DECISION MODELING WITH Prentice Hall Publishers and MICROSOFT EXCEL Chapter 4 Linear Optimization: Sensitivity Analysis Part 2 Copyright 2001 Prentice Hall Publishers and Ardith E. Baker

Sensitivity Analysis Using SolverTable SolverTable.xla is a _________-like macro to re-optimize and tabulate an LP model after each change in its________________. Similar to Excel’s DataTable, _________knows how to re-Solve the LP model for each change before tabulating any results. SolverTable can also ________the information in the Solver Sensitivity Report. SolverTable is not ___________to two-variable models.

Sensitivity Analysis with SolverTable To begin using SolverTable, open the SimpleOakProd.xls workbook. Open the add-in file___________. Click on the resulting Enable Macros button. SolverTable will install itself and be available as a menu item on the _______menu.

RHS Ranging with SolverTable To illustrate SolverTable, start with the Simplified Oak Products model.

First a range of _________parameter values for the constraint are entered as ____in a column (or a row). Next, enter a row of _________to model output cells.

Now, _______the table by click-dragging and choose SolverTable from the Tools pull-down menu.

In the resulting dialog, specify the ___________of the Long Dowel constraint’s RHS in the Input Column Cell edit field. Click OK to run Solver on the ________for each Long Dowels constraint RHS value (in this case for 11 optimizations).

SolverTable __________the requested model results referenced in the table’s____________. The ________changes abruptly each time a different set of __________combine to determine the optimal corner point.

Here are the corresponding GLP pictures of the Oak Products model for the Long Dowels Starting Inventory amounts (L). L = 400

L = 480

L = 800

L = 1100

L = 1320

Sweeping the values of L from 400 to 1350 causes the _______________to expand until the Long Dowels constraint becomes_____________. L = 1350

SolverTable can mimic DataTable 2 to tabulate simultaneous variations in______________, with the restriction that only ___output cell can be tabulated. To illustrate, a range of ____________values for the inventory constraint RHS values for both Long and Short Dowels will be analyzed. Using the Oak Products model, start by setting up the _______, in this case with a range of parameter values for both parameters.

Now, click on Tools – SolverTable and in the resulting dialog, specify cell $F$7 as the ______________and $F$6 as the____________________. Click OK to continue.

SolverTable will run Solver on the model for each paired combination of Long and Short Dowels ____________RHS values (108 optimizations in this case), and for each run, tabulate the single ________ result referenced in the table’s upper left corner.

Sensitivity Analysis Objective Function Coefficient Ranging with SolverTable Similar to ranging an RHS, first set up a table with __________of the objective function coefficient in a column or row.

As before, click on Tools – SolverTable and in the resulting dialog, specify cell $B$3 as the Input Column Cell. Click OK to continue.

Here are the results of the SolverTable analysis Here are the results of the SolverTable analysis. Notice that the _____________coefficients for profit per Captain chair are the coefficient values at which the LP solution ________(as shown by the Allowable Increase).

Here are the corresponding GLP pictures of the Oak Products model for the Captain objective function coefficient values (V). Note how the _____________ solution changes abruptly for critical values of V. V = 0

V = 13.334

V = 40

V = 80

Sweeping the values of V from 0 to 99999 causes the objective function to ______from horizontal to nearly vertical in slope. V = 99999

Sensitivity Analysis Technical Coefficient Ranging with SolverTable SolverTable can be used to investigate alternative ____________technologies. Suppose Oak Products were to consider the option of strengthening or slightly weakening a Mate chair by increasing or decreasing the number of ____________it uses. Let’s examine the ________effects of reducing the number of long dowels per Mate chair from the current 4 to 2, and increasing the number above 4.

As before, first set up a table in Excel and run SolverTable As before, first set up a table in Excel and run SolverTable. In the SolverTable dialog, specify $C$6 (no. of Mates in the Long Dowel constraint) as the Input Column Cell. Here is the resulting solution:

Increasing the number of long dowels per Mate chair from 4 to 6 reduces the ________number of Mates to produce (with an associated increase in Captains), with a ______________in Profit. Reducing the number of long dowels per Mate chair from 4 to 2 also reduces the optimal number of Mates (with an associated increase in Captains), but with a _______________in Profit.

Sensitivity Analysis Eastern Steel Example Ore from four different locations is blended to make a_____________. Each ore contains three essential _________ (A, B, and C) that must appear in the final blend at minimum threshold levels. Find the ________________blend by solving the following LP model (Ti = fraction of a ton of ore from location i).

Eastern Steel Symbolic Model Min 800T1 + 400T2 + 600T3 + 500T4 s.t. 10T1 + 3T2 + 8T3 + 2T4 > 5 (requirement on A) 90T1 + 150T2 + 75T3 + 175T4 > 100 (requirement on B) 45T1 + 25T2 + 20T3 + 37T4 > 30 (requirement on C) T1 + T2 + T3 + T4 = 1 (blend condition) T1 , T2 , T3 , T4 > 0 (nonnegativity constraints)

Here is the Excel spreadsheet: Next, run Solver and specify the Sensitivity Report.

