lecture 3 robert zimmer room 6, 25 st james. introduction to optimization modeling

Lecture 3Lecture 3Robert ZimmerRobert Zimmer

Room 6, 25 St JamesRoom 6, 25 St James

Introduction to Optimization ModelingIntroduction to Optimization Modeling

3.2 Introduction to 3.2 Introduction to OptimizationOptimization

► Common elements of all optimization problemsCommon elements of all optimization problems Decision VariablesDecision Variables - the variables whose values the - the variables whose values the

decision maker is allowed to choose.decision maker is allowed to choose. Objective Function Objective Function - value that is to be optimized – - value that is to be optimized –

maximized or minimizedmaximized or minimized Constraints Constraints that must be satisfiedthat must be satisfied

► Excel terminology for optimizationExcel terminology for optimization Decision variables = Decision variables = changing cellschanging cells Objective = Objective = target celltarget cell ConstraintsConstraints impose restrictions on the values in the impose restrictions on the values in the

changing cells.changing cells.

► A common form for a constraint is A common form for a constraint is nonnegativitynonnegativity► NonnegativityNonnegativity constraints imply that changing constraints imply that changing

cells must contain nonnegative values.cells must contain nonnegative values.► Two steps in solving an optimization problem.Two steps in solving an optimization problem.

Model developmentModel development – decide what the decision variables – decide what the decision variables are, what the objective is, which constraints are required are, what the objective is, which constraints are required and how everything fits togetherand how everything fits together

OptimizeOptimize – systematically choose the values of the – systematically choose the values of the decision variables that make the objective as large or small decision variables that make the objective as large or small as possible and cause all of the constraints to be satisfied.as possible and cause all of the constraints to be satisfied.

► A A feasible solutionfeasible solution is any set of values of the is any set of values of the decision variables that satisfies all of the decision variables that satisfies all of the constraints.constraints.

► The set of all feasible solutions is called the The set of all feasible solutions is called the feasible regionfeasible region..

► An An infeasible solution infeasible solution is a solution where at least is a solution where at least one constraint is not satisfied.one constraint is not satisfied.

► The The optimal solutionoptimal solution is the feasible solution that is the feasible solution that optimizes the objective.optimizes the objective.

► An An algorithmalgorithm is basically a “plan of attack”. It is basically a “plan of attack”. It

is a prescription for carrying out the steps is a prescription for carrying out the steps required to achieve some goal.required to achieve some goal.

► The The simplex methodsimplex method is an algorithm that is is an algorithm that is suitable for linear models.suitable for linear models.

► Excel’s Solver tool finds the best feasible Excel’s Solver tool finds the best feasible solution with the most suitable algorithm.solution with the most suitable algorithm.

► There is really a There is really a thirdthird step in the optimization step in the optimization process: process: sensitivity analysissensitivity analysis. This step . This step allows us to ask a number of what-if questions allows us to ask a number of what-if questions about the completed model.about the completed model.

Example 3.1 – Two Variable Example 3.1 – Two Variable ModelModel

► Maggie decided she must plan her desserts Maggie decided she must plan her desserts carefully. Maggie will allow herself no more then carefully. Maggie will allow herself no more then 450 calories and 25 grams of fat in her daily 450 calories and 25 grams of fat in her daily desserts. She requires at least 120 grams of desserts. She requires at least 120 grams of desserts a day. Each dessert also has a “taste desserts a day. Each dessert also has a “taste index”.index”.

► What should her daily dessert plan be to stay What should her daily dessert plan be to stay within her constraints and maximizes the total within her constraints and maximizes the total taste index of her dessert?taste index of her dessert?

► First step is to identify appropriate decision First step is to identify appropriate decision variables, the appropriate objective, the variables, the appropriate objective, the constraints and the relationships between them.constraints and the relationships between them.

Ex. 3.1(cont’d) - Algebraic ModelEx. 3.1(cont’d) - Algebraic Model

► Identify the decision variable, write expressions Identify the decision variable, write expressions fro the total taste index and the constraints in fro the total taste index and the constraints in terms of the terms of the x’sx’s. Then add explicit constraints . Then add explicit constraints to ensure that the to ensure that the x’sx’s are nonnegative. are nonnegative.

Maximize 37(85)Maximize 37(85)xx11 +65(95) +65(95)xx22

Subject to:Subject to:120120xx11+160+160xx22≤450≤450

55xx11+10+10xx22≤25≤253737xx11+65+65xx22≥120≥120

xx11,,xx22 ≥ 0 ≥ 0

Ex. 3.1(cont’d) - Graphical Ex. 3.1(cont’d) - Graphical ModelModel

► When there are only two decision variables the When there are only two decision variables the problem can be solved graphically.problem can be solved graphically.

