# if else statement linear programming

I am trying to work out an if else statement for the following problem, which should be mathematically linear programmed:

when both item 1 and item 2 are picked, both their costs are reduced with 20%. I have more items than only item 1 and 2, and there costs never change. if only one of item 1 or 2 are picked, there regular costs will be implemented.

I thought of the following, however I don't know how to rewrite it in a way it is mathematical and thus not contains any ifs and whens:

$$C_i$$: reduced cost of item $$i$$ for $$i \in \{1,2\}$$

$$K_i$$: regular cost of item $$i$$ for $$i \in \{1,2,3,4\}$$

$$X_i$$: whether or not item $$i$$ is picked for $$i \in \{1,2,3,4\}$$

the sum of $$X_i C_i$$ should be smaller than or equal to $$1300$$

$$C_1 = 0.8K_1$$ if $$X_1+X_2=2$$

$$C_2= 0.8K_2$$ if $$X_1+X_2=2$$

$$C_i=K_i$$ for $$i\in \{3,4\}$$

Is there anyone who can help me with this?

• For some basic information about writing mathematics at this site see, e.g., here, here, here and here. Commented Jun 14, 2022 at 9:43

Normally in these cases we use an auxiliar variable that takes values $$\{0,1\}$$. Thus we add the variable $$\delta \in \{0,1\}$$ that is $$1$$ if both items are bought and $$0$$ otherwise. Add the corresponding price to the objective function times $$\delta$$ and you should be done.
I agree with @Gowexx that you should use a binary variable $$Y_{1,2}$$. To enforce this, try using these constraints:
$$Y_{1,2} \ge (X_1 + X_2)-(Y_1 + Y_2)$$ $$Y_{1,2} \ge X_1 - Y_1$$ $$Y_{1,2} \ge X_2 - Y_2$$ $$|Y_1-X_2| =0$$ $$|Y_2-X_1| =0$$ $$Y_1,Y_2,X_1,X_2 \in \{0,1\}$$ $$Y_{1,2} \in \{0,2\}$$
$$Y_1$$ and $$Y_2$$ are used to account for one of the $$X$$'s being $$0$$ and the other being $$1$$. If one of the $$X$$'s is not turned on, the corresponding $$Y$$ will turn on and turn the RHS of Constraint 1 to 0. If both are turned on, RHS should be 2.
\begin{align} \sum_{i=1}^4 K_i X_i - 0.2(K_1+K_2)Y &\le 1300 \\ Y &\le X_1 \\ Y &\le X_2 \end{align}