 * x₂       to minimize

Subject to

8*x₁ + 2*x₂  ≥  16

x₁   +  x₂    ≥  5

2*x₁  +  7*x₂  ≥  20

x₁ ≥ 0     x₂  ≥ 0  both integers

As the constraints, all are of the form  ≥ we have to subtract surplus variables sₐ and add artificial variables Aₐ

sₐ   ≥  0     Aₐ  ≥ 0

The first table is

z         x₁      x₂      s₁     s₂    s₃    A₁     A₂    A₃    =  Cte

1       - 15    - 30     0     0     0      M     M     M    =    16

0         8         2      -1    0     0      1        0     0    =     5

0         1          1       0   -1      0      0      1       0   =   20

0         2         7       0    0    -1       0      0      1    =   20

Using AtoZmax and after 6 iterations we find

z (min) = 105

x₁ = 3

x₂ = 2

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step-by-step explanation: