1. LINEAR PROGRAMMING

2. Introduction QVery efficient solution procedure: simplex method.
QDevelopment of linear programming was among the most important scientific advances of mid‐20th cent.
QMost common type of applications: allocate limited resources to competing activities in an optimal way.
QLinear programming uses a mathematical model.
Linear because it requires linear functions.
Programming as synonymous of planning.
QVery efficient solution procedure: simplex method.

3. Prototype example
QWyndor Glass Co. produces glass products, including windows and glass doors.
QPlant 1 produces aluminum frames, Plant 2 produces wood frames and Plant 3 produces glass and assembles.
QTwo new products:
Glass door with aluminum framing (Plant 1 and 3).
New wood‐framed glass window (Plant 2 and 3).
QAs the products compete for Plant 3, the most profitable mix of the two products is needed.

4. Formulation of the LP problem
Q x1 = number of batches of doors produced per week Q x2 = number of batches of windows produced per week Q Z = total profit per week (in thousands)
maximize subject to:
Z = 3x1 + 5x2
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0.
Wyndor Glass Co. Product‐Mix Problem
Doors Windows
Profit Per Batch $3.000 $5.000 (Batches of 20)
Hours availabl
Hours Used Per Batch Produced
per week
Plant 1 1 3 2 4 12 18
Plant 2
Plant 3

5. Graphical solution Q Slope of the line is –3/5
Z = 3x1 + 5x2
3 1
x2 = − x1 + Z
5 5
Q Slope of the line is –3/5
Q Trial‐and‐error: Z = 10, 20, 36
(y = mx + b)
Feasible region: satisfies all constraints
maximize subject to:
Z = 3x1 + 5x2
x1 ≤ 4 2x2 ≤ 12
3x1 + 2x2 ≤ 18
and x1 ≥ 0,
x2 ≥ 0.

6. Excel Solution Wyndor Glass Co. Product‐Mix Problem Doors Windows
Profit Per Batch $3.000 $5.000
Range Name
Cells
BatchesProduced HoursAvailable HoursUsed
C12:D12 G7:G9 E7:E9
Hours
Hours
Hours Used Per Batch Produced Used
Available
Plant 1 1 2
<= 4
HoursUsedPerBatchProduced
C7:D9
Plant 2 12
ProfitPerBatch
C4:D4
Plant 3 3 18
TotalProfit
G12
Doors
Windows
Total Profit
Batches Produced 2 6
$36.000

7. Matlab solution
All solvers are designed to minimize the objective function in the optimization toolbox. To maximize f(x), minimize –f(x).
solution
João Miguel da Costa Sousa /

8. Generalizing the example
Prototype example
General problem
Production capacities of plants 3 plants
Production of products 2 products
Production rate of product j, xj
Resources m resources Activities n activities
Level of activity j, xj
Profit Z
Overall measure of performance Z
The most common type of application of LP involves
allocating resources to activities.

9. Linear programming model
QZ = value of overall performance measure.
Qxj = level of activity j, for j = 1, 2, …, n.
Qcj = parameters of Z related to xj.
Decision variables
Qbj = amount of resource i that is available for allocation of activities, for i = 1, 2, …, m.
Qaij = amount of resource i consumed by each unity of activity j.
Parameters

10. Standard form of the model
Select the values for x1, x2, …, xn to
maximize:
Z = c1 x1 + c2 x2 +…+ cn xn
subject to:
a11 x1 + a12 x2 +…+ a1n xn ≤ b1
a21 x1 + a22 x2 +…+ a2n xn ≤ b2
… am1 x1 + am2 x2 +…+ amn xn ≤ bm x1 ≥ 0, x2 ≥ 0,…, xn ≥ 0.
and

11. Allocation of resources to activities
Resource Usage per Unit of Activity
Activity
Amount of
Resource 1 2 … n
Resource available 1
a11
a12 …
a1n b1 2
… m
a21 …
am1
a22 …
am2 …
a2n …
amn b2 … bm
Contribution to Z per unit of activity c1 c2 … cn

12. Terminology for solutions
QSolution: any specification of values for the decision variables (x1, x2,…, xn).
QFeasible solution: solution for which all the
constraints are satisfied.
QInfeasible solution: solution for which at least one constraint is violated.
QFeasible region: collection of all feasible solutions.
QNo feasible solution: no solution satisfies all constraints.

