Introduction to sensitivity analysis BSAD 0 Dave Novak Summer 0 Overview Introduction to sensitivity analysis Range of optimality Range of feasibility Source: Anderson et al., 0 Quantitative Methods for Business th edition some slides are directly from J. Loucks 0 Cengage Learning Also called post-optimality analysis Specifically, consider how changes in the coefficients (within specified ranges) of a an LP impact the optimal solution What if this value or that value changes? Keep in mind that an LP is solved in terms of a specific set of objective function coefficients and constraint coefficients Recall the term deterministic The LP from Lecture Max x + x s.t. x < x + x < 9 x + x < x > 0 and x > 0 Objective Function Regular Non-negativity Important because decision-makers operate in dynamic environments with imperfect information In reality, the coefficients associated with the OF and the constraints are subject to uncertainty Allows decision-maker to ask what-if questions about the problem Prices of raw material change, demand for products change, production capabilities change, stock prices change, etc. If you are using an LP, you can generally expect some type of change over time How do these changes impact the optimal solution, and at what point do you need to completely revise and re-solve the LP?
In sensitivity analysis, we consider only ONE change at a time Change ONE OF coefficient Change ONE RHS constraint coefficient We cannot use basic sensitivity analysis to examine simultaneous changes We cannot use basic sensitivity analysis to examine changes in LHS constraint coefficients must reformulate and resolve the problem A basic LP Max: c x + c x +/-. c n x n s.t: a x + a x +/-. a n x n / /= b a x + a x +/-. a n x n / /= b Where: c i = OF coefficient x i = decision variable a i = constraint coefficient b i = RHS constraint coefficient A basic LP Max: x + x s.t: ) x + x ) x + x Where: and = x and x = decision variables and = LHS constraint coefficients for constraint # = RHS constraint coefficient for constraint # We will consider TWO different sensitivity analysis scenarios ) Changes in objective function coefficients (range of optimality ROO) ) Changes in RHS constraint coefficients (range of feasibility ROF) Changes in Changes in How changes in the objective function coefficients (c i values) might impact the optimal solution We want to find the range of OF coefficient values (for each coefficient) where the current solution will remain optimal Managers should focus on objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range Max x + x s.t. x < x + x < 9 x + x < x > 0 and x > 0 Objective Function Regular Non-negativity
Changes in Do not impact the feasible region or the existing extreme points in any way, Only impact the final profit or cost value associated with the optimal solution In a two-decision variable graphical problem, a change in an OF coefficient changes the slope of the OF line Regardless of how many decision variables are considered, changes in do not change the feasible region Changes in Referred to as the range of optimality Within the ROO, the current solution remains optimal Outside of this range, the slope of the OF line changes enough that a different extreme point in the feasible region becomes the new optimal solution Download Lecture.xlsx from the class website The problem is already solved Our Max: c x + c x +/-. c n x n s.t: a x + a x +/-. a n x n / /= b a x + a x +/-. a n x n / /= b X (# of units of product ) X (# of units of product ) Objective Function (Maximize Profit) LHS RHS ST: ) Constraint# ) Constraint # 9 9 ) Constraint # Max: x + x s.t: ) x ) x + x 9 ) x + x Final x i values $B$ X (# of units of product ) 0 0. $B$ X (# of units of product ) 0 0. $B$0 ) Constraint # LHS 0.. $B$ ) Constraint# LHS 0 E+0 $B$9 ) Constraint # LHS 9 9 b i c i Note that changes in the will NOT impact the feasible region, or the extreme points within the feasible region, as the constraints are not changed in any way Sensitivity focused on changes in OF coefficient values address the question at what point does a change in the profit or cost associated with a particular decision variable result in a different extreme point becoming the new optimal solution?
