# Formulating a coefficient from a table

We have a table of data as follows $$\begin{array}{c|c|c|c|c} & b \leqslant a & a < b \leqslant 2a & 2a < b \leqslant 3a & 3a < b \leqslant 4a \\ \hline i=1 & e & 2d & 2d & 2d \\ \hline i=2 & - & e & d & d \\ \hline i=3 & - & - & e & \frac{2}{3}d \\ \hline i=4 & - & - & - & e \\ \hline \end{array}$$

The data inside every cell of this table corresponds to a parameter of the problem which we show by $$c_i$$.

How can we formulate $$c_i$$ without adding another index to $$c$$?

There seems to be a connection between the value of $$i$$ and the interval to which $$b$$ belongs (the coefficient of $$a$$) but I don't know how to model it.

EDIT

I think I should have mentioned that $$a, b, e, d$$ and $$c_i$$ are all parameters of the problem.

$$z_{j} \in \{0,1\}$$ equals 1 represents variable $$b$$ belongs to interval $$j$$. You can linearize interval relation as follows: $$A_{j-1} \cdot z_{j} + \epsilon \cdot z_{j} \le b_{j} \le A_{j} \cdot z_{j} \quad \forall j \in {1,2,3,4}$$ Here, this leads to 4 binary variables $$z_{j}$$ and values of $${A_0}$$,$${A_1}$$,$${A_2}$$,$${A_3}$$,$${A_4}$$ are $$0$$, $$a$$, $$2a$$, $$3a$$ and $$4a$$ respectively (assuming 0 to be lower bound of variable $$b$$). $$\epsilon$$ is introduced on LHS to handle strict less than condition.
variable $$b$$ is to be defined as $$b = \sum_{j} b_{j}$$ and only one of the intervals has to be selected $$\sum_{j} z_{j} = 1$$ To achieve above linearization, you might consider using SOS Type-1 relation as well. If a is also a variable, you might have to write some Big-M constraints to linearize the same. finally, $$c_{i}$$ can be defined as follows: $$c_{i} = \sum_{j\mid j\ge i} W_{ij} z_{j}$$ where $$W_{ij}$$ represents tabular parameters corresponding to row $$i$$ and column $$j$$
• Thank you. As I mentioned in the edited question, $a,b,c,d,e$ are all parameters. Based on the problem instance, we know which column (interval) $b$ belongs to. Can't we simplify $c_i$ formulation? Jan 18, 2022 at 23:11
• That is correct. Both $c$ and $b$ are known in advance. $c_i$ is the coefficient of a decision variable and we do look up the values in the table but I was thinking about how to write it because for each $i$, for any interval of $b$, when $i$ is equal to the coefficient of $a$ on the RHS, $c_i=e$ but when $i$ is less than the coefficient of $a$, $c_i=\frac{2}{\text{coefficient of$a$on the LHS}}$. But I don't know to write this mathematically. Jan 19, 2022 at 9:46
• Can you tell the usage of $c_i$? Is it part of any optimization model? Jan 21, 2022 at 4:18