MGMT20005 Chap.5 Linear Programming Models
Linear Programming Models
Week 5 translates a business decision into a linear optimisation model. Decision variables name controllable quantities with units and a time basis. The objective is a linear expression to maximise or minimise. Constraints translate capacity, demand, policy, balance and mix restrictions, while sign or domain restrictions define allowable values.
A valid LP uses fixed coefficients multiplied by individual decision variables; products of variables, powers and other nonlinear functions do not belong. In two variables, every constraint defines a half-plane and their common intersection is the feasible region. An optimum occurs at an extreme point when a finite feasible optimum exists.
Graphical solution compares vertices or moves an objective line in the improving direction. Solver extends the same logic to larger models using Simplex LP.
What this chapter covers
- 01
Decision variables with units and time periods
- 02
Linear objective functions and fixed coefficients
- 03
Capacity, demand, policy and balance constraints
- 04
Non-negativity and other domain restrictions
- 05
Feasible regions and extreme-point solutions
- 06
Binding constraints, slack, surplus and special cases
Worked example · free
Solve a two-product plan at the corners
- 2Axis corners are (0,0), (20,0) and (0,25). Solve the two resource boundaries together for the remaining intersection.
- 2From y = 40 minus 2x, substitute into x + 2y = 50: x + 80 minus 4x = 50, so x = 10 and y = 20.
- 2Objective values are 0, 800, 750 and 1,000 at the four corners respectively.
- 2Choose x = 10 and y = 20 with z = 1,000. Both resource constraints are binding because each left-hand side equals its RHS.
Key terms
- Decision variable
- A controllable quantity whose value is chosen by the optimisation model.
- Objective function
- A linear expression of decision variables representing the quantity to maximise or minimise.
- Constraint
- A linear equality or inequality representing a resource, requirement, balance or policy restriction.
- Feasible region
- All variable combinations that satisfy every constraint and domain restriction simultaneously.
- Binding constraint
- A constraint whose left-hand side equals its right-hand side at the solution.
- Slack
- Unused capacity in an at-most constraint, calculated as RHS minus left-hand side.
- Extreme point
- A vertex of the feasible region where a finite LP optimum can be found.
Linear Programming Models FAQ
What makes an optimisation model linear?
Every objective and constraint term uses a fixed coefficient multiplied by one decision variable. The model contains no products between decision variables, no powers of variables and no nonlinear functions. A percentage rule can remain linear when the percentage multiplies a sum and the expression is rearranged without multiplying variables together.
Why must variables include units and a time basis?
The definition controls every coefficient and constraint. Ten products per week is different from ten labour hours per day. Writing the unit and period makes dimensional checks possible, prevents unlike quantities from being added and turns the Solver values back into a usable operational recommendation.
How is a graphical LP solved?
Graph each constraint boundary, identify the correct half-plane, preserve non-negativity and find the common feasible region. Determine its extreme points and evaluate the objective at each, or move an objective-value line parallel in the improving direction. The best finite feasible contact is the optimum.
What do infeasible and unbounded mean?
Infeasible means no point satisfies all constraints together. Unbounded means the feasible region allows the objective to improve without a finite limit. Before treating either result as a business fact, check sign restrictions, inequality directions and missing upper bounds because a formulation error can create the same Solver message.
Exam move
Use a formulation checklist before any calculation. Define every variable with meaning, unit and time period. State whether the objective is maximised or minimised and verify that each coefficient uses the objective unit per variable unit. Translate one business sentence into one constraint, circle at least and at most, and read every equation back in words. Add non-negativity or integrality explicitly.
For graphical problems, compute intercepts, shade each half-plane, list every feasible vertex and evaluate the objective at all of them. Report both the decision-variable values and the objective value, then name binding constraints and slack where requested. In Solver, make objective and constraint left-hand sides transparent SUMPRODUCT formulas and choose Simplex LP only after confirming the model is linear.
Keep a separate unit ledger beside each model. Record the unit of every decision variable, objective coefficient and constraint coefficient, then verify that each complete equation balances. After solving, substitute the chosen values into every constraint and name the slack.
This routine catches plausible-looking but dimensionally impossible formulations and supplies the evidence needed for a concise managerial recommendation.
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