Graphical solution Graphical solution x x Con : x + x < OF: Max x + x Con : x + x < Con : x < Con : x < Original Optimal Solution: x =, x = Con : x + x < 9 Con : x + x < 9 9 0 x 9 0 x Graphical solution x Con : x + x < Con : x < Con : x + x < 9 x 9 0 We want to know how much each one of the coefficients in the OF can change before the optimal solution mix changes (x, x ) Max x + x If the per unit profit associated with x (c ) changes from to some other value would the optimal solution still be (x =, x = )? The current optimal solution is to produce units of x and units of x at profit of $ and $ per unit respectively The OF coefficient associated with x (referred to as c ) can range between $. and $ before The optimal solution mix will change from (, ) to something else (another extreme point value in our feasible region) $B$ X (# of units of product ) 0 0. $B$ X (# of units of product ) 0 0. $B$0 ) Constraint # LHS 0.. $B$ ) Constraint# LHS 0 E+0 $B$9 ) Constraint # LHS 9 9 ariable Cells $B$ X (# of units of product ) 0 0. $B$ X (# of units of product ) 0 0. The current OF coefficient associated with x (referred to as c ) = The allowable increase is, so the upper bound of c = + = The allowable decrease is 0., so the lower bound of c = 0. =. As long as, c, which is the profit associated with x is between $. and $, the optimal solution will be (, ). If c changes to a value outside of this range, the optimal solution changes to another point in the feasible region
Likewise, (c ) the OF coefficient associated with x can range between $ and $.0 before the optimal solution mix will change from (, ) to something else (another extreme point value in our feasible region) $B$ X (# of units of product ) 0 0. $B$ X (# of units of product ) 0 0. $B$0 ) Constraint # LHS 0.. $B$ ) Constraint# LHS 0 E+0 $B$9 ) Constraint # LHS 9 9 The range of values the c and c can take on without changing the optimal solution is referred to as the range of optimality (ROO) ROO for c : $. $ ROO for c : $ $.0 Any change in an OF coefficient changes the slope of the OF line! Graphically, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines Mathematically find the range of optimality for OF coefficient c (the coefficient associated with x, which is ) Slope of current OF line = (-c /c ) = (- / ) Slope of an objective function line, Max c x + c x, is (-c /c ), and the slope of a constraint, a x + a x = b, is (-a /a ) Current optimal solution (, ) involves binding constraints Con and Con Con : x + x = 9 Slope of Con = (-a /a ) = (- / ) Find the range of values for c (while holding c constant, staying at $), such that the slope of the OF line stays between the binding constraints - < -c / < - / Slope of Con Con : x + x = Slope of Con = (-a /a ) = (- / ) = - Slope of Con Slope of OF with c represented as a variable
- < -c / < - / Find the range of optimality for c (the OF coefficient associated with x while holding c constant (fixed at ) - < - / c < - / Find the range of optimality for c (the OF coefficient associated with x while holding c constant (fixed at ) - < - / c < - / Assume the profit associated with x were to increase by $ per unit, would we would still produce units of x and units of x? Increase c from $/unit to $/unit We can answer this by looking at the Sensitivity Report The OF coefficient associated with x (referred to as c ) can range between $. and $ before The optimal solution mix will change from (, ) to something else (another extreme point value in our feasible region) $B$ X (# of units of product ) 0 0. $B$ X (# of units of product ) 0 0. $B$0 ) Constraint # LHS 0.. $B$ ) Constraint# LHS 0 E+0 $B$9 ) Constraint # LHS 9 9 If c increases from to, we would still produce units of x and units of x However, our total profit would now be $ instead of $ Before change: () + () = After change: () + () = As long as $. c $, the optimal solution mix of units of x and units of x will not change
If (c ) the profit associated with x decreases by $.0 per unit (from to.0), we would still produce units x and units of x However, our total profit would now be $.0 instead of $ () +.0() =.0 As long as $ c $.0, the optimal solution mix of units of x and units of x will not change If (c ) the profit associated with x were to decrease by $.0 per unit (from to.0), would we still produce units of x? No, decreasing x to.0 is OUTSIDE range of optimality (c <.) We know that the optimal solution will change to a different critical point! It is no longer (, ) x Con : x + x < Objective function line for.x + x Range of optimality That wraps up changes in the OF coefficients Next, changes in RHS values Feasible Region Con : x + x < 9 9 0 x Summary Introduction to sensitivity analysis Focus on change in OF coefficient values range of optimality