Spreadsheet Modeling & Decision Analysis:

Spreadsheet Modeling & Decision Analysis:
A Practical Introduction to Management Science, 3e by Cliff Ragsdale

Sensitivity Analysis and the Simplex Method
Chapter 4 Sensitivity Analysis and the Simplex Method Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Introduction When solving an LP model we assume that all relevant factors are known with certainty. Such certainty rarely exists. Sensitivity analysis helps answer questions about how sensitive the optimal solution is to changes in various coefficients in an LP model. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

General Form of a Linear Programming (LP) Problem
MAX (or MIN): c1X1 + c2X2 + … + cnXn Subject to: a11X1 + a12X2 + … + a1nXn <= b1 : ak1X1 + ak2X2 + … + aknXn <= bk am1X1 + am2X2 + … + amnXn = bm How sensitive is a solution to changes in the ci, aij, and bi? Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Approaches to Sensitivity Analysis
Change the data and re-solve the model! Sometimes this is the only practical approach. Solver also produces sensitivity reports that can answer questions about: amounts objective function coefficients can change without changing the solution. the impact on the optimal objective function value of changes in various constrained resources. the impact on the optimal objective function value of forced changes in certain decision variables. the impact changes in constraint coefficients will have on the optimal solution. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Software Note When solving LP problems, be sure to select the “Assume Linear Model” option in the Solver Options dialog box as this allows Solver to provide more sensitivity information than it could otherwise do. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Once Again, We’ll Use The Blue Ridge Hot Tubs Example...
MAX: 350X X2 } profit S.T.: 1X1 + 1X2 <= 200 } pumps 9X1 + 6X2 <= 1566 } labor 12X1 + 16X2 <= 2880 } tubing X1, X2 >= 0 } nonnegativity Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Subject to: D9:D11<=E9:E11 B5:C5>=0
Aqua-Spas Hydro-Luxes Number to make Total Profit Unit Profits $350 $300 $0 Constraints Used Available Pumps Req'd 1 200 Labor Req'd 9 6 1566 Tubing Req'd 12 16 2880 Blue Ridge Hot Tubs Maximize: D6 By changing: B5:C5 Subject to: D9:D11<=E9:E11 B5:C5>=0 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Aqua-Spas Hydro-Luxes Number to make 122 78 Total Profit Unit Profits
$350 $300 $66,100 Constraints Used Available Pumps Req'd 1 200 Labor Req'd 9 6 1566 Tubing Req'd 12 16 2712 2880 Blue Ridge Hot Tubs Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Answer Report Microsoft Excel 16.0 Answer Report Worksheet: [Fig4-1.xls]Production Report Report Created: 3/26/ :12:58 AM Result: Solver found a solution. All Constraints and optimality conditions are satisfied. Solver Engine Engine: Simplex LP Solution Time: 0 Seconds. Iterations: 2 Subproblems: 0 Solver Options Max Time 100 sec, Iterations 100, Precision Max Subproblems 5000, Max Integer Sols 5000, Integer Tolerance 5% Objective Cell (Max) Cell Name Original Value Final Value $D$6 Unit Profits Total Profit $66,100 Variable Cells Integer $B$5 Number to make Aqua-Spas 122 Contin $C$5 Number to make Hydro-Luxes 78 Constraints Cell Value Formula Status Slack $D$9 Pumps Req'd Used 200 $D$9<=$E$9 Binding $D$10 Labor Req'd Used 1566 $D$10<=$E$10 $D$11 Tubing Req'd Used 2712 $D$11<=$E$11 Not Binding 168 $B$5>=0 $C$5>=0 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Answer Report The Answer Report for the Blue Ridge Hot Tubs problem summarizes the solution to the problem, and is fairly self-explanatory. The first section of the report summarizes the original and final (optimal) value of the objective cell. The next section summarizes the original and final (optimal) values of the decision variable cells. The final section of this report provides information about the constraints. In particular, the Cell Value column shows the final (optimal) value assumed by each constraint cell. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Answer Report Note that these values correspond to the final value assumed by the LHS formula of each constraint The Formula column indicates the upper or lower bounds that apply to each constraint cell. The Status column indicates which constraints are binding and which are nonbinding. A constraint is binding if it is satisfied as a strict equality in the optimal solution; otherwise, it is nonbinding. Notice that the constraints for the number of pumps and amount of labor used are both binding, meaning that all the available pumps and labor hours will be used if this solution is implemented. Therefore, these constraints are preventing Blue Ridge Hot Tubs from achieving a higher level of profit Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

