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$Quick Tour of Microsoft Excel SolverMonthQ1Q2Q3Q4TotalSeasonality
Units Sold
Sales RevenueColor Coding
Cost of SalesGross Margin Target cell
Salesforce Changing cellsAdvertising
Corp Overhead ConstraintsTotal CostsProd. Profit
Profit Margin
Product PriceProduct CostbThe following examples show you how to work with the model above to solve for one value or severalevalues to maximize or minimize another value, enter and change constraints, and save a problem model.RowContainsExplanationFixed values:Seasonality factor: sales are higher in quarters 2 and 4,and lower in quarters 1 and 3.=35*B3*(B11+3000)^0.55Forecast for units sold each quarter: row 3 contains3the seasonality factor; row 11 contains the cost ofadvertising. =B5*$B$185Sales revenue: forecast for units sold (row 5) timesprice (cell B18). =B5*$B$195Cost of sales: forecast for units sold (row 5) timesproduct cost (cell B19).=B6B73Gross margin: sales revenues (row 6) minus cost ofsales (row 7).Sales personnel expenses.)Advertising budget (about 6.3% of sales).=0.15*B64Corporate overhead expenses: sales revenues (row 6)
times 15%.
=SUM(B10:B12)4Total costs: sales personnel expenses (row 10) plus,advertising (row 11) plus overhead (row 12).=B8B137Product profit: gross margin (row 8) minus total costs (row 13).=B15/B68Profit margin: profit (row 15) divided by sales revenue(row 6).Product price.
Product cost.fThis is a typical marketing model that shows sales rising from a base figure (perhaps due to the salesepersonnel) along with increases in advertising, but with diminishing returns. For example, the firstf$5,000 of advertising in Q1 yields about 1,092 incremental units sold, but the next $5,000 yields onlyabout 775 units more.bYou can use Solver to find out whether the advertising budget is too low, and whether advertising _should be allocated differently over time to take advantage of the changing seasonality factor.Solving for a Value to Maximize Another ValueeOne way you can use Solver is to determine the maximum value of a cell by changing another cell. Thehtwo cells must be related through the formulas on the worksheet. If they are not, changing the value in5one cell will not change the value in the other cell.`For example, in the sample worksheet, you want to know how much you need to spend on advertisingjto generate the maximum profit for the first quarter. You are interested in maximizing profit by changingadvertising expenditures.nJOn the Tools menu, click Solver. In the Set target cell box, type b15 or ) 8C FQselect cell B15 (firstquarter profits) on the worksheet. Select the Max option.F IUIn the By changing cells box, type b11 or select cell B11 (firstquarter advertising) # &on the worksheet. Click Solve. dYou will see messages in the status bar as the problem is set up and Solver starts working. After acmoment, you'll see a message that Solver has found a solution. Solver finds that Q1 advertising of*$17,093 yields the maximum profit $15,093.MAfter you examine the results, select Restore original values and click OK to& =H J<discard the results and return cell B11 to its former value.Resetting the Solver OptionshIf you want to return the options in the Solver Parameters dialog box to their original settings so that) :5you can start a new problem, you can click Reset All.+ 4.Solving for a Value by Changing Several ValuesgYou can also use Solver to solve for several values at once to maximize or minimize another value. Forkexample, you can solve for the advertising budget for each quarter that will result in the best profits forlthe entire year. Because the seasonality factor in row 3 enters into the calculation of unit sales in row 5gas a multiplier, it seems logical that you should spend more of your advertising budget in Q4 when the esales response is highest, and less in Q3 when the sales response is lowest. Use Solver to determinethe best quarterly allocation.POn the Tools menu, click Solver. In the Set target cell box, type f15 or select ) 8C FTcell F15 (total profits for the year) on the worksheet. Make sure the Max option isG JMselected. In the By changing cells box, type b11:e11 or select cells B11:E11 #. 5V(the advertising budget for each of the four quarters) on the worksheet. Click Solve.P ULAfter you examine the results, click Restore original values and click OK to% <G I@discard the results and return all cells to their former values.gYou've just asked Solver to solve a moderately complex nonlinear optimization problem; that is, to findgvalues for the four unknowns in cells B11 through E11 that will maximize profits. (This is a nonlinearaproblem because of the exponentiation that occurs in the formulas in row 5). The results of thisbunconstrained optimization show that you can increase profits for the year to $79,706 if you spend)$89,706 in advertising for the full year.fHowever, most realistic modeling problems have limiting factors that you will