Using the values of variable in the objective function before optimizing

Question

I am facing a problem to multiply a term in objective function. My objective function is :

\begin{equation} \min \sum_{t=1}^\top p_{t} \cdot (1+r) \cdot y_{t} \end{equation} where $p$ is the price, $y$ is the quantity (decision variable) and $r$ is the price surcharge. $t$ is the index for time periods.

The problem is that the value of $r$ depends on between which interval the value of $y$ (decision variable) lies as shown by the following piecewise function.

\begin{equation} r_t (Y_t) = \begin{cases} % s_0 0 & \mathrm{if }\; Y_t \leq b_0 \\ % s_1 0.05 & \mathrm{if }\; b_0 < Y_t \leq b_1 \\ % s_2 0.1 & \mathrm{if }\; b_1 < Y_t \leq b_2\\[6pt] ... & \mathrm{if }\; ... \end{cases} \end{equation} where $b_0$, $b_1$ etc. are set of quantity intervals = [0, 50, 100, 150,..]

Since we don't have the values of $y$ until at the point where objective function is specified (cause value of decision variables) are only available once we give call to optimize m.optimize(). So, how we could program this in Python interface.

Answer 1

There are many ways to do this. Here is a popular one: define a binary variable $x_i$ per interval $[b_i,b_{i+1}]$ and use the following constraints:

\begin{align*} 1&=x_0+x_1+\cdots+x_n \tag{1}\\ 0x_0 + (b_0+\epsilon)x_1 + \cdots+ (b_{n-1}+\epsilon)x_n \le Y_t &\le b_0x_0 + b_1x_1 + \cdots+ b_nx_n \tag{2}\\ r_t &\ge f_i - M_i(1-x_i) \tag{3}\\ x_i &\in \{0,1\} \end{align*}

Constraint $(1)$ enforces the solver to select exactly one interval. Constraint $(2)$ enforces $x_i=1 \; \Longrightarrow \;b_{i-1} < Y_t \le b_i$. For example, if $x_1=1$, $b_0 + \epsilon \le Y_t \le b_1$ ($\epsilon$ is your tolerance). And $(3)$ is a bigM constraint which defines the piecewise linear function $r_t$ accordingly ($f_0 = 0, f_1=0.05,...$). For example, if $x_1=1$, the constraint becomes $r_t \ge f_1 = 0.05$.

Note that the above solution holds for a given $t$. You will need to replicate the idea over your time span $t=1,..,T$.