all the available pumps and labor hours will be used,
Finally, the values in the Slack column indicate the difference between the LHS and RHS of each constraint. By definition, binding constraints have zero slack and nonbinding constraints have some positive level of slack. The values in the Slack column indicate that if this solution is implemented: all the available pumps and labor hours will be used, but 168 feet of tubing will be left over. The slack values for the non-negative conditions indicate the amounts by which the decision variables exceed their respective lower bounds of zero. The Answer Report does not provide any information that could not be derived from the solution shown in the spreadsheet model. However, the format of this report gives a convenient summary of the solution that can be incorporated easily into a word processing document as part of a written report to management. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Microsoft Excel 16.0 Sensitivity Report Figure 4.3
Worksheet: [Fig4-1.xls]Production Report Report Created: 3/26/ :12:59 AM Variable Cells Final Reduced Objective Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Number to make Aqua-Spas 122 350 100 50 $C$5 Number to make Hydro-Luxes 78 300 Constraints Shadow Constraint Price R.H. Side $D$9 Pumps Req'd Used 200 7 26 $D$10 Labor Req'd Used 1566 234 126 $D$11 Tubing Req'd Used 2712 2880 1E+30 168 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

CHANGES IN THE OBJECTIVE FUNCTION COEFFICIENTS
The Sensitivity Report for the Blue Ridge Hot Tub report, summarizes information about the variable cells and constraints for the model. This information is useful in evaluating how sensitive the optimal solution is to changes in various coefficients in the model. CHANGES IN THE OBJECTIVE FUNCTION COEFFICIENTS Chapter 2 introduced the level-curve approach to solving a graphical LP problem and showed how to use this approach to solve the Blue Ridge Hot Tubs problem. This graphical solution is repeated in Figure 4.4 (Fig4-4.xlsm). The slope of the original level curve in Figure 4.4 is determined by the coefficients in the objective function of the model (the values 350 and 300). In Figure 4.5, we can see that if the slope of the level curve were different, the extreme point represented by X1 =80, X2 = 120 would be the optimal solution. Of course, the only way to change the level curve for the objective function is to change the coefficients in the objective function. So, if the objective function coefficients are at all uncertain, we might be interested in determining how much these values could change before the optimal solution would change. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

original optimal solution
How Changes in Objective Coefficients Change the Slope of the Level Curve X2 250 original level curve 200 new optimal solution 150 original optimal solution 100 new level curve 50 50 100 150 200 250 X1 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

