If the phrase is extended as "constrained by reference to" then, for example, you may refer to page 28 (page 30 in the pdf file, second paragraph) in the following link

https://iiif.library.cmu.edu/file/Cooper_box00005_fld00022_bdl0001_doc0001/Cooper_box00005_fld00022_bdl0001_doc0001.pdf

where the phrase is used with respect to the linear model (26), its dual (27), and the refinement (29).

In linear programming, "constrained by reference" isn't a standard or widely recognized term. However, it seems to allude to how constraints in a linear program relate to and are defined by the problem's variables and their relationships. Let's break down the key aspects of constraints in linear programming to clarify this:

1. What are Constraints?

• Restrictions: Constraints are limitations or restrictions on the decision variables in a linear programming problem. These variables represent the quantities you can control to achieve your objective (e.g., how many units to produce, how much to invest).
• Mathematical Expressions: Constraints are expressed as linear equations or inequalities involving the decision variables. They define the feasible region, which is the set of all possible solutions that satisfy all the constraints.

2. How Constraints are Defined

• Problem Context: Constraints arise from the real-world limitations of the problem you're modeling. These could be:Resource Availability: Limited raw materials, labor hours, or machine capacity. Production Requirements: Minimum or maximum production levels, quality standards. Financial Limits: Budget constraints, investment limits.
• Variable Relationships: Constraints establish relationships between the decision variables. For example:x + y ≤ 100 (The sum of products x and y cannot exceed 100 units) 2x + 3z ≥ 50 (A combination of products x and z must meet a minimum requirement)

3. The Role of "Reference"

It appears "constrained by reference" is highlighting that constraints are not arbitrary; they are "referenced" or determined by:

• The problem's variables: Constraints are always expressed in terms of the decision variables you're trying to optimize.
• The relationships between variables: Constraints capture how the variables interact and the limits on those interactions.

In essence: Constraints in linear programming are always defined with respect to the variables and the context of the problem. They "refer" to these elements to establish the feasible region and guide the optimization process.

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