Chapter 37. Analyzing Data Using Goal Seeking and Solver

IN THIS CHAPTER

What-if analysis – in reverse

Single-cell goal seeking

Introducing Solver

Solver examples

The preceding chapter discusses what-if analysis — the process of changing input cells to observe the results on other dependent cells. This chapter looks at that process from the opposite perspective: finding the value of one or more input cells that produces a desired result in a formula cell.

This chapter covers two tools: Goal Seeking and the Solver add-in.

37.1. What-If Analysis, in Reverse

Consider the following what-if question: "What is the total profit if sales increase by 20 percent?" If you set up your worksheet model properly, you can change the value in one cell to see what happens to the profit cell. The examples in this chapter take the opposite approach. If you know what a formula result should be, Excel can tell you the values that you need to enter in one or more input cells to produce that result. In other words, you can ask a question such as "How much do sales need to increase to produce a profit of $1.2 million?" Excel provides two tools that are relevant:

• Goal Seek: Determines the value that you need to enter in a single input cell to produce a result that you want in a dependent (formula) cell.

• Solver: Determines the values that you need to enter in multiple input cells to produce a result that you want. Moreover, because you can specify certain constraints to the problem, you gain significant problem-solving ability.

37.2. Single-Cell Goal Seeking

Single-cell goal seeking is a rather simple concept. Excel determines what value in an input cell produces a desired result in a formula cell. The following example shows you how single-cell goal seeking works.

37.2.1. A goal-seeking example

Figure 37.1 shows the mortgage loan worksheet used in the preceding chapter. This worksheet has four input cells (C4:C7) and four formula cells (C10:C13). Originally, this worksheet was used for a what-if analysis example. This example demonstrates the opposite approach. Rather than supply different input cell values to look at the calculated formulas, this example lets Excel determine one of the input values that will produce the desired result.

Figure 37.1. This worksheet is a good demonstration of goal seeking.

Assume that you're in the market for a new home and you know that you can afford an $1,800 monthly mortgage payment. You also know that a lender can issue a 30-year fixed-rate mortgage loan for 6.50%, based on an 80% loan-to-value (that is, a 20% down payment). The question is "What is the maximum purchase price I can handle?" In other words, what value in cell C4 causes the formula in cell C11 to result in $1,800? In this simple example, you could plug values into cell C4 until C11 displays $1,800. With more complex models, Excel can usually determine the answer much more efficiently.

To answer the question posed in the preceding paragraph, first set up the input cells to match what you already know. Specifically:

• Enter 20% in cell C5 (the down payment percent)

• Enter 360 in cell C6 (the loan term, in months)

• Enter 6.5% in cell C7 (the annual interest rate)

Next, choose Data Data Tools What-If Analysis Goal Seek. Excel displays the Goal Seek dialog box, shown in Figure 37.2. Completing this dialog box is similar to forming a sentence. You want to set cell C11 to 1800 by changing cell C4. Enter this information in the dialog box either by typing the cell references or by pointing with the mouse. Click OK to begin the goal-seeking process.

Figure 37.2. The Goal Seek dialog box.

In less than a second, Excel displays the Goal Seek Status box, which shows the target value and the value that Excel calculated. In this case, Excel found an exact value. The worksheet now displays the found value in cell C4 ($355,974). As a result of this value, the monthly payment amount is $1,800. At this point, you have two options:

• Click OK to replace the original value with the found value.

• Click Cancel to restore your worksheet to the form that it had before you chose Goal Seek.

37.2.2. More about goal seeking

Excel can't always find a value that produces the result that you're seeking. Sometimes, a solution simply doesn't exist. In such a case, the Goal Seek Status box informs you of that fact.

Other times, however, Excel may report that it can't find a solution, but you're pretty sure that one exists. If that's the case, you can try the following options:

• Change the current value of the By Changing Cell field in the Goal Seek dialog box (refer to Figure 37.2) to a value that is closer to the solution and then reissue the command.

• Adjust the Maximum iterations setting on the Formulas tab of the Excel Options dialog box (choose File Excel Options). Increasing the number of iterations (or calculations) makes Excel try more possible solutions.

• Double-check your logic and make sure that the formula cell does, indeed, depend upon the specified changing cell.

37.3. Introducing Solver

The Excel Goal Seek feature is a useful tool, but it clearly has limitations. It can solve for only one adjustable cell, and it returns only a single solution. Excel's powerful Solver tool extends this concept by enabling you to do the following:

• Specify multiple adjustable cells.