For example, if the owner of Blue Ridge Hot Tubs does not have complete control over the costs of producing hot tubs (which is likely because he purchases the fiberglass hot tub shells from another company), the profit figures in the objective function of our LP model might not be the exact profits earned on hot tubs produced in the future. So before the manager decides to produce 122 Aqua-Spas and 78 Hydro-Luxes, he might want to determine how sensitive this solution is to the profit figures in the objective. That is, the manager might want to determine how much the profit figures could change before the optimal solution of X1 = 122, X2 = 78 would change. This information is provided in the Sensitivity Report. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The original objective function coefficients associated with the variable cells are listed in the Objective Coefficient column in Figure SR. The next two columns show the allowable increases and decreases in these values. For example, the objective function value associated with Aqua-Spas (or variable Xl) can increase by as much as $100 or decrease by as much as $50 without changing the optimal solution, assuming all other coefficients remain constant. (You can verify this by changing the profit coefficient for Aqua-Spas to any value in the range from $300 to $450 and re-solving the model.) Similarly, the objective function value associated with Hydro-Luxes (or variable X2) can increase by $50 or decrease by approximately $66.67 without changing the optimal values of the decision variables, assuming all other coefficients remain constant. (Again, you can verify this by re-solving the model with different profit values for Hydro-Luxes.) Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Changes in Objective Function Coefficients
Values in the “Allowable Increase” and “Allowable Decrease” columns for the Changing Cells indicate the amounts by which an objective function coefficient can change without changing the optimal solution, assuming all other coefficients remain constant. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Alternate Optimal Solutions
Values of zero (0) in the “Allowable Increase” or “Allowable Decrease” columns for the Changing Cells indicate that an alternate optimal solution exists. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Limits Report Microsoft Excel 16.0 Limits Report
Worksheet: [Fig4-1.xls]Production Report Report Created: 3/26/ :12:59 AM Objective Cell Name Value $D$6 Unit Profits Total Profit $66,100 Variable Lower Upper Limit Result $B$5 Number to make Aqua-Spas 122 23400 66100 $C$5 Number to make Hydro-Luxes 78 42700 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Limits Report The Limits Report for the original Blue Ridge Hot Tubs problem is shown above. This report lists the optimal value of the objective cell. It then summarizes the optimal values for each variable cell and indicates what values the objective cell assumes if each variable cell is set to its upper or lower limits. The values in the Lower Limits column indicate the smallest value each variable cell can assume while the values of all other variable cells remain constant and all the constraints are satisfied. The values in the Upper Limits column indicate the largest value each variable cell can assume while the values of all other variable cells remain constant and all the constraints are satisfied. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

CHANGES IN THE RHS VALUES
Constraints that have zero slack in the optimal solution to an LP problem are called binding constraints. Binding constraints prevent us from further improving (i.e., maximizing or minimizing) the objective function. For example, the Answer Report in Figure 4.2 indicates that the constraints for the number of pumps and hours of labor available are binding, whereas the constraint on the amount of tubing available is nonbinding. This is also evident in Figure 4.3 by comparing the Final Value column with the Constraint R.H. Side column. The values in the Final Value column represent the LHS values of each constraint at the optimal solution. A constraint is binding if its Final Value is equal to its Constraint R.H. Side value. After solving an LP problem, you might want to determine how much better or worse the solution would be if we had more or less of a given resource. For example, Howie Jones might wonder how much more profit could be earned if additional pumps or labor hours were available. The Shadow Price column in the Sensitivity Report Figure 4.3 provides the answers to such questions. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The shadow price for a constraint indicates the amount by which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant. If a shadow price is positive, a unit increase in the RHS value of the associated constraint results in an increase in the optimal objective function value. If a shadow price is negative, a unit increase in the RHS value of the associated constraint results in a decrease in the optimal objective function value. To analyze the effects of decreases in the RHS values, you reverse the sign on the shadow price. That is, the negated shadow price for a constraint indicates the amount by which the optimal objective function value changes given a unit decrease in the RHS value of the constraint, assuming all other coefficients remain constant. The shadow price values apply provided that the increase or decrease in the RHS value falls within the allowable increase or allowable decrease limits in the Sensitivity Report for each constraint. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