13. Terminology for solutions
QOptimal solution: feasible solution with the most favorable value of the objective function.
QMost favorable value: largest value if maximizing or
smallest value if minimizing.
QNo optimal solutions: (1) no feasible solutions or,
(2) unbounded Z (see next slide).
QMore than one solution: see next slide.

14. Examples of solutions Q More than one solution
Q Unbounded cost function

15. Corner‐point feasible solution
Q Solution that lies at a corner of feasible region. Best CPF solution must be an optimal solution..

16. Assumptions of linear programming
QProportionality: the contribution of each activity to the value of the objective function Z and left‐hand side of each functional constraint is proportional to the level of activity xj.
QViolations:
1) Setup costs.

17. Violations of proportionality
2) Increase of marginal return due to economy of scale (longer production runs, quantity discounts, learning‐curve effect, etc.).

18. Violations of proportionality
3) Decrease of marginal return due to e.g. marketing costs: to sell more the advertisement grows exponentially.

19. Assumptions of linear programming
QAdditivity: every function in a linear programming model is the sum of the individual contributions.
QDivisibility: decision variables can have any values.
If values are limited to integer values, the problem needs an integer programming model.
QCertainty: parameters of the model cj, aij and bi are assumed to be known constants.
In real applications parameters have always some degree of uncertainty and sensitivity analysis must be conducted (identify sensitive parameters).

20. The assumptions in perspective
QMathematical model is idealized representation of real problem.
QApproximations and simplifying assumptions are required for the model to be tractable.
QA reasonably high correlation between prediction of model and reality is required.
QIt is common in real applications of LP that almost none of the 4 assumptions holds completely.
QIt is important to analyze how large the disparities are, and maybe use alternative models.

21. Conclusions QLinear programming models can be solved using:
Spreadsheet as Excel Solver for relatively simple problems
Problems with thousands (or even millions!) of decision variables and/or functional constraints must use modeling languages such as CPLEX/MDL and LINGO/LINDO.
Matlab Optimization toolbox (GUI or code).
QLinear programming solves many problems.
QHowever, when one or more assumptions are violated seriously, other programming techniques are needed.

22. SIMPLEX METHOD

23. Example revisited
QWyndor Glass Co.

24. Setting up the simplex method
Q Original form
Q Augmented form
Maximize
subject to
Z = 3x1 + 5x2
Maximize
subject to
Z = 3x1 + 5x2
x1 ≤ 4 2x2 ≤ 12
3x1 + 2x2 ≤ 18
and x1 ≥ 0, x2 ≥ 0.
(1)
(2)
(3)
x1 + x3 = 4
2x2 + x4 = 12
3x1 + 2x2 + x5 = 18
and x ≥ 0, j = 1,2,3,4,5. j
Convert functional inequality constraints to equivalent equality constraints using slack variables

25. Setting up the simplex method
QSlack variable is:
Zero: solution lies on constraint boundary;
Positive: solution lies on feasible side of constraint;
Negative: solution lies on infeasible side of constraint
QExample problem has 5 variables and 3 nonredundant equations, and so 2 (5 – 3) degrees of freedom.
QTwo variables have an arbitrary value. Simplex uses the zero value for these nonbasic variables.
QThe solution for the other three variables (basic variables) is a basic solution.
QBasic solutions have thus a number of properties.

26. Properties of a basic solution
Q Each variable is basic or nonbasic.
Q Number of basic variables is equal to number of functional constraints (equations).
Q The nonbasic variables are set to zero.
Q Values of basic variables are obtained solving system of equations (functional constraints in augmented form).
Q Set of basic variables are called the basis.
Q If basic variables satisfy the nonnegativity constraints, basic solution is a BF solution.

27. Final form
QConsider the objective function as the new constraint equation:
Maximize Z
subject to (0)
(1)
Z − 3x1 − 5x2 = 0
x1 + x3 = 4
(2)
(3)
2x2 + x4 = 12
3x1 + 2x2 + x5 = 18
and x j ≥ 0, j = 1,…,5.

