First, let's recap your OPL model for clarity:

dvar float+ x; /* daily quantity of t-shirts*/
dvar float+ y; /* daily quantity of shorts */
maximize 6*x + 10*y; /* maximize the daily profit from t-shirts and shorts*/
subject to{
20/60*x + 10/60*y <=176; /* cutting department labour constraint*/
20/60*x + 50/60*y <=400; /* sewing department labour constraint*/
10/60*x + 10/60*y <= 96; /* packaging department labour constraint*/
x >= 100; /* minimum daily demand for t-shirts*/
}

Part 1: Manual Calculation of Optimal Coefficient Ranges #

First, solving your LP gives the optimal solution: x=160 t-shirts, y=416 shorts. At this point, the sewing and packaging constraints are binding (they hit their maximum limits), while the cutting constraint has unused capacity and the minimum t-shirt demand is exceeded.

For this optimal solution to stay unchanged (i.e., we keep producing 160 t-shirts and 416 shorts), the unit profit coefficients need to stay within a range where the current binding constraints still define the maximum profit point. Here's how to calculate those ranges:

Range for T-shirt Unit Profit (c_x, originally 6) #

• If we decrease c_x too much, the optimal solution switches to producing the minimum required t-shirts (100) and more shorts (440). Setting the profit of both points equal gives us the lower bound: c_x = 4.
• If we increase c_x too much, the optimal solution switches to producing more t-shirts (480) and fewer shorts (96). Setting profits equal here gives us the upper bound: c_x ≈ 8.67 (or exactly 26/3).

So the valid range for c_x is [4, 8.67].

Range for Shorts Unit Profit (c_y, originally 10) #

• If we decrease c_y too much, the optimal solution switches to producing more t-shirts (480) and fewer shorts (96). Setting profits equal gives the lower bound: c_y =6.
• If we increase c_y too much, the optimal solution switches to producing the minimum t-shirts (100) and more shorts (440). Setting profits equal here gives the upper bound: c_y≈10.27 (or exactly 113/11).

So the valid range for c_y is [6, 10.27].

Part 2: Viewing Sensitivity Analysis in ILOG CPLEX Optimization Studio #

You don't have to calculate these ranges manually—CPLEX can generate them for you directly. Here's how:

Step 1: Enable Sensitivity Analysis in Run Configurations #

1. Click the dropdown next to the green Run button and select Edit to open the run configuration.
2. Navigate to the CPLEX tab in the configuration window.
3. Under the Solution section, check the box labeled Sensitivity analysis (you can choose to include both variable and constraint sensitivity data).
4. Click Apply then re-run your model.

Step 2: Access the Results #

Once the model solves again, you can view the sensitivity data in two places:

• Output Tab: Look for a section titled Objective Coefficient Ranges—it will list each variable's current coefficient, allowable increase/decrease, and the full valid range.
• Solution Explorer: Expand the Solution node, then go to Sensitivity → Objective Coefficients to see a formatted table with the ranges for x and y.

Example CPLEX Output #

You’ll see results matching our manual calculation:

Variable x:
  Current coefficient: 6.0
  Allowable increase: 2.6667 (to 8.6667)
  Allowable decrease: 2.0 (to 4.0)

Variable y:
  Current coefficient:10.0
  Allowable increase:0.2727 (to10.2727)
  Allowable decrease:4.0 (to6.0)

内容的提问来源于stack exchange，提问作者Charuka Deshan