► To graph this, consider the associate equality To graph this, consider the associate equality (120(120xx11+160+160xx22=450) and find where the associated =450) and find where the associated line crosses the axes.line crosses the axes.

► Graph the constraints on the figure as shown on the Graph the constraints on the figure as shown on the next slide.next slide.

Ex. 3.1(cont’d) - Graphical Ex. 3.1(cont’d) - Graphical ModelModel

► To see which feasible point maximizes the To see which feasible point maximizes the objective, draw a sequence of lines where, for objective, draw a sequence of lines where, for each, the objective is a constant.each, the objective is a constant.

► The last feasible point that it touches is the The last feasible point that it touches is the optimal point.optimal point.

Ex. 3.1(cont’d) - Spreadsheet Ex. 3.1(cont’d) - Spreadsheet ModelModel

► Common elements in all LP spreadsheet models Common elements in all LP spreadsheet models are:are: InputsInputs – all numeric data given in the statement of the – all numeric data given in the statement of the

problem (Blue border)problem (Blue border) Changing cellsChanging cells – the values in these cells can be changed – the values in these cells can be changed

to optimize the objective (Red border)to optimize the objective (Red border) Target(objective) cellTarget(objective) cell – contains the value of the – contains the value of the

objective (Double line black boarder)objective (Double line black boarder) ConstraintsConstraints – specified in the Solver dialog box – specified in the Solver dialog box NonnegativityNonnegativity – check an option in a Solver dialog box to – check an option in a Solver dialog box to

indicate nonnegative changing cellsindicate nonnegative changing cells


► Three stages of the complete solution:Three stages of the complete solution: Model development stage – enter all inputs, trial values for Model development stage – enter all inputs, trial values for

the changing cells, and formulas relating these in the changing cells, and formulas relating these in spreadsheetspreadsheet

Invoke Solver – designate the objective cell, changing cells, Invoke Solver – designate the objective cell, changing cells, the constraints and selected options, and tell Solver to find the constraints and selected options, and tell Solver to find the the optimal optimal solution.solution.

Sensitivity analysis – see how the optimal solution changes Sensitivity analysis – see how the optimal solution changes as the selected inputs varyas the selected inputs vary


► Solver dialog box for this model.Solver dialog box for this model.


► Optimal Solution for the Dessert ModelOptimal Solution for the Dessert Model


► In this solution the calorie and fat constraints have In this solution the calorie and fat constraints have been met exactly, thus they are been met exactly, thus they are bindingbinding. The . The constraint on grams in constraint on grams in nonbindingnonbinding, the positive , the positive difference in grams is calleddifference in grams is called slack slack..

3.4 Sensitivity Analysis3.4 Sensitivity Analysis

► Often it is useful to perform sensitivity analysis to Often it is useful to perform sensitivity analysis to see how (or if) the optimal solution changes as one see how (or if) the optimal solution changes as one or more inputs change.or more inputs change.

► The Solve dialog box offers you the option to The Solve dialog box offers you the option to obtain a sensitivity report.obtain a sensitivity report.

► Solver’s sensitivity report performs two types of Solver’s sensitivity report performs two types of sensitivity analysis:sensitivity analysis:1.1. on the coefficients of the objectives, the on the coefficients of the objectives, the c’sc’s, and, and

2.2. on the right hand sides of the constraints, the on the right hand sides of the constraints, the b’sb’s..

► The sensitivity report has two sections The sensitivity report has two sections corresponding to the two types of analysis. corresponding to the two types of analysis. Example 3.1’s sensitivity report.Example 3.1’s sensitivity report.

► The The reduced costreduced cost for any decision not currently in for any decision not currently in the optimal solution indicates how much better that the optimal solution indicates how much better that coefficient must be before that variable will enter at coefficient must be before that variable will enter at a positive level.a positive level.

► The term The term shadow priceshadow price is an economic term. It is an economic term. It indicates the change in the optimal value of the indicates the change in the optimal value of the objective function when the right-hand side of some objective function when the right-hand side of some constraint changes by a given amount.constraint changes by a given amount.

► The SolverTable Add-in allows us to ask sensitivity The SolverTable Add-in allows us to ask sensitivity questions about questions about anyany of the input variables. of the input variables.

► SolverTable’s can be used in two ways:SolverTable’s can be used in two ways: One-way tableOne-way table – single input cell and any number of – single input cell and any number of

output cellsoutput cells Two-way tableTwo-way table – two input cells and one or more outputs – two input cells and one or more outputs

► The results are easily interpreted.The results are easily interpreted.