28. Tabular form (simplex tableau)
(a) Algebraic form
(b) Tabular Form
Coefficient of:
Z x1 x2 x3 x4 x5
Basic variable
Right side
Eq.
(0) Z – 3x1 – 5x2
= 0 Z
(0)
1 ‐3 ‐
(1) x1
+ x3
= 4 x3
(1) 4
(2)
2x2 + x4
= 12 x4
(2) 12
(3) 3x1 + 2x2
+ x5 = 18 x5
(3) 18

29. Simplex method in tabular form
QInitialization. Introduce slack variables. Select decision variables to be initial nonbasic variables (=0) and slack variables to be initial basic variables.
❖Initial solution of example: (0, 0, 4, 12, 18)
QOptimality test. Current BF solution is optimal iff every coefficient in row (0) is nonnegative. If yes, stop; otherwise, iterate to obtain the next BF solution.
❖Example: coefficients of x1 and x2 are –3 and –5.

30. Simplex method in tabular form
QIteration.
Step 1: Determine the entering basic variable, which is the nonbasic variable with the largest negative
coefficient.
pivot column
entering basic variable
Coefficient of:
Z x1 x2 x3
‐5 0
0 1
2 0
Basic variable
Right side
Eq.
x4 x5 Z
(0) 1 ‐3
x3 (1) 0 1
x4 (2) 0 0 4 1 12 x5
(3) 3 2 1 18

31. Iteration
Step 2: Determine the leaving basic variable, by applying minimum ratio test:
Coefficient of:
Basic variable
Eq.
Right side Z
x1 x2
‐3 ‐5 0
x3 x4 x5 Z
(0) 1
0 0
pivot row
x3 x4 x5
(1) 1 1
0 4
(2) 2
→ 12/2 = 6 → minimum
(3) 3
1 18 → 18/2 = 9
pivot number (PN)
leaving basic variable
replace leaving BV for entering BV in the BV column of the next simplex tableau

32. Iteration
QStep 3: solve for the new BF solution by using elementary row operations.
Iteration Basic Eq. variable Z
Coeffici
x1 x2
ent o x3 f: x4
Rig
x5 sid
Z (0) 1 ‐3 ‐5
0 0
0 x3 (1)
0 4
x4 (2)
x5 (3)
(0)
2.5 30
(1) 4 x2
(2)
0.5 6 x5
(3) 3 ‐1
ht e
→ row with negative coef.
→ row with positive coef.
÷2 (PN) 2

33. Iteration 2: Steps 1 and 2
Optimality test: solution still not optimal → one more iteration…
Coefficient of:
Basic variable
Right side
Iteration
Eq.
Ratio Z x
x2 x3 x
0 1
0 0 x 1 4 5 Z
(0) 1 ‐3
x3 x2 x5 1
(1) 1
4 → 4/1 = 4
(2) 3 6
(3) ‐1 1
6 → 6/3 = 2
minimum

34. Complete set of simplex tableau
Coefficient of:
Right
Iteration
Basic variable
Eq.
side Z x
x2 x3 x x 1 4 5
‐3 1 3 Z
x3 x4 x5
(0)
(1)
(2)
(3) 1 ‐5 1 1 1 4 12 18 2
‐3 1 Z
x3 x2
(0)
(1)
(2) 1 1 1
2.5
0.5 30 4 6 1 3 x5
(3) ‐1 1 6 Z
x3 x2 x1
(0)
(1)
(2)
(3) 1 1 1 1
1.5 1
1/3 ‐1/3
0.5 0
‐1/3 1/3 36 2 6 2

35. Tie breaking in simplex method
QTie for entering basic variable:
arbitrary choice;
QTie for leaving basic variable:
QNo leaving basic variable:
unbounded Z, i.e., error in model or computation;
QMultiple optimal solutions:
simplex finds one optimal BF solution; find the others by continuing the algorithm choosing a nonbasic variable with a zero coefficient as the entering BV.

36. Adapting to other model forms
QEquality constraints
There is no obvious initial BF solution, so artificial variables and the Big M method are used.
QNegative right‐hand sides:
Multiply inequality by ‐1;
QFunctional constraints in ≥ form
Surplus and artificial variables / big M method are used.
QMinimization
Maximize –Z.

