Design of a Linear Programming Model on MATLAB

Defining the Problem

I started by defining decision variables to represent the land allocation for each tree species at each site. The objective function was formulated to maximize total revenue, computed as the sum of allocated areas multiplied by their respective revenue coefficients. The model also had to satisfy key constraints, including:

• Land availability constraints, ensuring that the allocated area at each site did not exceed its total available land.

• Minimum yield constraints, ensuring that the total expected yield for each species met the given thresholds.

• Non-negativity constraints, enforcing that allocated areas could not be negative.

The Problem defined as a LP is: 

Xij: i=The index of the site  

   j=The type of tree 

Maximize 

16X11+12X12+20X13+18X14+14X21+13X22+24X23+20X24+17X31+10X32+28X33+20X34+ 12X41+11X42+18X43+17X44 

Subject To: 

1X11+1X12+1X13+1X14 ≤ 1500 

1X21+1X22+1X23+1X24 ≤ 1700 

1X31+1X32+1X33+1X34 ≤ 900 

1X41+1X42+1X43+1X44 ≤ 600 

17X11+15X21+13X31+10X41 ≥22.5 

14X12+16X22+12X32+11X42 ≥ 9 

10X13+12X23+14X33+8X43 ≥ 4.8 

9X14+11X24+8X34+6X44 ≥ 3.5 

Xij ≥ 0 

I programmed the linear programming problem using the Simplex Method in MATLAB to solve an optimization problem with multiple constraints and objectives. The code was structured into four main parts:

1. Main Code:
   The main code served as the controller, initiating the process by calling functions for Phase 1, Phase 2, and Bland’s Anti-Cycling Rule. It set up the problem, including the objective function, constraints, and starting solution, before moving through Phase 1 and Phase 2 to find the optimal solution.

2. Phase 1 Function:
   Phase 1 ensured the problem had a feasible solution by introducing artificial variables and checking for feasibility. It removed redundant constraints and evaluated termination conditions to either find a feasible solution or stop if no solution existed, before proceeding to Phase 2.

3. Phase 2 Function:
   Once feasibility was confirmed, Phase 2 started optimizing the objective function by pivoting through solutions. It checked for optimality and continued pivoting until the best possible solution was found, improving the objective value at each iteration.

4. Bland’s Anti-Cycling Rule:
   To avoid cycling, which could cause infinite loops in the Simplex Method, Bland’s Rule was implemented. It ensured that in case of ties in pivoting, the smallest-indexed variable was chosen, guaranteeing that the algorithm progressed toward the optimal solution.

By coding this process in MATLAB, I leveraged its built-in functionality and efficient handling of matrices to effectively solve the linear programming problem while avoiding cycling and ensuring the optimal outcome.