For example, Figure 4.3 indicates that the shadow price for the labor constraint is Therefore, if the number of available labor hours increased by any amount in the range from 0 to 234 hours, the optimal objective function value changes (increases) by $16.67 for each additional labor hour. If the number of available labor hours decreased by any amount in the range from 0 to 126 hours, the optimal objective function value changes (decreases) by -$16.67 for each lost labor hour. A similar interpretation holds for the shadow price for the constraint on the number of pumps. (It is coincidental that the shadow price for the pump constraint (200) is the same as that constraint's RHS and Final Values.) Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Changes in Constraint RHS Values
The shadow price of a constraint indicates the amount by which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant. Shadow prices hold only within RHS changes falling within the values in “Allowable Increase” and “Allowable Decrease” columns. Shadow prices for nonbinding constraints are always zero. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Comments About Changes in Constraint RHS Values
Shadow prices only indicate the changes that occur in the objective function value as RHS values change. Changing a RHS value for a binding constraint also changes the feasible region and the optimal solution (see graph on following slide). To find the optimal solution after changing a binding RHS value, you must re-solve the problem. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

X2 How Changing the RHS Value of a Constraint Can Change the Feasible Region and Optimal Solution 250 Suppose available labor hours increase from 1,566 to 1,728 200 150 old optimal solution old labor constraint 100 new optimal solution 50 new labor constraint 50 100 150 200 250 X1 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

End of Slides for Hot Tub Chapter 4 Analysis
SENG 5332 SP 2019 End of Slides for Hot Tub Chapter 4 Analysis Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Other Uses of Shadow Prices
Suppose a new Hot Tub (the Typhoon-Lagoon) is being considered. It generates a marginal profit of $320 and requires: 1 pump (shadow price = $200) 8 hours of labor (shadow price = $16.67) 13 feet of tubing (shadow price = $0) Q: Would it be profitable to produce any? A: $320 - $200*1 - $16.67*8 - $0*13 = -$13.33 = No! Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Meaning of Reduced Costs
The Reduced Cost for each product equals its per-unit marginal profit minus the per-unit value of the resources it consumes (priced at their shadow prices). Optimal Value of Optimal Value of Type of Problem Decision Variable Reduced Cost at simple lower bound <=0 Maximization between lower & upper bounds =0 at simple upper bound >=0 at simple lower bound >=0 Minimization between lower & upper bounds =0 at simple upper bound <=0 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Key Points The shadow prices of resources equate the marginal value of the resources consumed with the marginal benefit of the goods being produced. Resources in excess supply have a shadow price (or marginal value) of zero. The reduced cost of a product is the difference between its marginal profit and the marginal value of the resources it consumes. Products whose marginal profits are less than the marginal value of the goods required for their production will not be produced in an optimal solution. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Analyzing Changes in Constraint Coefficients
Q: Suppose a Typhoon-Lagoon required only 7 labor hours rather than 8. Is it now profitable to produce any? A: $320 - $200*1 - $16.67*7 - $0*13 = $3.31 = Yes! Q: What is the maximum amount of labor Typhoon-Lagoons could require and still be profitable? A: We need $320 - $200*1 - $16.67*L3 - $0*13 >=0 The above is true if L3 <= $120/$16.67 = $7.20 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Simultaneous Changes in Objective Function Coefficients
The 100% Rule can be used to determine if the optimal solutions changes when more than one objective function coefficient changes. Two cases can occur: Case 1: All variables with changed obj. coefficients have nonzero reduced costs. Case 2: At least one variable with changed obj. coefficient has a reduced cost of zero. In Case 1 the current solution remains optimal provided the obj. coefficient changes are all within their Allowable Increase or Decrease. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Simultaneous Changes in Objective Function Coefficients
In Case 2, for each variable compute: If more than one objective function coefficient changes, the current solution will remain optimal provided that the rj sum to <= 1. (Note that if the rj sum to > 1, the current solution, might remain optimal, but this is not guaranteed.) Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

A Warning About Degeneracy
The solution to an LP problem is degenerate if the Allowable Increase of Decrease on any constraint is zero (0). When the solution is degenerate: 1. The methods mentioned earlier for detecting alternate optimal solutions cannot be relied upon. 2. The reduced costs for the changing cells may not be unique. Additionally, in this case, the objective function coefficients for changing cells must change by at least as much as (and possibly more than) their respective reduced costs before the optimal solution would change. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