37. Artificial variables and Big M method
Q The real problem Q The artificial problem
Define x5 = 18 − 3x1 − 2x2 .
Maximize
subject to
Z = 3x1 + 5x2
Maximize Z = 3x1 + 5x2 − Mx5 ,
subject to
(1)
(2)
(3)
x1 ≤ 4 2x2 ≤ 12
3x1 + 2x2 = 18
x ≤ 4 1
2x ≤ 12 2
3x1 + 2x2 ≤ 18
and x1 ≥ 0,
x2 ≥ 0.
(so 3x1 + 2x2 + x5 = 18)
and x1 ≥ 0, x2 ≥ 0, x5 ≥ 0.
from equation (0) for proper form.
Algebraically eliminate x5

38. Postoptimality analysis
QReoptimization: when the problem changes slightly, new solution is derived from the final simplex tableau.
QShadow price for resource i (denoted yi ) measures *
the marginal value of this resource, i.e., how much Z can be increased by (slightly) increasing the amount of this resource bi.
❖Example: recall bi values of Wyndor Glass problem.
The final tableau gives y1* = 0, y * = 1.5, y * = 1.
2 3

39. Graphical check
Q Increasing b2 in 1 unit, the new profit is 37.5

40. Shadow prices QIncreasing b2 in 1 unit increases the profit in $1500.
Should this be done? Depends on the marginal profitability of other products using that hour time.
QShadow prices are part of the sensitivity analysis.
QConstraint on resource 1 is not binding the optimal solution. There is a surplus of this resource. Resources as this are free goods.
QConstraints on resources 2 and 3 are binding constraints. They have positive shadow prices and no surplus. Resources are scarce goods.

41. Sensitivity analysis
QIdentify sensitive parameters (those that cannot be changed without changing optimal solution).
Parameters bi can be analyzed using shadow prices.
Parameters ci for 2 variables can be analyzed graphically.
Parameters aij can also be analyzed graphically.
QFor problems with a large number of variables usually only bi and ci are analyzed, because aij are determined by the technology being used.

42. Sensitivity analysis of parameters ci
Q Graphical analysis of Wyndor example:
c1 can have values in [0, 7.5] without changing optimal solution.
c2 can have any value greater than 2 without changing optimal solution.

43. Theory of simplex method
QRecall the standard form:
Maximize Z = c1 x1 + c2 x2 +…+ cn xn
subject to
a11 x1 + a12 x2 +…+ a1n xn ≤ b1
a21 x1 + a22 x2 +…+ a2n xn ≤ b2 …
am1 x1 + am2 x2 +…+ amn xn ≤ bm
and x1 ≥ 0, x2 ≥ 0,…, xn ≥ 0.

44. Augmented form Maximize Z = c1 x1 +…+ cn xn + cn+1 xn+1 +…+ cn xn+m
subject to
a11 x1 + a12 x2 +…+ a1n xn + xn+1 = b1
a21 x1 + a22 x2 +…+ a2n xn + xn+2 = b2 …
am1 x1 + am2 x2 +…+ amn xn + xn+m = bm
j = 1,2,…,n.
and x j ≥ 0,

45. Matrix form (standard form)
Maximize Z = cx
subject to
Ax ≤ b
and x ≥ 0
c = [c1 , c2 ,…,cn ],
⎡ a11 a12 22
⎡ x1 ⎤
⎡ b1 ⎤ …
a … a
a1n ⎤
⎢ x ⎥ ⎢ b ⎥ ⎢ a ⎥
2n ⎥.
x = ⎢ 2 ⎥ , b = ⎢ 2 ⎥ , A = ⎢ 21
⎢ # ⎥
⎢ # ⎥
⎢ #
# % # ⎥
am2 …
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣xn ⎦
⎣bm ⎦
⎣am1
amn ⎦

46. ⎡ xn+1 ⎤ x = ⎢ xn+2 ⎥ ⎢ # ⎥ Matrix augmented form
QIntroduce the column vector of slack variables:
⎡ xn+1 ⎤
⎢ ⎥
x = ⎢ xn+2 ⎥ s
⎢ # ⎥
⎢ ⎥
⎣xn+m ⎦
QThe constraints are (with I (m×m) and 0 (1×(n+m)) ):
⎡x ⎤
[A,I] = b and
⎡x ⎤ ≥ 0
⎢ ⎥ ⎢ ⎥
⎣xs ⎦
⎣xs ⎦

47. Solving for a BF solution
QOne iteration of simplex:
The n nonbasic variables are set to zero.
The m basic variables are in a column vector denoted as xB.
The basic matrix B is obtained from [A, I] by eliminating the columns with coefficients of nonbasic variables.
The problem becomes: BxB = b.
The solution for basic variables and the value of the objective function for this basic solution are:
x = B−1b and Z = c x = c B−1b
B B B B

48. Revised simplex method
Initialization (as original): Introduce slack variables
xs. Decision variables are initial nonbasic variables.
Iteration:
Step 1 (as original): Determine entering basic variable (nonbasic variable with the largest negative coefficient).
Step 2: Determine leaving basic variable. As original, but computations are simplified.
Step 3: Determine new BF solution setting xB = B–1b.
Optimality test: solution is optimal if every coefficient in Z is ≥ 0. Computations are simplified.