► For the dessert model, check how sensitive the For the dessert model, check how sensitive the optimal dessert plan and total taste index are tooptimal dessert plan and total taste index are to(1) changes in the number of calories(1) changes in the number of calories(2) the number of daily dessert calories allowed.(2) the number of daily dessert calories allowed.

► The solution to question (1) can be solved by The solution to question (1) can be solved by selecting the Data/SolverTable menu item and selecting the Data/SolverTable menu item and select a one-way table in the first dialog box. The select a one-way table in the first dialog box. The second dialog box should be completed as shown second dialog box should be completed as shown on the next slide.on the next slide.

► The second question asks us to vary two inputs The second question asks us to vary two inputs simultaneously. This requires a two-way SolverTable. simultaneously. This requires a two-way SolverTable. Select the two-way option in the first SolverTable Select the two-way option in the first SolverTable dialog box to get the two-way table dialog box.dialog box to get the two-way table dialog box.

3.5 Properties of Linear 3.5 Properties of Linear ModelsModels

► Linear programming is an important subset of a Linear programming is an important subset of a larger class of models called larger class of models called mathematical mathematical programming modelsprogramming models..

► Three important properties that LP models possessThree important properties that LP models possess Proportionality Proportionality

► If a level of any activity is multiplied by a constant factor, the If a level of any activity is multiplied by a constant factor, the contribution of this activity to the objective, or to any of the contribution of this activity to the objective, or to any of the constraints in which the activity is involved, is multiplied by constraints in which the activity is involved, is multiplied by the same factor.the same factor.

AdditivityAdditivity► This property implies that the sum of the contributions from This property implies that the sum of the contributions from

the various activities to a particular constraint equals the total the various activities to a particular constraint equals the total contribution to that constraint.contribution to that constraint.

DivisibilityDivisibility► This property means that both integer and noninteger levels This property means that both integer and noninteger levels

of the activities are allowed.of the activities are allowed.

► How can you recognize whether a model satisfies How can you recognize whether a model satisfies proportionality and additivity?proportionality and additivity?

► Not easy to recognize in a spreadsheet model Not easy to recognize in a spreadsheet model because the logic of the model can be embedded because the logic of the model can be embedded in a series of cell formulas.in a series of cell formulas.

► Often it is easier to recognize when a model is not Often it is easier to recognize when a model is not linear. Two situations that lead to nonlinear models linear. Two situations that lead to nonlinear models are whenare when1.1. there are products or quotients of expressions involving there are products or quotients of expressions involving

changing cells, andchanging cells, and

2.2. there are nonlinear functions, such as squares, square there are nonlinear functions, such as squares, square roots, or logarithms, of changing cells.roots, or logarithms, of changing cells.

► Real-life problems are almost never exactly linear. Real-life problems are almost never exactly linear. However, a linear approximation often yields very However, a linear approximation often yields very useful results.useful results.

► In terms of Solver, if the model is linear the Assume In terms of Solver, if the model is linear the Assume Linear Model box must be checked in the Solver Linear Model box must be checked in the Solver Options dialog box.Options dialog box.

► Check the Assume Linear Model box even if the Check the Assume Linear Model box even if the divisibility property is violated.divisibility property is violated.

► If the Solver returns a message that “the condition If the Solver returns a message that “the condition for Assume Linear Model are not satisfied” itfor Assume Linear Model are not satisfied” it can indicate a logical error in your formulation.can indicate a logical error in your formulation. can also indicate that Solver erroneously thinks the can also indicate that Solver erroneously thinks the

linearity conditions are not satisfied.linearity conditions are not satisfied.

► Try not checking the Assume Linear model box and Try not checking the Assume Linear model box and see if that works. In any case it always helps to see if that works. In any case it always helps to have a well-scaled model.have a well-scaled model.

3.6 Infeasibility and 3.6 Infeasibility and UnboundednessUnboundedness

► It is possible that there are no feasible solutions to It is possible that there are no feasible solutions to a model. There are generally two possible reasons a model. There are generally two possible reasons for this:for this:1.1. There is a mistake in the model (an input entered There is a mistake in the model (an input entered

incorrectly) orincorrectly) or

2.2. the problem has been so constrained that there are no the problem has been so constrained that there are no solutions left.solutions left.

► In general, there is no foolproof way to find the In general, there is no foolproof way to find the problem when a “no feasible solution” message problem when a “no feasible solution” message appears.appears.

► A second type of problem is A second type of problem is unboundednessunboundedness..► Unboundedness is that the model can be made as Unboundedness is that the model can be made as

large as possible. If this occurs it is likely that a large as possible. If this occurs it is likely that a wrong input has been entered or forgotten some wrong input has been entered or forgotten some constraints.constraints.