When the solution is degenerate (cont’d):
3) The allowable increases and decreases for the objective function coefficients still hold and, in fact, the coefficients may have to be changed substantially beyond the allowable increase and decrease limits before the optimal solution changes. 4) The given shadow prices and their ranges may still be interpreted in the usual way but they may not be unique. That is, a different set of shadow prices and ranges may also apply to the problem (even if the optimal solution is unique). Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Limits Report See file Fig4-1.xls
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Sensitivity Assistant
An add-in on the CD-ROM for this book that allows you to create: Spider Tables & Plots Summarize the optimal value for one output cell as individual changes are made to various input cells. Solver Tables Summarize the optimal value of multiple output cells as changes are made to a single input cell. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Sensitivity Assistant
See files: Fig4-11.xls & Fig4-13.xls Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

The Simplex Method To use the simplex method, we first convert all inequalities to equalities by adding slack variables to <= constraints and subtracting slack variables from >= constraints. For example: ak1X1 + ak2X2 + … + aknXn <= bk is converted to: ak1X1 + ak2X2 + … + aknXn + Sk = bk And: ak1X1 + ak2X2 + … + aknXn >= bk is converted to: ak1X1 + ak2X2 + … + aknXn - Sk = bk Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

For Our Example Problem...
MAX: 350X X2 } profit S.T.: 1X1 + 1X2 + S1 = 200 } pumps 9X1 + 6X2 + S2 = } labor 12X1 + 16X2 + S3 = 2880 } tubing X1, X2, S1, S2, S3 >= 0 } nonnegativity If there are n variables in a system of m equations (where n>=m) we can select any m variables and solve the equations (setting the remaining n-m variables to zero.) Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Possible Basic Feasible Solutions
Basic Nonbasic Objective Variables Variables Solution Value 1 S1, S2, S3 X1, X2 X1=0, X2=0, S1=200, S2=1566, S3=2880 0 2 X1, S1, S3 X2, S2 X1=174, X2=0, S1=26, S2=0, S3=792 60,900 3 X1, X2, S3 S1, S2 X1=122, X2=78, S1=0, S2=0, S3=168 66,100 4 X1, X2, S2 S1, S3 X1=80, X2=120, S1=0, S2=126, S3=0 64,000 5 X2, S1, S2 X1, S3 X1=0, X2=180, S1=20, S2=486, S3=0 54,000 6* X1, X2, S1 S2, S3 X1=108, X2=99, S1=-7, S2=0, S3=0 67,500 7* X1, S1, S2 X2, S3 X1=240, X2=0, S1=-40, S2=-594, S3=0 84,000 8* X1, S2, S3 X2, S1 X1=200, X2=0, S1=0, S2=-234, S3=480 70,000 9* X2, S2, S3 X1, S1 X1=0, X2=200, S1=0, S2=366, S3= ,000 10* X2, S1, S3 X1, S2 X1=0, X2=261, S1=-61, S2=0, S3= ,300 * denotes infeasible solutions Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Basic Feasible Solutions & Extreme Points
250 200 150 100 50 5 2 3 4 1 Basic Feasible Solutions 1 X1=0, X2=0, S1=200, S2=1566, S3=2880 2 X1=174, X2=0, S1=26, S2=0, S3=792 3 X1=122, X2=78, S1=0, S2=0, S3=168 4 X1=80, X2=120, S1=0, S2=126, S3=0 5 X1=0, X2=180, S1=20, S2=486, S3=0 Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

Simplex Method Summary
The simplex method operates by first identifying any basic feasible solution (or extreme point) for an LP problem, then moving to an adjacent extreme point, if such a move improves the value of the objective function. When no adjacent extreme point has a better objective function value, the current extreme point is optimal and the simplex method terminates. The process of moving from one extreme point to an adjacent one is accomplished by switching one of the basic variables with one of the nonbasic variables to create a new basic feasible solution that corresponds to the adjacent extreme point. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning

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