• Linear Programming Problem (LPP) Formulation:
• Objective Function: The linear mathematical function $Z = c_1 x_1 + c_2 x_2 + \dots + c_n x_n$ to be maximized or minimized.
• Decision Variables: The unknown quantities ($x_1, x_2, \dots, x_n$) whose optimal values are to be determined.
• Constraints: The system of linear inequalities or equalities ($a_{i1} x_1 + \dots + a_{in} x_n \le b_i$) representing physical, economic, or resource restrictions imposed on the decision variables.
• Conclusion: The restrictions are known as Constraints (Option 1).

Correct option: (1) Constraints

• Determining the Optimal Number of Clusters:
• In cluster analysis and principal component analysis, the Scree plot (introduced by Cattell) graphs the eigenvalues or within-cluster sum of squares (WSS) against the number of components/clusters.
• The "elbow" point where the curve flattens out (the transition from steep cliff to gentle scree) identifies the optimal number of clusters/components.
• Other Plots:
• Scatter plot: Displays bivariate relationships.
• Biplot: Superimposes observations and variables in PCA.
• Normal probability plot: Tests distributional normality.
• Conclusion: Judged based on the Scree plot (Option 1).

Correct option: (1) Sctee plot

• Comparing Before-and-After Observations:
• When observations (thrips incidence) are recorded on the exact same sampling units before and after a treatment spray, the two sets of observations are inherently correlated and non-independent.
• The appropriate statistical test is the Paired $t$-test, which analyzes the individual differences $d_i = y_{i,\text{after}} - y_{i,\text{before}}$:

• Applying an independent two-sample $t$-test or standard one-way ANOVA is invalid because it ignores intra-unit pairing and inflates the residual error variance.
• Conclusion: The best test is Paired test (Option 3).

Correct option: (3) Paired test