49. Fundamental insight ⎡−c 0 0⎤ ⎢T⎥ = ⎢ A I b⎥
Initial tableau
Row 0:
t = [−c | 0 | 0]
Other rows:
T = [A |I|b]
⎡ t ⎤
⎡−c 0 0⎤
Combined:
⎢T⎥ = ⎢ A I b⎥
⎣ ⎦ ⎣ ⎦
Final tableau
Row 0:
t* = t + y * T = [y * A − c | y*| y *b],
y* = c B−1 B
Other rows:
T* = S* T = [S* A | S*|S*b],
S* = B −1
⎡ t * ⎤ ⎡y * A − c y * y *b⎤
Combined:
⎢ ⎥ = ⎢ ⎥
⎣T *⎦
S* A S* S*b⎦ ⎣

50. Fundamental insight
QOptimal performance index: Z*=y*b QOptimal solution: xB=S*b, xnB=0 QShadow prices: y*
QAfter any iteration, the coefficients of the slack variables in each equation immediately reveal how that equation has been obtained from the initial equations.

51. DUALITY THEORY AND SENSITIVITY ANALYSIS

52. Duality theory QEvery linear programming problem (primal) has a
dual problem.
QMost problem parameters are estimates, others (as resource amounts) represent managerial decisions.
QThe choice of parameter values is made based on a
sensitivity analysis.
QInterpretation and implementation of sensitivity analysis are key uses of duality theory.

53. Primal and Dual Problems
Primal problem
(standard form)
Dual problem n
Maximize Z = ∑c j x j m
Minimize W = ∑ yi bi
j =1
subject to n
i =1
subject to m
∑ yi aij ≥ c j ,
∑aij x j ≤ bi , for i = 1,2,…,m
j =1
and x ≥ 0 for j = 1 2 n
for j = 1,2,…,n
i =1
and yi ≥ 0, ,
, ,…, .
for i = 1,2,…,m. j

54. Example: Wyndor Glass Co.
Primal problem
Dual problem
⎡ 4 ⎤
Maximize Z = [3 5]⎡x1 ⎤ ,
⎢ ⎥
⎣x2 ⎦
Minimize Z = ⎡⎣ y
subject to
y y ⎤⎦ ⎢12⎥ ,
⎢ ⎥ 1
2 3
⎢⎣18⎥⎦
subject to
⎡1 0⎤ ⎡ 4 ⎤
⎢ ⎥ ⎡x1 ⎤ ⎢ ⎥
⎡1 0⎤
⎡⎣ y y y ⎦⎤ ⎢0 2⎥ ≥ [3 5
⎢0 2⎥ ⎢ ⎥ ≤ ⎢12⎥
⎢⎣3 2⎥⎦ ⎣x2 ⎦ ⎢⎣18⎥⎦ 1
2 3
⎢ ⎥
⎢⎣3 2⎥⎦
y2 y3 ⎤⎦ ≥ [ ].
and ⎡x1 ⎤ ≥ ⎡0⎤.
⎢ ⎥ ⎢ ⎥
⎣x ⎦ ⎣0⎦
and ⎡⎣ y1 2

55. Primal‐dual table for LP
Primal problem
Coefficient of:
Right side x
x … x 1 2 n
y1 y2
a11 a21
a12 a22
a1n a2n …
≤ b1
≤ b2
Coefficients of objective function (Minimize)
Coefficient of:
Dual problem … ym …
am1 …
am2 … …
amn …
≤ bm
Right side …
c1 c2 … cn
Coefficients of objective function (Maximize) ≥ ≥ ≥

56. Example: Wyndor Glass Co.
Right side
y1 y2 1 2
≤ 4
≤ 12 y3 3 2
≤ 18
Right side ≥ ≥ 3 5

57. Duality theory QGeneral relations between primal and dual problems:
The parameters for a (functional) constraint in either problem are the coefficients of a variable in the other problem.
The coefficients in the objective function of either problem are the right‐hand sides for the other problem.