• Specify constraints on the values that the adjustable cells can have.

• Generate a solution that maximizes or minimizes a particular worksheet cell.

• Generate multiple solutions to a problem.

Although goal seeking is a relatively simple operation, using Solver can be much more complicated. In fact, Solver is probably one of the most difficult (and potentially frustrating) features in Excel. I'm the first to admit that Solver isn't for everyone. In fact, most Excel users have no use for this feature. However, many users find that having this much power is worth spending the extra time to learn about it.

37.3.1. Appropriate problems for Solver

Problems that are appropriate for Solver fall into a relatively narrow range. They typically involve situations that meet the following criteria:

• A target cell depends upon other cells and formulas. Typically, you want to maximize or minimize this target cell or set it equal to some value.

• The target cell depends on a group of cells (called changing cells) that Solver can adjust to affect the target cell.

• The solution must adhere to certain limitations, or constraints.

After you set up your worksheet appropriately, you can use Solver to adjust the changing cells and produce the result that you want in your target cell — and simultaneously meet all the constraints that you defined.

37.3.2. A simple Solver example

I start with a simple example to introduce Solver and then present some increasingly complex examples to demonstrate what this feature can do.

Figure 37.3 shows a worksheet that is set up to calculate the profit for three products. Column B shows the number of units of each product, Column C shows the profit per unit for each product, and Column D contains formulas that calculate the total profit for each product by multiplying the units by the profit per unit.

Figure 37.3. Use Solver to determine the number of units to maximize the total profit.

You don't need an MBA degree to realize that the greatest profit comes from Product C. Therefore, to maximize total profit, the logical solution is to produce only Product C. If things were really this simple, you wouldn't need tools such as Solver. As in most situations, this company has some constraints that must be met:

• The combined production capacity is 300 total units per day.

• The company needs 50 units of Product A to fill an existing order.

• The company needs 40 units of Product B to fill an anticipated order.

• Because the market for Product C is relatively limited, the company doesn't want to produce more than 40 units of this product.

These four constraints make the problem more realistic and a bit more challenging. In fact, it's a perfect problem for Solver.

I go into more detail in a moment, but here is the basic procedure for using Solver:

To start Solver to tackle this example, choose Data Analysis Solver. Excel displays its Solver Parameters dialog box, shown in Figure 37.4.

Figure 37.4. The Solver Parameters dialog box.

In this example, the target cell is D6 — the cell that calculates the total profit for three products.

Figure 37.6. Solver displays this dialog box when it finds a solution to the problem.

At this point, you have the following options:

• Keep the solution that Solver found.

• Restore the original changing cell values.

• Create any or all three reports that describe what Solver did.

• Click the Save Scenario button to save the solution as a scenario so that Scenario Manager can use it (see Chapter 36).

The Reports section of the Solver Results dialog box lets you select any or all of three optional reports. If you specify any report options, Excel creates each report on a new worksheet, with an appropriate name. Figure 37.7 shows an Answer Report, in the form of a collapsible outline (I chose the Outline Reports check box in the Solver Results dialog box). In the Constraints section of the report, three of the four constraints are binding, which means that these constraints were satisfied at their limit with no more room to change.

This simple example illustrates how Solver works. The fact is, you could probably solve this particular problem manually just as quickly. That, of course, isn't always the case.

Figure 37.7. One of three reports that Solver can produce.

When you close the Solver Results dialog box (by clicking either OK or Cancel) the Undo stack is cleared. In other words, you cannot undo any changes that Solver makes to your workbook.

37.3.3. More about Solver

Before presenting more complex examples, this section discusses the Solver Options dialog box. From this dialog box, you control many aspects of the solution process, as well as load and save model specifications in a worksheet range.

Usually, you want to save a model only when you're using more than one set of Solver parameters with your worksheet. This is because Excel saves the first Solver model automatically with your worksheet (using hidden names). If you save additional models, Excel stores the information in the form of formulas that correspond to the specifications. (The last cell in the saved range is an array formula that holds the options settings.)

It's not unusual for Solver to report that it can't find a solution, even when you know that one should exist. Often, you can change one or more of the Solver options and try again. When you click the Options button in the Solver Parameters dialog box, Excel displays the Solver Options dialog box, shown in Figure 37.8.

Figure 37.8. You can control many aspects of how Solver solves a problem.

