Modelling a fixed cost problem as an integer linear program requires

Question

BlackTom AI · Accepted Answer

Fixed costs are modeled by introducing binary (0-1) decision variables that indicate whether the fixed-cost activity is active.

Relevant concepts: integer linear programming (ILP) vs. linear programming (LP), fixed (startup or setup) costs, binary indicator variables, and linking (Big‑M) constraints. A fixed cost applies only when a specific decision is taken (for example, opening a facility). To represent that "pay fixed cost f only if activity occurs," we need a binary variable y ∈ {0,1} where y = 1 means the activity is turned on and y = 0 means it is off. The objective includes f·y plus any variable costs. A linking constraint (e.g., x ≤ M·y) ensures that continuous/activity variables x can be positive only when y = 1.

Step-by-step reasoning and formulation:
1. Let x ≥ 0 be the continuous flow or production quantity; let y ∈ {0,1} indicate whether the facility or activity is used.
2. Objective: minimize total cost = variable_cost_per_unit · x + fixed_cost · y.
3. Linking constraint: x ≤ M · y, where M is an upper bound on x (e.g., maximum capacity). If y = 0, x ≤ 0 so x = 0; if y = 1, x can take its feasible values.
4. Solve as an ILP so y remains integer. This models the conditional nature of fixed costs correctly.

Why the correct answer is "using 0-1 variables.": Without binary variables, the objective cannot conditionally charge the fixed cost only when the activity occurs. Simply adding fixed costs to the objective without a binary switch would charge them always or require non-linear expressions. Multiple-choice constraints are unrelated structural constraints; LP relaxation removes integrality and would fail to enforce the on/off nature. Therefore the correct modeling device is 0-1 (binary) variables, matching the provided answer "using 0-1 variables."

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