► Infeasibility and unboundedness are quite different. Infeasibility and unboundedness are quite different. It is possible for a model to have no feasible It is possible for a model to have no feasible solution but no realistic model can have an solution but no realistic model can have an unbounded solution.unbounded solution.

Example 3.2 – Product Mix Example 3.2 – Product Mix ModelModel

► The product mix problem is basically to select the The product mix problem is basically to select the optimal mix of products to produce to maximize profit.optimal mix of products to produce to maximize profit.

► The Monet company produces four types of picture The Monet company produces four types of picture frames. The four types differ with respect to size, shape frames. The four types differ with respect to size, shape and materials used.and materials used.

► Each frame requires a certain amount of skilled labor, Each frame requires a certain amount of skilled labor, metal and glass. They also all have different selling metal and glass. They also all have different selling prices.prices.

► Monet can produce in the coming week but they do not Monet can produce in the coming week but they do not want any inventory at the end of the week.want any inventory at the end of the week.

► What should the company do to maximize its profit for What should the company do to maximize its profit for this week?this week?


MaximizeMaximize 66xx11 + 2 + 2xx22 + 4 + 4xx33 + 3 + 3xx44 (profit objective) (profit objective)

Subject toSubject to 22xx11 + + xx22 + 3 + 3xx33 + 2 + 2xx44 4000 (labor constraint) 4000 (labor constraint)

44xx11 + 2 + 2xx22 + + xx33 + 2 + 2xx44 10,000 (glass constraint) 10,000 (glass constraint)

xx11 1000 (frame 1 sales 1000 (frame 1 sales constraints)constraints)




xx11, , xx22, , xx33, , xx44 0 (nonnegativity constraint) 0 (nonnegativity constraint)


► To develop the spreadsheet model follow these To develop the spreadsheet model follow these steps:steps: InputsInputs - Enter the various inputs in the shaded ranges. - Enter the various inputs in the shaded ranges.

Enter only numbers, not formulas in the input cells. Enter only numbers, not formulas in the input cells. Range namesRange names – Name the ranges as indicated. – Name the ranges as indicated. Changing cellsChanging cells - Enter any four values in the range - Enter any four values in the range

named Produced. named Produced. Resources usedResources used - Enter the formula - Enter the formula

=SUMPRODUCT (B9:E9,Produced)=SUMPRODUCT (B9:E9,Produced) in cell in cell B21 and copy it to the rest of the Used range. B21 and copy it to the rest of the Used range.

Revenues, costs, and profitsRevenues, costs, and profits – Enter the formulas to – Enter the formulas to calculate these values.calculate these values.

► The optimal solution for the product mix model The optimal solution for the product mix model is shown on the next slide.is shown on the next slide.

► The sensitivity analysis allows us to experiment The sensitivity analysis allows us to experiment with different inputs to this problem. Simply with different inputs to this problem. Simply change the inputs and then rerun Solver.change the inputs and then rerun Solver.

► Use SolverTable to perform a more systematic Use SolverTable to perform a more systematic sensitivity analysis on one or more input sensitivity analysis on one or more input variables.variables.

► Additional insight can be gained from Solver’s Additional insight can be gained from Solver’s sensitivity report.sensitivity report.

Example 3.3 – Another Product Example 3.3 – Another Product Mix ModelMix Model

► Pigskin company must decide how many footballs Pigskin company must decide how many footballs to produce each month. It has decided to us a 6-to produce each month. It has decided to us a 6-month planning horizon.month planning horizon.

► Pigskin wants to determine the production schedule Pigskin wants to determine the production schedule that minimizes the total production and holding that minimizes the total production and holding costs.costs.

► By modeling this type of problem, one needs to be By modeling this type of problem, one needs to be very specific about the very specific about the timingtiming events. events.

► By modifying the timing assumptions in this type of By modifying the timing assumptions in this type of model, one can get alternative – and equally model, one can get alternative – and equally realistic – models with very different solutions.realistic – models with very different solutions.


► The decision variables are the production quantities The decision variables are the production quantities for the 6 months (for the 6 months (PP11 through through PP66). ). II11 through through II66 is the is the corresponding end-of-month inventories.corresponding end-of-month inventories.

► The obvious constraints are on the production and The obvious constraints are on the production and inventory storage capacities for each month, inventory storage capacities for each month, jj..

► In addition, "balance” constraints that relate to In addition, "balance” constraints that relate to P’sP’s and and I’s I’s are needed. The balance equation for the are needed. The balance equation for the month month j is Ij is Ijj--1 + 1 + PPjj = = DDjj + + IIjj. .