Look at the Reduced Cost column:

The __________of any particular decision variable is defined to be the amount by which the _________of that variable in the objective function would have to change in order to have a ________optimal value for that variable.

The Reduced Cost of a decision variable (whose optimal value is currently______) is the rate (per unit amount) at which the _________value is hurt as that variable is “forced into” a previously optimal solution.

Sensitivity Analysis Sensitivity Report Interpretation for Alternative LP Models In this example, the Friendly Loan Company has an annual $15 million loan budget. __________is generated by the annual interest income from three types of loans: Real Estate (First Mortgage; 7%) Furniture Loans (12%) Signature Loans (15%) In addition, Friendly requires at least ______First Mortgage loans and no more than _____ Signature loans.

Here is the spreadsheet model: Note how compact the model is. The constraints are immediately _________to the quantities they affect and are ________formatted to include the inequality while still being read as a number. Empty cells are _________in order to focus attention on the important things.

Here are the spreadsheet formulas:

Using Solver, specify the parameters and solve the model. Be sure to specify the Sensitivity Report.

The resulting Solver analysis shows that all $15 million will be loaned out ($9 million into First Mortgage Loans, $1.5 million into Signature loans, and $4.5 million into Furniture loans). The annual Total interest income will be $1,395,000 with an average return of 9.3%. All three constraints are___________.

Here is the Sensitivity Report for the model: The Shadow Price of .12 indicates that a 12% return can be achieved on any budget_________. In addition, the _____________________value indicates that we can increase the budget as much as we want (_________).

Verify Solver’s Sensitivity Report by typing a new ____________into the spreadsheet and Solving. Notice that the Avg. Return for this model is still 9.3%. This indicates that the _____________for the extra $5 million is actually 9.3% and not 12% as indicated by the previous Sensitivity Analysis.

The Sensitivity Analysis for this model shows a _______________of The Sensitivity Analysis for this model shows a _______________of .12 (12%), the same as the previous model. To understand what is happening, re-formulate the model using the recommended rules from Chapter 3.

Here is the reformulated spreadsheet model:

And the resulting Sensitivity Analysis from Solver: Note the presence of 3 _______________and the correct Shadow Price of 9.3%. The solution is not ____________and none of the constraints are binding.

Sensitivity Analysis Simple Upper and Lower Bounds Now that we have looked at both spreadsheet models (the compact model vs. the recommended LP model), it would seem that they give different results. However, both models are completely_________, and neither Sensitivity Report contains any errors. To understand the differences, look at simple upper and lower_________.

The time and memory requirements for Solver to ________a model are determined primarily by the size of the coefficient ________of cells making up the LHS of the set of___________. The size of the constraint coefficient matrix is ___________to the product of the number of decision variables and___________. This size effects speed of optimization. In addition to ________________constraints, Solver allows any upper or lower constraint bounds directly on the _________________to be honored without actually considering them as constraints. This keeps the coefficient matrix smaller, allowing larger LP models to be___________.

However, the only ____________information available for any simple upper and lower bound __________are their shadow prices. Solver places any _________shadow price on an upper or lower bound constraint into the ______________column next to the relevant decision variable. The Reduced Cost numbers for Solver LP models containing simple upper and lower bounds are the ____________for whichever bound, if any, is binding on that decision variable.

Variable at Optimality The table below gives values the ___________ shadow price entry may have in Solver models containing simple upper and lower__________. Value of Decision Variable at Optimality Reduced Cost Entry, Maximization Model Minimization Model Lower Bound (>) Binding Zero or Negative Shadow Zero or Positive Shadow Price Price Upper Bound (<) Binding Zero or Positive Shadow Zero or Negative Shadow Neither Bound Binding Zero Shadow Price Zero Shadow Price

Solver invokes its special bounding procedure whenever it sees “________________” cell references in the Subject to the Constraints: box of the Solver Parameters dialog. Solver __________evoke this procedure if the upper or lower bound on any decision variable is specified indirectly on the ____________. This “_____________” can be achieved by the use of some intervening formula, such as the ______________formula.

Sensitivity Analysis Shadow Price Interpretation Although the shadow price given in the two different models was correct, the _____________of that price was incorrect. Remember, a shadow price is the change in the LP’s __________of change in a given constraint’s RHS value holding all other data, including the other RHS’s, constant.

So, for example, the correct interpretation of the shadow price of So, for example, the correct interpretation of the shadow price of .120 should be : Holding the Loan Limit RHS’s for Signature and First Mortgage loans at their original dollar amount bounds of $1500 and $9000, respectively, the improvement in the objective function value is .12 for each additional budget dollar. The use of simple upper and lower bounds and the use of formulas on RHS’s of LP formulations can lead to more compact and managerially appealing spreadsheet formulations of LP’s.