## 5 BLMOPP with Multi-Choices Interval-Type  
In many real-life problems, there may be a situation where the parameters of the optimization problem are generally not precisely known but can be defined in interval multi-choices. Moreover, instead of a single choice, it may be possible that there are many choices associated with the parameters of the problem. In such a situation, the decision-maker may be confused in selecting the proper choice of the parameters of the problem. To overcome the complexities of the problem, we use the multi-choices interval form. Here we are discussing four different cases with interval-type multi-choices parameters, which follow:  
**Case I. ** In this case, we have considered a BLMOPP in which all cost coefficients \(C_{lmj}^{k}(k\ =\ 1,2,\ldots K;l\ =\ 1,2;m\ =\ 1,2,\ldots M;j\ =\ 1,2,\ldots J)\) are multi-choice interval type. Then the model is;  
\[\underset{x_{1}}{Max}\,Z_{lm}(X))\ =\ \underset{j\ =\ 1}{ \overset{J}{\sum}}\ (C_{lmj}{}^{(1)},C_{lmj}{}^{(2)},\ldots,C_{lmj}{}^{(k)})x_{j}, \tag{5}\] \[\underset{x_{2}}{Max}\,Z_{lm}(X))\ =\ \underset{j\ =\ 1}{ \overset{J}{\sum}}\ (C_{lmj}{}^{(1)},C_{lmj}{}^{(2)},\ldots,C_{lmj}{}^{(k)})x_{j}\] \[k\ =\ 1,2,\ldots,K,\ l\ =\ 1,2,\ m\ =\ 1,2,\ldots,M;\ j\ =\ 1,2,\ldots,J\] \[\underset{j\ =\ 1}{\overset{n}{\sum}}\ a_{ij}x_{j}\leq b_{i}\,i\ =\ 1,2,\ldots,I;\ j\ =\ 1,2,\ldots,J\] \[x_{j}\geq 0,\]  
where \(C_{lmj}\) are in the multi-choice interval type from \(k\) = 1,2,...,\(K\). **Case II. ** In this case, we have considered a BLMOPP where, \(a_{ij}\) (\(i\) = 1 2,...,\(I;j\) = 1,2,...,\(J\)) are in the multi-choice interval types. After that, the model is rewritten as:\[ \begin{array}{c}Max_{x_{1}}Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj})x_{j},\\ Max_{x_{2}}Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj})x_{j}\\ l\ =\ 1,2;\ m\ =\ 1,2,\ldots,M;\ j\ =\ 1,2,\ldots,J\\ \sum\limits_{j\ =\ 1}^{J}\left(a_{ij}{}^{(1)},a_{ij}{}^{(2)},\ldots,a_{ij}{}^{(k)} \right)x_{j}\leq(b_{i}),\ i\ =\ 1,2,\ldots,I;\ j\ =\ 1,2,\ldots,J\end{array}  \tag{6}\]  
where \(a_{ij}\) are in the multi-choice interval type from \(k\) = 1,2,...,\(K\). **Case III. ** In this case, we have considered a BLMOPP where \(b_{i}(i\ =\ 1,2,\ldots,I)\) are in multi-choice interval type. The model is further rewritten as:  
\[ \begin{array}{c}Max_{x_{1}}\ Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj})x_{j},\\ Max_{x_{2}}\ Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj})x_{j}\\ l\ =\ 1,2;\ m\ =\ 1,2,\ldots,M;\ j\ =\ 1,2,\ldots,J\\ \sum\limits_{j\ =\ 1}^{J}\left(a_{ij}\right)x_{j}\leq\left(b_{i}{}^{(1)},b_{i}{}^{(2)}, \ldots,b_{i}{}^{(k)}\right),\ i\ =\ 1,2,\ldots,I;\ j\ =\ 1,2,\ldots,J\end{array}  \tag{7}\]  
where \(b_{i}\) are in the multi-choice interval type from \(k\ =\ 1,\ 2,\ldots,\ K\). **Case IV. ** In the final case, we consider a BLMOPP where all the parameters of the problem \((C_{lmj},a_{ij},b_{i})\) are in the multi-choice interval type. The mathematical model of the problem given by:  
\[ \begin{array}{c}Max_{x_{1}}\ Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj}{}^{(1)},C_{lmj}{}^{(2)},\ldots,C_{lmj}{}^{(k)})x_{j},\\ Max_{x_{2}}\ Z_{lm}(X))\ =\ \sum\limits_{j\ =\ 1}^{J}(C_{lmj}{}^{(1)},C_{lmj}{}^{(2)}, \ldots,C_{lmj}{}^{(k)})x_{j}\\ k\ =\ 1,2,\ldots,K,\ l\ =\ 1,2,\ m\ =\ 1,2,\ldots,M;\ j\ =\ 1,2,\ldots,J\\ \sum\limits_{j\ =\ 1}^{J}\left(a_{ij}{}^{(1)},a_{ij}{}^{(2)},\ldots,a_{ij}{}^{(k)} \right)x_{j}\leq\left(b_{i}{}^{(1)},b_{i}{}^{(2)},\ldots,b_{i}{}^{(k)}\right),\ i\ =\ 1,2,\ldots,I;\ j\ =\ 1,2,\ldots,J\end{array}  \tag{8}\]  
where \(k\), (\(k\) = 1,2,...,\(K\)) is the number of alternatives choices for the cost coefficient \(C_{lmj}(l\ =\ 1,2;m\ =\ 1,2,\ldots,M;\ j\ =\ 1,2,\ldots,J)\), number of alternatives choices for the \(a_{ij}(i\ =\ 1,2,\ldots,q;\ j\ =\ 1,2,\ldots,n)\) and number of alternatives choices for them \(b_{i}(i\ =\ 1,2,\ldots,I)\). In all these four types of cases, some of the parameters of the problem are in the multi-choice interval type. For each multi-choice parameter, the feasible region will be different.