The definition of weak key in the question is a mixup with that of semi-weak key pair, or/and semi-weak key. Let's put these definitions straight.
We'll consider a block cipher $E$ over set $M$, with a key set $\mathcal K$.
The defining property of a weak key $K\in\mathcal K$ is: $\forall x\in M,\,E_K(E_K(x))=x$. In other words, encryption and decryption with $K$ are the same.
The defining property of a semi-weak key pair $(K_1,K_2)\in\mathcal K^2$ is: $\forall x\in M,\,E_{K_1}(E_{K_2}(x))=x$. In other words, encryption with one key in the pair is the same as decryption with the other.
The defining property of a semi-weak key $K\in\mathcal K$ is that there exists $K_2\in\mathcal K$ such that $(K,K_2)$ is a pair of semi-weak keys. In other words, the key occurs in a semi-weak key pair.
The M-DES block cipher in the question is a symmetric Feistel cipher with 4-bit message space $M$ and 4-bit key space $\mathcal K$; and a structure similar to that of DES, but simplified to caricature. In particular
- The S-boxes have 1-bit input and 1-bit output (one is bit inversion, the other has constant output 1). Complete linearity follows: within polarity, any ciphertext bit is the XOR of whatever few plaintext and key bits influence it.
- Expansion E is removed; and, beyond a mysterious and probably unrelated $|P|=2^4$, we are not told about a permutation P. Complete lack of diffusion between odd and even bits of the message space would follow that both are gone.
Reducing M-DES to equations, then applying the definition to find weak keys, is left as an exercise to the OP. A computer is not useful.