58. Primal‐dual relationships
QWeak duality property: If x is a feasible solution for the primal problem and y is a feasible solution for the dual problem, then
cx ≤ yb
QStrong duality property: If x* is an optimal solution for the primal problem and y* is an optimal solution for the dual problem, then
cx* = y*b

59. Primal‐dual relationships
QComplementary solutions property: At each iteration, the simplex method identifies a CPF solution x for the primal problem and a complementary solution y for the dual problem, where
cx = yb
If x is not optimal for the primal problem, then y is
not feasible for the dual problem.

60. Primal‐dual relationships
QComplementary optimal solutions property: At the final iteration, the simplex method identifies an optimal solution x* for the primal problem and a complementary optimal solution y* for the dual problem, where
cx* = y*b
The yi* are the shadow prices for the primal problem.

61. Primal‐dual relationships
QSymmetry property: For any primal problem and its dual problem, all relationships between them must be symmetric because the dual of this dual problem is this primal problem.
QDuality theorem: The following are the only possible relationships between the primal and dual problems:
1. If one problem has feasible solutions and a bounded objective function, then so does the other problem, so both the weak and strong duality properties are applicable.

62. Primal‐dual relationships
If one problem has feasible solutions and an unbounded objective function, then the other problem has no feasible solution.
If one problem has no feasible solutions, then the other problem has either no feasible solutions or an unbounded objective function.

63. Primal‐dual relationships
Q Relationships between complementary basic solutions
Primal basic solution
Complementary dual basic solution
Both basic solutions
Primal feasible?
Dual feasible?
Suboptimal
Superoptimal
Yes No
Optimal
Neither feasible nor superoptimal

64. Primal‐dual relationships

65. Applications of duality theory
QIf the number of functional constraints m is bigger than the number of variables n, applying the SM directly to the dual problem will achieve a substantial reduction in computational effort.
QEvaluating a proposed solution for the primal problem. i) if cx=yb, x and y must be optimal even without applying the SM; ii) if cx<yb, yb provides an upper bound on the optimal value of Z.
QEconomic interpretation…

66. Duality theory in sensitivity analysis
QSensitivity analysis investigates the effect in the optimal solution of changing parameters aij, bi, cj.
QIt can be easier to study these effects in the dual problem:
Optimal solution of the dual problem are the shadow prices of the primal problem.
Changes in coefficients of a nonbasic variable. Only changes one constraint in the dual problem. If this constraint is satisfied the solution is still optimal.
Introducing a new variable. Only introduces a new constraint in the dual problem. If the dual problem is still feasible, solution is still optimal.

67. The essence of sensitivity analysis
QOne assumption of LP is that all the parameters of the model (aij, bi and cj) are known constants.
QActually:
The parameters values used are just estimates based on a prediction of future conditions;
The data may represent deliberate overestimates or underestimates to protect the interests of the estimators.
QAn “optimal” solution is optimal only with respect to the specific model being used to represent the real problem → sensitivity analysis is crucial!

68. Simplex tableau with parameter changes
QWhat changes in the simplex tableau if changes are made in the model parameters, namely b→b, c→ c, A→ A ?
Eq.
Coefficient of:
Right side Z
Original variables
Slack variables
New initial tableau
(0) 1 −c
(1,2,…,m) A I b
Revised final tableau
z * −c = y * A − c y*
Z* = y *b
A* = S* A S*
b* = S*b

69. Applying sensitivity analysis
QChanges in bi:
Allowable range: range of values for which the current optimal BF solution remains feasible (find range of bi such that b* = S*b ≥ 0 , assuming this is the only change in the model). The shadow price for bi remains valid if bi stays within this interval.
The 100% rule for simultaneous changes in RHS: the shadow prices remain valid as long as the changes are not too large. If the sum of the % changes does not exceed 100%, the shadow prices will definitely still be valid.

70. Applying sensitivity analysis
QChanges in coefficients of nonbasic variable:
Allowable range to stay optimal: range of values over which the current optimal solution remains optimal (find c j ≤ y * A j , assuming this is the only change in the model).
The 100% rule for simultaneous changes in objective function coefficients: if the sum of the % changes does not exceed 100%, the original optimal solution will definitely still be valid.
QIntroduction of a new variable:
Same as above.

71. Other simplex algorithms
QDual simplex method
Modification useful for sensitivity analysis, based on the duality theory.
QParametric linear programming
Extension for systematic sensitivity analysis. Vary one or more parameters continuously over some interval(s) to see when the optimal solution changes.
QUpper bound technique
A simplex version for dealing with decision variables having upper bounds: xj ≤ uj, where uj is the maximum feasible value of xj.