want to apply to certainivalues. These constraints may be applied to the target cell, the changing cells, or any other value that*is related to the formulas in these cells.Adding a ConstraintgSo far, the budget recovers the advertising cost and generates additional profit, but you're reaching aapoint of diminishing returns. Because you can never be sure that your model of sales response tofadvertising will be valid next year (especially at greatly increased spending levels), it doesn't seem6prudent to allow unrestricted spending on advertising.eSuppose you want to maintain your original advertising budget of $40,000. Add the constraint to the Oproblem that limits the sum of advertising during the four quarters to $40,000.IOn the Tools menu, click Solver, and then click Add. The Add Constraint 0 3: HKdialog box appears. In the Cell reference box, type f11 or select cell F11 *5 8V(advertising total) on the worksheet. Cell F11 must be less than or equal to $40,000.TThe relationship in the Constraint box is <= (less than or equal to) by default, so "* ,Tyou don't have to change it. In the box next to the relationship, type 40000. ClickH MOK, and then click Solve. NAfter you examine the results, click Restore original values and then click OK% <L Cto discard the results and return the cells to their former values.bThe solution found by Solver allocates amounts ranging from $5,117 in Q3 to $15,263 in Q4. Total `Profit has increased from $69,662 in the original budget to $71,447, without any increase in theadvertising budget.Changing a ConstraintdWhen you use Microsoft Excel Solver, you can experiment with slightly different parameters to decidedthe best solution to a problem. For example, you can change a constraint to see whether the resultseare better or worse than before. In the sample worksheet, try changing the constraint on advertising:dollars to $50,000 to see what that does to total profits.GOn the Tools menu, click Solver. The constraint, $F$11<=40000, should 2 >Malready be selected in the Subject to the constraints box. Click Change. In 5B HK
the Constraint box, change 40000 to 50000. Click OK, and then click Solve. $ )2 4E JJClick Keep solver solution and then click OK to keep the results that are * ,displayed on the worksheet.hSolver finds an optimal solution that yields a total profit of $74,817. That's an improvement of $3,370oover the last figure of $71,447. In most firms, it's not too difficult to justify an incremental investment ofh$10,000 that yields an additional $3,370 in profit, or a 33.7% return on investment. This solution also< iresults in profits of $4,889 less than the unconstrained result, but you spend $39,706 less to get there.Saving a Problem Model\When you click Save on the File menu, the last selections you made in the Solver Parameters J [_dialog box are attached to the worksheet and retained when you save the workbook. However, youacan define more than one problem for a worksheet by saving them individually using Save Model in S ]^the Solver Options dialog box. Each problem model consists of cells and constraints that you ,entered in the Solver Parameters dialog box. ]When you click Save Model, the Save Model dialog box appears with a default selection, based )don the active cell, as the area for saving the model. The suggested range includes a cell for each aconstraint plus three additional cells. Make sure that this cell range is an empty range on the
worksheet.KOn the Tools menu, click Solver, and then click Options. Click Save Model. 0 7@ JIIn the Select model area box, type h15:h18 or select cells H15:H18 on the # *worksheet. Click OK. dNote You can also enter a reference to a single cell in the Select model area box. Solver will use= Nhthis reference as the upperleft corner of the range into which it will copy the problem specifications.^To load these problem specifications later, click Load Model on the Solver Options dialog box,2 <D R\type h15:h18 in the Model area box or select cells H15:H18 on the sample worksheet, and then hclick OK. Solver displays a message asking if you want to reset the current Solver option settings with Athe settings for the model you are loading. Click OK to proceed.3 5?Example 1: Product mix problem with diminishing profit margin.SYour company manufactures TVs, stereos and speakers, using a common parts inventoryZof power supplies, speaker cones, etc. Parts are in limited supply and you must determine[the most profitable mix of products to build. But your profit per unit built decreases withRvolume because extra price incentives are needed to load the distribution channel.TV setStereoSpeakerNumber to Build> Part Name InventoryNo. UsedChassisPicture TubeDiminishingSpeaker ConeReturnsPower Supply Exponent:ElectronicsProfits:
By ProductTotal eThis model provides data for several products using common parts, each with a different profit marginjper unit. Parts are limited, so your problem is to determine the number of each product to build from the/inventory on hand in order to maximize profits.Problem SpecificationsTarget CellD18Goal is to maximize profit.Changing cellsD9:F9Units of each product to build.ConstraintsC11:C15<=B11:B15*Number of parts used must be less than or *equal to the number of parts in inventory.D9:F9>=0Number to build value must be greater than orequal to 0.iThe formulas for profit per product in cells D17:F17 include the factor ^H15 to show that profit per unitcdiminishes with volume. H15 contains 0.9, which makes the problem nonlinear. If you change H15 tob1.0 to indicate that profit per unit remains constant with volume, and then click Solve again, theR WIoptimal solution will change. This change also makes the problem linear.#Example 2: Transportation Problem.bMinimize the costs of shipping goods from production plants to warehouses near metropolitan demandbcenters, while not exceeding the supply available from each plant and meeting the demand from eachmetropolitan area.=Number to ship from plant x to warehouse y (at intersection):Plants:San FranDenverChicagoDallasNew YorkS. Carolina TennesseeArizonaTotals:Demands by Whse >Supply=Shipping costs from plant x to warehouse y (at intersection): Shipping:eThe problem presented in this model involves the shipment of goods from three plants to five regionalawarehouses. Goods can be shipped from any plant to any warehouse, but it obviously costs more tobship goods over long distances than over short distances. The problem is to determine the amounts`to ship from each plant to each warehouse at minimum shipping cost in order to meet the regional/demand, while not exceeding the plant supplies.Target cellB20(Goal is to minimize total shipping cost.C8:G10'Amount to ship from each plant to each
warehouse.B8:B10<=B16:B18,Total shipped must be less than or equal to supply at plant.C12:G12>=C14:G14,Totals shipped to warehouses must be greater&than or equal to demand at warehouses. C8:G10>=0Number to ship must be greater than or equal to 0._You can solve this problem faster by selecting the Assume linear model check box in the Solver 3 FX ^bOptions dialog box before clicking Solve. A problem of this type has an optimum solution at which# (Wamounts to ship are integers, if all of the supply and demand constraints are integers.7Example 3: Personnel scheduling for an Amusement Park.bFor employees working five consecutive days with two days off, find the schedule that meets demand6from attendance levels while minimizing payroll costs.Sch. Days off EmployeesSunMonTueWedThuFriSat ASunday, Monday BMonday, Tuesday C
Tuesday, Wed. DWed., Thursday EThursday, Friday FFriday, Saturday GSaturday, SundaySchedule Totals:
Total Demand:Pay/Employee/Day:
Payroll/Week:jThe goal for this model is to schedule employees so that you have sufficient staff at the lowest cost. Ingthis example, all employees are paid at the same rate, so by minimizing the number of employees workingceach day, you also minimize costs. Each employee works five consecutive days, followed by two daysoff.D20!Goal is to minimize payroll cost.D7:D13Employees on each schedule. D7:D13>=01Number of employees must be greater than or equalD7:D13=Integer'Number of employees must be an integer.F15:L15>=F17:L172Employees working each day must be greater than orequal to the demand.Possible schedules Rows 71311 means employee on that schedule works that day.lIn this example, you use an integer constraint so that your solutions do not result in fractional numbers of_employees on each schedule. Selecting the Assume linear model check box in the Solver Options + >P ^Mdialog box before you click Solve will greatly speed up the solution process. !'Example 4: Working Capital Management.PDetermine how to invest excess cash in 1month, 3month and 6month CDs so as toVmaximize interest income while meeting company cash requirements (plus safety margin).YieldTermPurchase CDs in months: 1mo CDs:1, 2, 3, 4, 5 and 6Interest 3mo CDs:1 and 4 Earned: 6mo CDs:1Month:Month 1Month 2Month 3Month 4Month 5Month 6End
Init Cash:
Matur CDs: Interest:
Cash Uses: End Cash:oIf you're a financial officer or a manager, one of your tasks is to manage cash and shortterm investments in akway that maximizes interest income, while keeping funds available to meet expenditures. You must trade offrthe higher interest rates available from longerterm investments against the flexibility provided by keeping fundsin shortterm investments.hThis model calculates ending cash based on initial cash (from the previous month), inflows from maturingkcertificates of deposit (CDs), outflows for new CDs, and cash needed for company operations for each month.jYou have a total of nine decisions to make: the amounts to invest in onemonth CDs in months 1 through 6;hthe amounts to invest in threemonth CDs in months 1 and 4; and the amount to invest in sixmonth CDs inmonth 1.H8$Goal is to maximize interest earned.B14:G14$Dollars invested in each type of CD.