Ex. 3.3(cont’d) - Algebraic Ex. 3.3(cont’d) - Algebraic ModelModel

► By putting all variables (P’s and I’s) on the left and all By putting all variables (P’s and I’s) on the left and all known values on the right (a standard LP convention), known values on the right (a standard LP convention), these balance constraints becomethese balance constraints become

P1 – I1 = 100-50P1 – I1 = 100-50

I1 + P2 – I2 = 150I1 + P2 – I2 = 150

I2 + P3 – I3 = 300I2 + P3 – I3 = 300

I3 + P4 – I4 = 350I3 + P4 – I4 = 350

I4 + P5 – I5 = 250I4 + P5 – I5 = 250

I5 + P6 – I6 = 100I5 + P6 – I6 = 100► The goal is to minimize the sum of production and The goal is to minimize the sum of production and

holding costs. It is the sum of unit production costs holding costs. It is the sum of unit production costs multiplied by multiplied by P’s, P’s, plus until holding costs multiplied by plus until holding costs multiplied by I’sI’s..


► The difference between this model from the The difference between this model from the product mix model is that some of the constraints product mix model is that some of the constraints are built into the spreadsheet itself by means of are built into the spreadsheet itself by means of the formulas.the formulas.

► The only changing cells are production quantities. The only changing cells are production quantities. ► The decision variables in an algebraic model are The decision variables in an algebraic model are

not necessarily the same as the changing cells in not necessarily the same as the changing cells in an equivalent spreadsheet model.an equivalent spreadsheet model.

► To develop the spreadsheet model:To develop the spreadsheet model: InputsInputs - Enter the inputs in the shaded ranges. - Enter the inputs in the shaded ranges. Name rangesName ranges – Name ranges indicated. – Name ranges indicated.


Production quantitiesProduction quantities - Enter any values in the range - Enter any values in the range Produced as the production quantities. As always, you can Produced as the production quantities. As always, you can enter values that you believe are good, maybe even enter values that you believe are good, maybe even optimal. optimal.

On-hand inventoryOn-hand inventory - Enter the formula - Enter the formula =B4 + B12=B4 + B12 in in cell B16. This calculates the first month on-hand inventory cell B16. This calculates the first month on-hand inventory after production. Then enter the “typical” formula after production. Then enter the “typical” formula =B20 + =B20 + C12C12 for on-hand inventory after production in month 2 in for on-hand inventory after production in month 2 in cell C16 and copy it across row 16.cell C16 and copy it across row 16.

Ending inventoriesEnding inventories - Enter the formula - Enter the formula =B16 – B18=B16 – B18 for for ending inventory in cell B20 and copy it across row 20.ending inventory in cell B20 and copy it across row 20.

Production and holding costs -Production and holding costs - Enter the formula Enter the formula calculate the monthly holding costs. Finally, calculate the calculate the monthly holding costs. Finally, calculate the cost totals in column H by summing with the SUM function.cost totals in column H by summing with the SUM function.


► The optimal solution from Solver.The optimal solution from Solver.


► SolverTable can be used to perform a number of SolverTable can be used to perform a number of interesting sensitivity analyses.interesting sensitivity analyses.

► In multiperiod models, the company has to make In multiperiod models, the company has to make forecasts about the future, such as the level of forecasts about the future, such as the level of demand. The length of the planning horizon is demand. The length of the planning horizon is usually the length of time for which the company usually the length of time for which the company can make reasonably accurate forecasts.can make reasonably accurate forecasts.

3.10 Decision Support 3.10 Decision Support SystemSystem

► Many people who are not experts need to use Many people who are not experts need to use models. models.

► It is useful to provide these users with a It is useful to provide these users with a decision decision support system support system (DSS) that can help them solve (DSS) that can help them solve problems without having to worry about technical problems without having to worry about technical details.details.

► The users sees a “front end” and a “back end”. The users sees a “front end” and a “back end”. The front end allows them to select input values. The front end allows them to select input values. The back end then produces a report that explains the The back end then produces a report that explains the

optimal policy in nontechnical terms.optimal policy in nontechnical terms.

► A “front-end” for a problem similar to the A “front-end” for a problem similar to the

Pigskin model.Pigskin model.

► A “back-end” for a problem similar to the Pigskin model.A “back-end” for a problem similar to the Pigskin model.

lecture 3 robert zimmer room 6, 25 st james. introduction to optimization modeling

Documents

appropriate decision

explicit constraints

minimized constraints

decision maker

changing cells objective

appropriate objective

best feasible solution

target cell constraints