Ch 12 - Linear Programming12th Mathematics
Definition: Linear Programming Problem (LPP)
What mathematical components and constraints formally define a Linear Programming Problem?
What mathematical components and constraints formally define a Linear Programming Problem?
A Linear Programming Problem (LPP) seeks to strictly optimise (maximise or minimise) a linear function of variables subject to a set of linear inequalities and non-negative restrictions.
- Used for finding maximum profit, minimum cost, or optimal resource allocation.
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Linear Objective Function
Linear Objective Function
State the general algebraic form of the objective function in a two-variable LPP.
Derivation
Constructed by summing the individual unit contributions ( and ) multiplied by their respective quantities ( and ).
Component Anatomy:
- : The quantity to be optimised (e.g., profit or cost).
- : Known constant real coefficients.
- : Decision variables.
Boundary Conditions / Domain:
- The function MUST be strictly linear (degree 1).
- Expressions like , , or are strictly prohibited.
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Decision Variables in LPP
In the context of LPP, what do the variables and represent, and what are they formally called?
In the context of LPP, what do the variables and represent, and what are they formally called?
In the objective function , the unknowns and are called decision variables.
- They represent the actual physical quantities the decision-maker has control over (e.g., number of chairs to build, kg of food to buy).
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Also in this chapter
28 more cards to study in Prism
- 4The Non-Negative Restriction Trap
- 5Translating Constraints: 'At Most' vs 'At Least'
- 6LPP Formulation Strategy (DOC-N)
- 7Tabular Presentation of Formulation Data
- 8Diet Problem Archetype
- 9Manufacturing Problem Archetype
- 10Geometric Definition: Feasible Region
- 11Convexity of the Feasible Region
- 12Feasible vs. Infeasible Solutions
- 13Empty Feasible Region (Infeasible Problem)
- 14The Origin Test for Shading Inequalities
- 15Fundamental Theorem of LPP (Theorem 1)
- 16Guarantee of Optima (Theorem 2)
- 17Corner-Point Method (CPM) SOP
- 18Exact Vertex Calculation via System of Equations
- 19The Integer Solution Rounding Trap
- 20The Origin Minimum Shortcut Trap
- 21The Unbounded Region Uncertainty
- 22Verifying Maximum in Unbounded Regions
- 23Verifying Minimum in Unbounded Regions
- 24Open Half-Plane Intersection Trap
- 25Multiple Optimal Solutions (Tied Corners)
- 26Slope Condition for Multiple Optimal Solutions
- 27Objective Parameter Equality (MCQ Trap)
- 28Geometric Trap: Redundant Constraints
- 29Objective Function Level Curves
- 30Iso-Profit Method SOP (Maximisation)
- 31Iso-Cost Method SOP (Minimisation)
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