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This is my homework, I have try my best to solve it, but It seems very hard to find out the x1, x2, and x3. Can anyone help me? Thank you very much!
Can anyone just write down the constraints or tips, I hope I can do the rest..Thx!
Tasty Cereal Pty Limited makes three types of cereals: Regular, Deluxe and Super using four different ingredients. The selling prices (per kg) for Regular, Deluxe and Super cereals are 6.50, 7.90 and 8.90 respectively. Availability, costs and usage of the ingredients are given in the table below.Tasty Cereal Pty Limited makes three types of cereals: Regular, Deluxe and Super using four different ingredients. The selling prices (per kg) for Regular, Deluxe and Super cereals are 6.50, 7.90 and 8.90 respectively. Availability, costs and usage of the ingredients are given in the table below.
Ingredient| Availability (kg/day)|Fat (units/kg) | Protein (units/kg) | Calories per kg | cost (dollar/per kg)
1 |1800 |12 |25 |180 |3.90
2 |2500 |9 |32 |220 |4.30
3 |2200 |15 |28 |160 |3.20
4 |1900 |18 |20 |130 |2.90
The Regular Cereals must have at most 18 units of fat, at least 24 units of protein and between 140 and 175 calories per kg. The Deluxe Cereals must have at most 15 units of fat, at least 27 units of protein and between 160 and 190 calories per kg. The Super Cereals must have at most 12 units of fat, at least 30 units of protein and between 180 and 200 calories per kg. It iscompany policy that at least 30% production should be Regular Cereal and no more than 25% is Super Cereal.
Develop a linear programming model for the above problem, Allocation of each type of ingredient to produce each type of cereal and Production levels of each type of cereal.
Cheers
Benny
I think we should develop 2 linear programming, one for allocation of each type of ingredient to produce each type of cereal, the other one for production levels of each type of cereal.
I have try my best to do some, but not sure it is correct.
For the allocation of each type of ingredient to produce each type of cereal
Min 3.9x1+4.3x2+3.2x3+2.9x4
s.t.
12x1+9x2+15x3+18x4<=18
25x1+32x2+28x3+20x4>=24
180x1+220x2+160x3+130x4<=140
180x1+220x2+160x3+130x4>=175
x1+x2+x3+x4=1
The same as deluxe and super, not sure it is correct.
Try to discuss and share, I know we are classmate
Hi Benny
Don't worry about the posting of the problem: we are not going to give away answers, and you did not ask for answers, but for guidance.
In ANY word problem, whether it is beginning algebra, calculus, or linear programming, the FIRST step is to identify all POTENTIALLY relevant variables and name them, preferably in writing so you do not forget them or mix them up. This should be taught over and over in first year algebra, but it is never too late to learn it. It is particularly important in linear programming, where there can be many, many variables.
You were on the right path in realizing that this problem has many more variables than the three implied in your first post. Let's think about what they are. Obviously, the quantities produced of each type of cereal, right? So that is three variables right there. Let them be x_{1}, x_{2}, and x_{3}. How about the quantities used of the four ingredients; how many of them are there? Are there any other variables of POTENTIAL relevance to your constraints?
In a linear programming problem, there is always an objective variable, the variable that is defined by the objective function and is to be optimized. What is that variable in this problem?
Before discussing the next steps, let's see the complete list of potentially relevant variables from you. (See how we provide help, but you do the work solving the problem.)
Not only can you, you must. How could you possibly know how much of a particular ingredient you will need if you do not know how much you are going to produce in each type of cereal?
It may help you see how everything comes together if you specify what your objective variable is and what the objective function is. That is critical in a linear programming problem.
all right, now I have some idea.
1. Decision variables: xr1, xr2, xr3, xr4, xd1, xd2...etc ---how many kgs of each ingredient in each type of cereal.
Objective function
min cost, thus min 3.9x1+4.3x2+3.2x3+2.9x4
s.t.
12x1+9x2+15x3+18x4<=18
25x1+32x2+28x3+20x4>=24
180x1+220x2+160x3+130x4<=140
180x1+220x2+160x3+130x4>=175
x1+x2+x3+x4=1
then, I will find out Allocation of each type of ingredient to produce each type of cereal
2. Decision variables: x1, x2, x3 for production level
objective function: max 6.50x1+7.90x2+ 8.90X3 (max revenue)
s.t.
Availability
and company policy.
In the right track?
Regards
Benny
You are still trying to allocate ingredients without knowing what you are making.
Try a different objective than minimizing cost. If you were to plan on maximizing profit, how would that change your objective function?
Moreover, you are underestimating the number of constraints. There are constraints on fat, protein, and calories. How are you going to address those?
I am going to give you an additional hint
Let:
$x_1 = \#\ of\ kg\ of\ regular\ cereal\ produced;$
$x_2 = \#\ of\ kg\ of\ deluxe\ cereal\ produced;$
$x_3 = \#\ of\ kg\ of\ super\ cereal\ produced;$
$x_4 = \#\ of\ kg\ of\ ingredient\ 1\ allocated\ to\ production\ of\ regular\ cereal\;$
$x_5 = \#\ of\ kg\ of\ ingredient\ 2\ allocated\ to\ production\ of\ regular\ cereal\;$
$x_6 = \#\ of\ kg\ of\ ingredient\ 3\ allocated\ to\ production\ of\ regular\ cereal\;$
and so on. You will end up with fifteen decision variables to use in your constraints and your objective function. Of course, not every variable will be used in each constraint.
This is a big problem. Do you see now why I said to start by identifying and naming all the potentially relevant variables?