B15, E15, B16
B14:G14>=02Investment in each type of CD must be greater than<
B15:B16>=0or equal to 0.E15>=0B18:H18>=100000,Ending cash must be greater than or equal to $100,000.jThe optimal solution determined by Solver earns a total interest income of $16,531 by investing as much asopossible in sixmonth and threemonth CDs, and then turns to onemonth CDs. This solution satisfies all of theconstraints.bSuppose, however, that you want to guarantee that you have enough cash in month 5 for an equipmentjpayment. Add a constraint that the average maturity of the investments held in month 1 should not be morethan four months.iThe formula in cell B20 computes a total of the amounts invested in month 1 (B14, B15, and B16), weightedpby the maturities (1, 3, and 6 months), and then it subtracts from this amount the total investment, weighted byp4. If this quantity is zero or less, the average maturity will not exceed four months. To add this constraint,g
restore the original values and then click Solver on the Tools menu. Click Add. Type b20 in the Cell + 19 >L OW Zb fc Reference box, type 0 in the Constraint box, and then click OK. To solve the problem, click Solve. '< >] bnTo satisfy the fourmonth maturity constraint, Solver shifts funds from sixmonth CDs to threemonth CDs. Thejshifted funds now mature in month 4 and, according to the present plan, are reinvested in new threemonth hCDs. If you need the funds, however, you can keep the cash instead of reinvesting. The $56,896 turningjover in month 4 is more than sufficient for the equipment payment in month 5. You've traded about $460 in)interest income to gain this flexibility.&Example 5: Efficient stock portfolio.\Find the weightings of stocks in an efficient portfolio that maximizes the portfolio rate ofYreturn for a given level of risk. This worksheet uses the Sharpe singleindex model; youIcan also use the Markowitz method if you have covariance terms available.Riskfree rateMarket varianceMarket rateMaximum weightBetaResVarWeight*Beta*Var.Stock AStock BStock CStock DTbillsReturnVariancePortfolio Totals: Maximize Return: A21:A29Minimize Risk: D21:D29kOne of the basic principles of investment management is diversification. By holding a portfolio of severalfstocks, for example, you can earn a rate of return that represents the average of the returns from theSindividual stocks, while reducing your risk that any one stock will perform poorly.kUsing this model, you can use Solver to find the allocation of funds to stocks that minimizes the portfolio`risk for a given rate of return, or that maximizes the rate of return for a given level of risk.iThis worksheet contains figures for beta (marketrelated risk) and residual variance for four stocks. Innaddition, your portfolio includes investments in Treasury bills (Tbills), assumed to have a riskfree rate ofjreturn and a variance of zero. Initially equal amounts (20 percent of the portfolio) are invested in each security.oUse Solver to try different allocations of funds to stocks and Tbills to either maximize the portfolio rate ofrreturn for a specified level of risk or minimize the risk for a given rate of return. With the initial allocationeof 20 percent across the board, the portfolio return is 16.4 percent and the variance is 7.1 percent.E18%Goal is to maximize portfolio return.E10:E14Weight of each stock.
E10:E14>=0+Weights must be greater than or equal to 0.E16=1Weights must equal 1.
G18<=0.071Variance must be less than or equal to 0.071.Beta for each stockB10:B13Variance for each stockC10:C13gCells D21:D29 contain the problem specifications to minimize risk for a required rate of return of 16.4bpercent. To load these problem specifications into Solver, click Solver on the Tools menu, click B HP U]Options, click Load Model, select cells D21:D29 on the worksheet, and then click OK until the Q S`Solver Parameters dialog box is displayed. Click Solve. As you can see, Solver finds portfolio2 7Oallocations in both cases that surpass the rule of 20 percent across the board.jYou can earn a higher rate of return (17.1 percent) for the same risk, or you can reduce your risk withoutQgiving up any return. These two allocations both represent efficient portfolios.eCells A21:A29 contain the original problem model. To reload this problem, click Solver on the Tools Q W_ dbmenu, click Options, click Load Model, select cells A21:A29 on the worksheet, and then click OK. %] _jSolver displays a message asking if you want to reset the current Solver option settings with the settings4for the model you are loading. Click OK to proceed.& (9Example 6: Value of a resistor in an electrical circuit.YFind the value of a resistor in an electrical circuit that will dissipate the charge to 1Zpercent of its original value within onetwentieth of a second after the switch is closed.Switch>q0 =voltsq[t] =t =secondsBattery
Capacitor (C)Inductor (L)L =henrysC =faradsResistorR =ohms (R)1/(L*C)(R/(2*L))^2
SQRT(B15B16)
COS(T*B17)
R*T/(2*L)Q0*EXP(B19)`This model depicts an electrical circuit containing a battery, switch, capacitor, resistor, and dinductor. With the switch in the left position, the battery charges the capacitor. When the switch_is thrown to the right, the capacitor discharges through the inductor and the resistor, both of"which dissipate electrical energy.^Using Kirchhoff's second law, you can formulate and solve a differential equation to determine`how the charge on the capacitor varies over time. The formula relates the charge q[t] at time tMto the inductance L, resistance R, and capacitance C of the circuit elements._Use Solver to pick an appropriate value for the resistor R (given values for the inductor L and[the capacitor C) that will dissipate the charge to one percent of its initial value within >onetwentieth of a second after the time the switch is thrown.G15 Goal is to set to value of 0.09.
Changing cellG12 Resistor.D15:D20&Algebraic solution to Kirchhoff's law.`This problem and solution are appropriate for a narrow range of values; the function representedHby the charge on the capacitor over time is actually a damped sine wave.VEEe,FF]HHIJLZ!O xQ.SiVyYD\(_Ta
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