This list describes Solver's options:

• Constraint Precision: Specify how close the Cell Reference and Constraint formulas must be to satisfy a constraint. Excel may solve the problem more quickly if you specify less precision.

• Use Automatic Scaling: Use when the problem involves large differences in magnitude — when you attempt to maximize a percentage, for example, by varying cells that are very large.

• Show Iteration Results: Instruct Solver to pause and display the results after each iteration by selecting this check box.

• Ignore Integer Constraints: When this check box is selected, Solver ignores constraints that specify that a particular cell must be an integer. Using this option may allow Solver to find a solution that cannot be found otherwise.

• Max Time: Specify the maximum amount of time (in seconds) that you want Solver to spend on a problem. If Solver reports that it exceeded the time limit, you can increase the amount of time that it spends searching for a solution.

• Iterations: Enter the maximum number of trial solutions that you want Solver to perform.

• Max Subproblems: For complex problems. Specify the maximum number of subproblems that may be explored by the Evolutionary algorithm.

• Max Feasible Solutions: For complex problems. Specify the maximum number of feasible solutions that may be explored by the Evolutionary algorithm.

37.4. Solver Examples

The remainder of this chapter consists of examples of using Solver for various types of problems.

37.4.1. Solving simultaneous linear equations

This example describes how to solve a set of three linear equations with three variables. Here's an example of a set of linear equations:

4x + y −2z =0
2x −3y +3z =9
−6x −2y +z = 0

The question that Solver will answer is What values of x, y, and z satisfy all three equations?

Figure 37.9 shows a workbook set up to solve this problem. This workbook has three named cells, which makes the formulas more readable:

• x: C11

• y: C12

• z: C13

The three named cells are all initialized to 1 (which certainly doesn't solve the equations).

Figure 37.9. Solver will attempt to solve this series of linear equations.

The three equations are represented by formulas in the range B6:B8:

• B6: =(4*x)+(y)−(2*z)

• B7: =(2*x)−(3*y)+(3*z)

• B8: =-(6*x)−(2*y)+(z)

These formulas use the values in the x, y, and z named cells. The range C6:C8 contains the "desired" result for these three formulas.

Solver will adjust the values in x, y, and z — that is, the changing cells in C11:C13 — subject to these constraints:

• B6=C6

• B7=C7

• B8=C8

Figure 37.10 shows the solution. The x (0.75), y (−2.0), and z (0.5) values satisfy all three equations.

Figure 37.10. Solver solved the simultaneous equations.

37.4.2. Minimizing shipping costs

This example involves finding alternative options for shipping materials, while keeping total shipping costs at a minimum (see Figure 37.11). A company has warehouses in Los Angeles, St. Louis, and Boston. Retail outlets throughout the United States place orders, which the company then ships from one of the warehouses. The company wants to meet the product needs of all six retail outlets from available inventory and keep total shipping charges as low as possible.

Figure 37.11. This worksheet determines the least expensive way to ship products from warehouses to retail outlets.

This workbook is rather complicated, so each part is explained individually:

• Shipping Costs Table: This table, in range B2:E8, is a matrix that contains per-unit shipping costs from each warehouse to each retail outlet. The cost to ship a unit from Los Angeles to Denver, for example, is $58.

• Product needs of each retail store: This information appears in C12:C17. For example, Denver needs 150 units, Houston needs 225, and so on. C18 contains a formula that calculates the total needed.

• Number to ship from: Range D12:F17 holds the adjustable cells that Solver varies. These cells are all initialized with a value of 25 to give Solver a starting value. Column G contains formulas that sum the number of units the company needs to ship to each retail outlet.

• Warehouse inventory: Row 21 contains the amount of inventory at each warehouse, and row 22 contains formulas that subtract the amount shipped (row 18) from the inventory.

• Calculated shipping costs: Row 24 contains formulas that calculate the shipping costs. Cell D24 contains the following formula, which is copied to the two cells to the right of Cell D24:

  =SUMPRODUCT(C3:C8,D12:D17)

  
  

Cell G24 is the bottom line, the total shipping costs for all orders.

