Chocolatier sampler boxes problem: applying goal programming and mixed-integer programing to optimally compromise goals.

Question

QUESTION:

A boutique chocolatier is planning to make a number of sampler boxes, each containing $36$ chocolates. (Therefore the total number of chocolates should be divisible by $36$.) The chocolatier would like to achieve the following goals:

• include at least $300$ Hazelnut Glories
• include at least as many Vanilla Surrenders as Hazelnut Glories,
• include at least $240$ Caramel Desires,
• include at least $400$ Dark Mysteries,
• produce no more than $27$ sampler boxes of chocolates overall.

Assume the goals are equally important.

(a) Apply goal programming methods to formulate a mixed-integer program to find the optimal compromise between these goals.

(b) Solve your mixed-integer program in Excel and interpret the solution. Your answer in your main report should include the IP formulation, the optimal solution, and an interpretation of the optimal solution for the boutique chocolatier. You should also submit the Excel file you used to solve the IP.

Can someone help me figuring out how to calculate this question?

In this question, I have 4 variables (I think it would be more easily to calculate), and 5 goals.

SO for the 27 boxes, I multiply the number of 36 = 972.

Then, it subject to 5 constraints, and 4 variables are integers and all decision values is large than or equal to 0.

Then when I enter the question into Excel Solver, I got two different answers and both are quite reasonable.

The first answer is equal to: 300, 32, 240, 400.

And for the second answer is equal to: 32, 300, 240, 400.

The only different is that for the first answer, I have click the box of all variables are non-negative.

SO, I don't know which answer is the best or actually i do it wrong?? Thanks for your help!!!

Answer 1

I don´t know which program you have. And I cannot interpret your solutions.

I would transform the inequalities into equalities by using slack variables ($y_i$).

$h+v+c+d+y_1=972$ (Whitout the slack variable, this is also a inequality.)

If the sum of the number of chocolates is divisible by 36, then $y_1$ has to be divisible by 36:

$\frac{y_1}{36}=z_1$

$h-y_2=300$

$v-y_3=h$

$c-y_4=400$

$d-y_5=500$

The objective function is the sum of $y_i$:

$\texttt{min} \ \ y_1+y_2+y_3+y_4+y_5 $

The values of the variables have to be natural numbers:

$h,v,c,d,z_1\in \mathbb N_0$ and $y_1,y_2,y_3,y_4,y_5 \geq 0$