Lecture notes of the Quantative Modeling lectures given by Ipek Seyran Topan in BOM (module 2 of IBA at the UT). Includes all the useful formulas and explanations relevant for test 2. Covers lectures 7,8,9,10,11,12
Test 2 QuaMo
Lecture 7 – Modeling with linear programming
Decision variables mathematical symbols representing levels of activity of a firm.
Objective function a linear mathematical relationship describing an objective of the firm,
in terms of decision variables - this function is to be maximized or
minimized.
Constraints requirements or restrictions placed on the firm by the operating
environment, stated in linear relationships of the decision variables.
[Using ≥, ≤, = , >, < signs]
Parameters numerical coefficients and constants used in the objective function and
constraints.
• A constraint is binding if the left-hand side and the right- hand side of the constraint
are equal when the optimal values of the decision variables are substituted into the
constraint. 120 ≥ 120
• A constraint is nonbinding if the left-hand side and the right-hand side of the
constraint are unequal when the optimal values of the decision variables are
substituted into the constraint. 130 ≥ 120
Slack variable - Maximization
Standard form requires that all constraints be in the form of equations (equalities).
- A slack variable is added to a £ constraint (weak inequality) to convert it to an
equation (=).
- A slack variable typically represents an unused resource.
- A slack variable contributes nothing to the objective function value.
,Surplus variable - Minimization
- A surplus variable is subtracted from a ³ constraint to convert it to an equation (=).
- A surplus variable represents an excess above a constraint requirement level.
- A surplus variable contributes nothing to the calculated value of the objective
function.
Problems
Special types of problems may occur for LP problems, include those with:
- Multiple optimal solutions
- Infeasible solutions
- Unbounded solutions
Characteristics of Linear Programming problems:
- A decision amongst alternative courses of action is required.
- The decision is represented in the model by decision variables.
- The problem encompasses a goal, expressed as an objective function, that the
decision maker wants to achieve.
- Restrictions (represented by constraints) exist that limit the extent of achievement
of the objective.
- The objective and constraints must be definable by linear mathematical functional
relationships
- Proportionality - The rate of change (slope) of the objective function and constraint
equations is constant.
- Additivity - Terms in the objective function and constraint equations must be
additive.
- Divisibility - Decision variables can take on any fractional value and are therefore
continuous as opposed to integer in nature.
- Certainty - Values of all the model parameters are assumed to be known with
certainty (non-probabilistic).
,Lecture 8 – Modeling with linear programming
Linear programming: note on appropriate constraints
Standard form requires all variables in the constraint equations to appear on the left of the
inequality (or equality) and all numeric values to be on the right-hand side.
Example of Lindo output:
Blue squares give the, for us, useful information.
Sensitivity Analysis
- Sensitivity analysis determines the effect on the optimal solution of changes in
parameter values of the objective function coefficients and right-hand side (RHS)
values of constraint equations.
- Changes may be reactions to anticipated uncertainties in the parameters or to new
or changed information concerning the model. It is important to the manager who
must operate in a dynamic environment with imprecise estimates of the coefficients.
- Sensitivity analysis allows the manager to ask certain what-if questions about the
problem.
, Sensitivity range:
- The sensitivity range for an objective function coefficient is the range of values over
which the current optimal solution point, x1 and x2, will remain optimal.
Sensitivity range for the right-hand-side:
- The sensitivity range for a right-hand-side value is the range of values over which the
quantity’s value can change without changing the solution variable mix =current
basis remains optimal, including the slack variables.
Example of Lindo output:
Red squares give the, for us, useful information. Related to the previous Lindo example.
Optimality and reduced cost:
- The range of optimality for each coefficient provides the range of values over which
the current solution remains optimal.
- Reduced cost = the amount by which an objective function coefficient would have to
improve (increase/decrease for a MAX/MIN problem) before it would be possible for
the corresponding variable to assume a positive value in the optimal solution. (By
definition, reduced cost is a non-negative number!)
- Managers should focus on those objective coefficients that have a narrow range of
optimality and coefficients near the endpoints of the range.
Shadow price:
- The shadow price for the ith constraint of an LP to be the amount by which the
optimal z-value is improved (increased in a max problem and decreased in a min
problem) if the right-hand side of the ith constraint is increased by 1 (only if current
basis remains optimal).
- As the RHS increases, other constraints will become binding and limit the change in
the value of the objective function.
- The dual price/shadow price for a nonbinding constraint is 0.
- Dual price/shadow price (definition in Lindo)= the amount by which the optimal z-
value of the LP is improved if the right-hand side of the ith constraint is increased by
one unit (assuming this change leaves the current basis optimal).
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