Solver fills in values in the range D12:F17 in such a way that minimizes shipping costs while still supplying each retail outlet with the desired number of units. In other words, the solution minimizes the value in cell G24 by adjusting the cells in D12:F17, subject to the following constraints:

• The number of units needed by each retail outlet must equal the number shipped. (In other words, all the orders are filled.) These constraints are represented by the following specifications:

  C12=G12   C14=G14   C16=G16
  C13=G13   C15=G15   C17=G17

  
  

• The adjustable cells can't be negative because shipping a negative number of units makes no sense. These constraints are represented by the following specifications:

  D12>=0   E12>=0   F12>=0
  D13>=0   E13>=0   F13>=0
  D14>=0   E14>=0   F14>=0
  D15>=0   E15>=0   F15>=0
  D16>=0   E16>=0   F16>=0
  D17>=0   E17>=0   F17>=0

  
  

• The number of units remaining in each warehouse's inventory must not be negative (that is, they can't ship more than what's available). This is represented by the following constraint specifications:

  D22>=0   E22>=0   F22>=0

  
  

Setting up the problem is the difficult part. For example, you must enter 27 constraints. When you have specified all the necessary information, click the Solve button to put Solver to work. Solver displays the solution shown in Figure 37.12.

www.solver.com

The total shipping cost is $55,515, and all the constraints are met. Notice that shipments to Miami come from both St. Louis and Boston.

Figure 37.12. The solution that was created by Solver.

37.4.3. Allocating resources

The example in this section is a common type of problem that's ideal for Solver. Essentially, problems of this sort involve optimizing the volumes of individual production units that use varying amounts of fixed resources. Figure 37.13 shows an example for a toy company.

This company makes five different toys, which use six different materials in varying amounts. For example, Toy A requires 3 units of blue paint, 2 units of white paint, 1 unit of plastic, 3 units of wood, and 1 unit of glue. Column G shows the current inventory of each type of material. Row 10 shows the unit profit for each toy.

Figure 37.13. Using Solver to maximize profit when resources are limited.

The number of toys to make is shown in the range B11:F11. These are the values that Solver determines (the changing cells). The goal of this example is to determine how to allocate the resources to maximize the total profit (B13). In other words, Solver determines how many units of each toy to make. The constraints in this example are relatively simple:

• Ensure that production doesn't use more resources than are available. This can be accomplished by specifying that each cell in column I is greater than or equal to 0 (zero).

• Ensure that the quantities produced aren't negative. This can be accomplished by specifying that each cell in row 11 be greater than or equal to 0.

Figure 37.14 shows the results that are produced by Solver. It shows the product mix that generates $12,365 in profit and uses all resources in their entirety, except for glue.

Figure 37.14. Solver determined how to use the resources to maximize the total profit.

37.4.4. Optimizing an investment portfolio

This example demonstrates how to use Solver to help maximize the return on an investment portfolio. A portfolio consists of several investments, each of which has a different yield. In addition, you may have some constraints that involve reducing risk and diversification goals. Without such constraints, a portfolio problem becomes a no-brainer: Put all your money in the investment with the highest yield.

This example involves a credit union (a financial institution that takes members' deposits and invests them in loans to other members, bank CDs, and other types of investments). The credit union distributes part of the return on these investments to the members in the form of dividends, or interest on their deposits.

This hypothetical credit union must adhere to some regulations regarding its investments, and the board of directors has imposed some other restrictions. These regulations and restrictions comprise the problem's constraints. Figure 37.15 shows a workbook set up for this problem.

Figure 37.15. This worksheet is set up to maximize a credit union's investments, given some constraints.

The following constraints are the ones to which you must adhere in allocating the $5 million portfolio:

• The amount that the credit union invests in new-car loans must be at least three times the amount that the credit union invests in used-car loans. (Used-car loans are riskier investments.) This constraint is represented as

  C5>=C6*3

  
  

• Car loans should make up at least 15 percent of the portfolio. This constraint is represented as

  D14>=.15

  
  

• Unsecured loans should make up no more than 25 percent of the portfolio. This constraint is represented as

  E8<=.25

  
  

• At least 10 percent of the portfolio should be in bank CDs. This constraint is represented as:

  E9>=.10

  
  

• The total amount invested is $5,000,000.

• All investments should be positive or zero. In other words, the problem requires five additional constraints to ensure that none of the changing cells goes below zero.

The changing cells are C5:C9, and the goal is to maximize the total yield in cell D12. Starting values of 1,000,000 have been entered in the changing cells. When you run Solver with these parameters, it produces the solution shown in Figure 37.16, which has a total yield of 9.25 percent.

Figure 37.16. The results of the portfolio optimization.