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I have to formulate (linearly) the following problem mathematically:


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What I tried:


1. Variables

Let $x_{ijk} = 1$ if, in month k, product i should be made in production line j, where $i=1-3$, $j=1-2$ and $k=1-6$.

Let $d_{ik}$ represent how much of product i should be made in month k.

Let $c_{ij}$ represent unit production cost of producing product i in production line j.

Let $p_{ij}$ represent production rate of producing product i in production line j.

Let $s_{ij}$ represent cost of switching...to product i? or from?


2. Objective Function

Min

$$z = 0.5\sum_{k=1}^{6} x_{1,j,k} + 0.35\sum_{k=1}^{6} x_{2,j,k} + 0.45\sum_{k=1}^{6} x_{3,j,k} + \sum_i \sum_j \sum_k c_{ij} p_{ij} x_{ijk}$$

$$- \sum_i \sum_j \sum_k s_{ijk} y_{ijk} x_{ijk}$$

where $y_{ijk} = 1$ if, in month k, production line j switches (to or from?) product i and $0$ otherwise.

Or is that the other way around?


3. Constraints:

For all k, all j and all i,

$$\text{Constraints} \ 1-36: \sum_{m=1}^{k} p_{ij}x_{ijm} - d_{i,0} \ge \sum_{m=1}^{k} d_{im}$$

$$\text{Constraints} \ 37-72: x_{ijk} \le My_{ijk}$$

where $d_{i,0}$ represents how much of product i is in the initial inventory.


Where did I go wrong?


Other questions:


  1. To what is $s_{ijk}$ supposed to refer? Cost of switching to product i? from?

Does it matter?

  1. Is my $y_{ijk}$ wrong?

From Chapter 3 here.

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