Math Is Fun Forum

spikep

Guest

Juice company production

Hi

Please help with this problem:

<BEGIN>

Juice company produces pear, orange, lemon, tomato and apple juices, and these combined (from the previous flavors) juices H and G.

FRUIT           MAX Availability (Kg)   COST ($ /Kg.)    Sell price ($/liter)
Orange (O)             32000                                 94              129
Pear ( P)                  25000                                 87              125
Lemon (L)                 21000                                 73              110
Tomato (T)               18000                                   47              88
Apple (A)            27000                                68              97

Combined juices selling price and info:

COMBINED       SPEC                                Sell price ($l/liter)
       H               No more than 50 % of A                     100
                        No more than 20 % of P
                        No more than 10 % of L

       G               40 % of O                                      120
                        35 % of L
                        25 % of P

Company wants to sell all the production.  Each kg of fruit produces 1 liter of juice respectively.
Determine the production levels and types of juice, obtaining the maximum benefit.

<END>

I started with this exercise like this:

Z = 35x1 + 38x2 + 37x3 + 41x4 + 29x5 + 26.7x6 + 35.1x7

Constraints:

0.4x1+ 1x1                <= 32000
0.2x2 + 0.25x5 + 1x5 <= 25000
0.1x3 + 0.35x3 + 1x3 <= 21000
0.5x5 + 1x5              <= 27000
1x4                         <= 18000

But I was told that I have to have more variables.

Thanks

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Juice company production

hi spikep

Welcome to the forum.

I'm having trouble following where your constraint inequalities came from.  Please would you define your variables and then state which constraint leads to each inequality.  It'll be easier then to see if you have everything covered.

Thanks,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

spikep

Member

Registered: 2015-10-25

Posts: 1

Re: Juice company production

I thought about it from a different perspective, so I did this table:

            O        P         L           T         A         Price

H                   x1        x2                   x3         100
G         x4      x5        x6                               120
O         x7                                                    129
P                   x8                                          125
L                               x9                              110
T                                         x10                   88
A                                                   x11         97

Avail.  32000  25000  21000  18000  27000    Availability

Cost     94       87       73        47       68
per kg

Revenue per juice

Juice H
C1 = 100 - 87 = 13
C2 = 100 - 73 = 27
C3 = 100 - 68 = 32

Juice G
C4 = 120 - 94 = 26
C5 = 120 - 87 = 33
C5 = 120 - 73 = 47

Juice O
C7 = 129 - 94 = 35

Juice P
C8 = 125 - 87 = 38

Juice L
C9 = 1|0 - 73 = 37

Juice T
C10 = 88 - 47 = 41

Juice A
C11 = 97 - 68 = 29

So the goal function whould be:

Z = 13x1 + 27x2 + 32x3 + 26x4 + 33x5 + 47x6 + 35x7 + 38x8 + 37x9 + 41x10 + 29x11

Contraints or restrictions:

x4 + x7         <= 32000
x1 + x5 + x8 <= 25000
x2 + x6 + x9 <= 21000
x10              <= 18000
x3 + x11       <= 27000

And here's where I think I have the problem:

x1 <= 0.20x1         If I work this ecuation
x1 - 0.20x1 <= 0
0.80x1 <= 0          <- this would be the result

The next ones already worked out:
0.9x2 >= 0   

0.50x3 <= 0

0.60x4 <= 0

0.75x5 <= 0

0.65x6 <= 0

If I enter those values in QM for Windows, the value for X is 0.
I think the problem is with the restrictions, but I don't know what else to do...

Thanks!

Last edited by spikep (2015-10-25 09:50:05)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Juice company production

hi spikep

Thanks.  That is much clearer.

Revenue per juice

Juice H
C1 = 100 - 87 = 13
C2 = 100 - 73 = 27
C3 = 100 - 68 = 32

Juice G
C4 = 120 - 94 = 26
C5 = 120 - 87 = 33
C5 = 120 - 73 = 47

COMBINED       SPEC                                Sell price ($l/liter)
       H               No more than 50 % of A                     100
                        No more than 20 % of P
                        No more than 10 % of L

       G               40 % of O                                      120
                        35 % of L
                        25 % of P

I think the juice G revenue should take account of the percentages.  H is harder as you're only told 'no more than'.  And those percentages don't add up to 100% anyway.  What goes in to make the rest?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob