Sensitivity Analysis with Excel


 Kory Pearson
 1 years ago
 Views:
Transcription
1 Sensitivity Analysis with Excel 1 Lecture Outline Sensitivity Analysis Effects on the Objective Function Value (OFV): Changing the Values of Decision Variables Looking at the Variation in OFV: Excel One and TwoWay Data Tables, and Scenario Manager Finding the Optimum OFV: Excel Solver Effects on the Optimum: Making One Change to the Model Parameters: Changing a Decision Variable Coefficient in a Constraint (Graphical intuitions) Changing the Right Hand Side (RHS) of a Constraint: Slack & Shadow Price Changing a Decision Variable Coefficient in the Objective Function: Reduced Cost Making Multiple Changes to the Model Parameters The 100% Rule SolverTable 2 1
2 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 3 Problem Statement Boxers and Briefs Example  Revisited Champion Sports manufactures two types of custom men's underwear: boxers and briefs. How many boxers and how many briefs should be produced per week, to maximize profits, given the following constraints The (profit) contribution per boxer is $3.00, compared to $4.50 per brief. Briefs use 0.5 yards of material; boxers use 0.4 yards. 300 yards of material are available. It requires 1 hour to manufacture one pair of boxers and 2 hours for one pair of briefs. 900 labors hours are available. There is unlimited demand for boxers but total demand for briefs is 375 units per week. Each boxer uses 1 insignia logo and 600 insignia logos are in stock. 4 2
3 Algebraic Formulation 5 The Algebraic LP Formulation Revisited Variables: number of Boxers, number of Briefs. Objective function: Objective Function Coefficients maximize ( $3.00 x Boxers ) + ( $4.50 x Briefs ) Constraint Coefficients Constraints: Material: (.4 x Boxers ) + (.5 x Briefs ) <= 300 yards Labor: ( 1 x Boxers ) + ( 2 x Briefs ) <= 900 hrs Demand: ( 0 x Boxers ) + ( 1 x Briefs ) <= 375 units Logos: ( 1 x Boxers ) + ( 0 x Briefs ) <= 600 NonNegativity: Boxers >= 0 Briefs >= 0 Constraints Right Hand Sides (RHS) 6 3
4 Graphical Formulation 7 The Graphical LP Formulation  Revisited Optimal Solution Boxers F Hours E D A C Logos Demand Optimum Material B Briefs $3.00 $4.50 Total Boxers Briefs Profit A $1, B $1, C $2, D $2, E $2, F $1,
5 Excel Formulation 9 Excel Formulation  Revisited Boxers and Briefs Example: Profit Maximization See boxers_and_briefs_example.xls in the downloads for today s class. 10 5
6 Excel Tools for Sensitivity Analysis Oneway Data Table, TwoWay Table, Scenario Manager Solver Answer Report Slack Pertains to changing RHS of a constraint Solver Sensitivity Report Shadow Price, Allowable Increase / Decrease Reduced Cost, Allowable Increase / Decrease SolverTable Pertains to changing Objective Function Coefficients 11 Changing the Value of Decision Variables With Two Way Data Tables The twoway data table below shows the effect, on profit, of changing the two decision variables (i.e. the number of boxers produced, and the number of briefs produced). This initial analysis assumes there are no constraints. Clearly, if constraints are taken into account, the maximum profit in the data table shown below ($4350, for 700 boxers and 500 briefs) would change since that is not a feasible solution. 12 6
7 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 13 Changing the Value of Decision Variables With Two Way Data Tables The twoway data table below shows the effect on profit of changing the number of boxers and briefs produced. Conditional formatting has been used to highlight infeasible solutions (in red) and the optimal feasible solution (in green). Don t panic! You aren t expected to know how to construct the table / formula shown below its only shown to illustrate how complex it is to create a table that shows the feasible and optimal solutions! Notice that we were lucky in this instance that the optimum solution falls at a point that was in our table: had we chosen broader intervals for number of boxers and briefs we would have found a good, but not optimal, solution. 14 7
8 Excel Answer Report  Revisited Boxers and Briefs Example: Profit Maximization Fortunately, as we ve seen Excel provides us with an easy mean of finding the optimal feasible solution: simply use Excel Solver. Make sure Solver is available: If not, Tools AddIns Solver Addin Use it: Tools Solver Read the Answer Report to find the optimal value, and the decision variable settings at this optimum. 15 Excel Answer Report Optimum Objective Function Value Optimum Product Mix Status of the Constraints 16 8
9 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 17 Answer Report The Answer Report also shows us which constraints are binding and nonbinding at the optimum. Slack indicates the spare capacity on a nonbinding constraint at the optimum. Slack = 0 implies constraint is binding: resources are exhausted Slack 0 implies constraint is nonbinding: there are leftover resources We can see below, for instance, that: getting additional logos wouldn t help us improve profit since we already have 100 unused logos at the optimum advertising our briefs to stimulate demand (at the current price) would be money wasted, since demand for briefs is already greater than the number of briefs we should produce at the optimum. material and labor constraints are cramping our profit. 18 9
10 Slack and Binding Slack measures unused available resources Binding constraints Optimal solution lies on binding constraints Multiple binding constraints means solution is at intersection of constraints: a vertex Slack and binding constraints Slack = 0 : the resource is exhausted, the constraint is binding, the optimal vertex includes this constraint. Slack 0 : spare capacity is available, constraint is not binding 19 Excel Sensitivity Analysis Report Choose Sensitivity to see a more detailed Sensitivity Report... Pertain to Objective Function Coefficient Ranging Pertain to Right Hand Side (RHS) Ranging on Constraints 20 10
11 Excel Sensitivity Analysis Report The Excel Sensitivity Analysis Report allows you to see: over what range and under what conditions the components of a solution remain unchanged how sensitive a solution is to changes in the data, and to get an insight into how technological improvements may affect optimum values. 21 Boxers Sensitivity Analysis Changing a Decision Variable CoEfficient in a Constraint Graphical Intuitions The Sensitivity Report doesn t tell us anything about what happens when the coefficients of a decision variable in a constraint change, but here are some graphical intuitions Hours Logos Demand Assume we changed the amount of material required per brief from 0.5 yards to 1 yard. Notice how the shape of the feasible region changes, and the optimal solution changes. Material Boxers Hours Logos Demand Material Briefs Briefs 11
12 Boxers Sensitivity Analysis Report Changing the Right Hand Side (RHS) of a Constraint Graphical Intuitions Changing the RHS of a constraint causes the constraint line to shift left or right, but does not alter the slope of the line! Hours Demand Logos Material Assume an extra 100 yards of material became available (increasing our total available to 400 yards). Notice how the material constraint shifts right. The change is greater than the allowable increase (see later) and the optimum solution vertex changes. Boxers Hours Demand Logos Material Briefs Briefs Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint The Shadow Price What is meant by the optimal vertex? The optimal vertex is : what intersection of constraints is the optimal solution to be found at?. From the previous slide, we saw that the original optimal vertex was: at the intersection of the materials and hours constraints. However, when we increased the RHS of the materials constraint beyond the maximum allowable increase (i.e. by 100 yards), the optimal vertex shifts to: at the intersection of the logos and hours constraints. Had the change in materials been within the allowable increase, then the optimal vertex would have stayed the same (i.e. it would still have been at the intersection of the materials and hours constraints) but the optimal product mix and the optimal solution value would have changed
13 Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint Graphical Intuitions Changing the RHS of a constraint causes the constraint line to shift to a parallel position to the left or right, but does not alter the slope of the line! Assume a extra 100 hours of labor became available (increasing our total available to 1000 hours). Notice how the labor constraint shifts right, and the optimal solution value and optimal product mix change. However, the change is less than the allowable increase for this constraint and the optimum solution vertex stays the same: the optimal solution is still at the intersection of the materials and labor hours constraints. Boxers Hours Demand Logos Material Briefs Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint Tighten constraint: make feasible region smaller. Optimal value can only get worse (fewer choices) Loosen (relax) constraint: make feasible region larger. Optimal value can only do better (more choices) Assume b is a positive number and the constraint line has the form y = ax + b (i.e. y ax = b) then increasing the RHS (i.e. b) will cause the line to shift vertically up and will: Tighten the constraint if it s a lowerbound constraint Loosen the constraint if its an upperbound constraint So notice that increasing the RHS may have positive or negative effects on the optimum, depending on the type of constraint! In contrast, if b is a positive number and the constraint line has the form y = ax b (i.e. ax y = b) then increasing the RHS (i.e. b) will cause the line to shift vertically down! 26 13
14 Sensitivity Analysis Report Changing the Right Hand Side (RHS) of a Constraint The Shadow Price The Shadow Price for a constraint is the change in the optimal objective function value per unit increase in the Right Hand Side (RHS) of a given constraint. The Shadow Price only remains valid within the Allowable Increase and Decrease shown for that Shadow Price. 27 Shadow Prices Example A stain is found on 15 yards of material, reducing material from 300 to 285 yards. How does this affect optimal profit? New optimal profit = Old optimal profit  (shadow price x yards) = $2,400 ($5/yard x 15 yards) = $2,325 Notice the Allowable range for the Materials Usage constraint. What could you say about a stain on 60 yards? 28 14
15 Shadow Prices Example Labor is willing to negotiate 100 additional hours of production work. How much of a premium should management pay for overtime hours? Addition to optimal profit = shadow price x hours = $1/hour x 100 hours = $100 If $9.28 for 900th hour, then $10.28 for 901st hour. But remember, the productivity rate will change when people work longer hours 29 Increase Labor by 100 hours Boxers F Hours Logos E D G A Demand C B Material Briefs $3.00 $4.50 Total Boxers Briefs Contribution A $1, B $1, C $2, D $2, E $2, F $1, G $2,
16 Shadow Prices Example How much would you pay for one additional insignia logo? Nothing: logos are not constraining the solution! Notice the range for logos: what is 1E+30? 31 Shadow Price / RHS Ranging Shadow price Shadow price = marginal change to objective function value of increasing constraint RHS by 1 unit Scaling issues: changing one model unit (e.g. model may be in millions of units ) Slack and shadow price Shadow price = 0 implies constraint is not binding Shadow price 0 implies Slack = 0 Slack and allowable increase/decrease How much you must tighten a constraint to make it bind Notice that, for a nonbinding constraint, either the allowable increase or the allowable decrease will be equal to the slack, since either adding or subtracting the slack to / from the RHS will make the constraint bind
17 Shadow Price / RHS Ranging Shadow price on a constraint: Change in optimal objective function value per unit change in righthandside of the constraint zero if constraint is nonbinding Shadow price and RHS ranging: Allowable increase / decrease = Range of RHS coefficients for which shadow prices remain valid. Optimal value (production mix) will change: If binding constraints are moved, the optimal mix changes The objective function value changes The shadow price allows us to predict new optimal value. Need to resolve the model to get the new mix (decision variables). 33 Allowable Increase / Decrease for the RHS of a Constraint Allowable increase/decrease in constraints: How much you can tighten or relax a constraint RHS and remain binding (or nonbinding) (see also: Slack) Range in RHS coefficients for which the shadow price remains valid (see also: Shadow Price) Feasible Region: How much you can reshape the feasible region without changing the optimal vertex. (The optimal mix will always change, but the optimal vertex stays the same within the allowable increase/decrease.) 34 17
18 Allowable Increase / Decrease for the RHS of a Constraint Change within allowable increase/decrease Optimal vertex (intersection of the binding constraints) is unchanged. Optimal value of objective function is updated by Shadow Price. Optimal product mix (value of decision variables) is calculated by retesting the model. Change outside allowable increase/decrease Start over. 35 Shadow Price, Slack, and Binding 36 18
19 Shadow Price in Maximization vs Minimization Problems For maximization problems: increase in the objective function value is good (+ve shadow price is good, and helps the optimum solution) decrease in the objective function value is bad (ve shadow price is bad, and hurts the optimum solution) For minimization problems: increase in the objective function value is bad (+ve shadow price is bad, and hurts the optimum solution) decrease in the objective function value is good (ve shadow price is good, and helps the optimum solution) 37 Shadow Prices in a Minimization Problem Our objective in the Big Mac Attack Problem is to achieve leastcost in meeting our Recommended Daily Allowance (RDA) requirements: including both upper and lower limits
20 Shadow Prices in a Minimization Problem Sodium was an upper bound constraint. It has a negative shadow price: increasing the RHS of the sodium constraint would loosen the constraint and allow us to achieve a better optimum: i.e. a lower meal cost. Vitamin C was an lower bound constraint. It has a positive shadow price: increasing the RHS of the Vitamin C constraint would tighten the constraint and force us to a worse optimum: i.e. a higher meal cost. Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$21 TOTAL Protein E E+30 $E$21 TOTAL Fat E E $F$21 TOTAL Sodium E $G$21 TOTAL VitaminA E $H$21 TOTAL VitaminC E E $I$21 TOTAL VitaminB E E+30 $J$21 TOTAL VitaminB E E+30 $K$21 TOTAL Niacin E E+30 $L$21 TOTAL Calcium E $M$21 TOTAL Iron E Sensitivity Analysis Report Changing a Decision Variable CoEfficient in the Objective Function The allowable increase and decrease for the Adjustable Cells (i.e. for the objective function coefficients) tells us how much the objective function coefficients can change before the optimal solution vertex changes. Note that the optimal solution value changes as the objective function coefficients change, but the optimal product mix and vertex stays the same within the objective function coefficient s allowable increase/decrease range
21 Sensitivity Analysis Report Changing an Objective Function Coefficient Graphical Intuitions Changing an objective function coefficient changes the slope of the objective function. Optimal solution value always changes but, within the allowable increase / decrease, the optimal product mix will not: within the allowable increase decrease the isoprofit line just swivels around a single point! If the slope of the objective function changes beyond the allowable increase / decrease then the optimal vertex will change, and the optimal product mix will change, as illustrated to the right. At right we see the relative contribution of briefs (i.e. the coefficient of briefs in the objective function) increasing, causing the objective function to become steeper, until eventually (beyond the allowable increase) the optimal product mix and optimal vertex shift. The new optimal solution favors more briefs in the optimal product mix. Boxers Hours Demand Logos Material Briefs 41 Sensitivity Analysis Report Changing an Objective Function Coefficient Graphical Intuitions Multiple Optima Notice that, for a certain combination of objective function coefficients, the objective function can becomes tangent to a segment of a constraint line (in this case, to a segment of the Hours constraint). In this case, every point along the tangential line segment is an optimum, so multiple optimal product mixes are available, all with the same optimal solution value. Notice that for an isoprofit line with multiple optima, the allowable increase and decrease are zero, since it is impossible for the line to swivel around a single point and any change in the objective function coefficients will cause the optimal vertex to shift! Boxers Hours Demand Logos Material Briefs 42 21
22 Summary Changing a Decision Variable CoEfficient in the Objective Function A coefficient is associated with each decision variable The allowable increase / decrease for each coefficient is the range over which the coefficient can vary without changing the product mix (i.e. without changing the vertex at which the optimal solution is found) The following do change: Objective Function Value Shadow Prices Reduced Costs Need to resolve the model to find this information Can use to understand flexibility in relative pricing of a product. 43 Summary Changing a Decision Variable CoEfficient in the Objective Function Changes slope of isoprofit / isocost curve (in 2D) Changes decision variable contribution to objective function Does NOT change shape of feasible region In 2dimensional case, the slope of the objective function changes, but, within the allowable increase / decrease range, the optimal solution still resides at the same extreme vertex of the feasible region 44 22
23 Changing a Decision Variable CoEfficient in the Objective Function Example A management consultant offers to improve efficiency in the production of boxers. This would increase the contribution by $0.50 to $3.50. What is the new mix? What is the increase in weekly profit? No change in the product mix. Change in weekly profit = $0.50 x 500 boxers/week = $250 ($2,650 total) Note the range: What could you say about $1 increase? 45 Changing a Decision Variable CoEfficient in the Objective Function Example The contribution of briefs decreases by $0.75 to $3.75. What is the new mix? What is the decrease in weekly profit? No change in the product mix. Change in weekly profit = $0.75 x 200 briefs/week = $150 ($2,250 total) 46 23
24 Reduced Cost Associated with each decision variable. Amount by which profit contribution of variable must be improved before the variable will have a positive value in the solution. Or, rate at which the objective function value will deteriorate if a variable currently at zero is forced to increase by a small amount. Zero if the variable already appears in the optimal solution. 47 Reduced Cost (RC) Unit cost (penalty or loss) in optimal objective function value of forcibly including a Decision Variable (DV) not in the optimal solution. Necessary change in DV coefficient ( reduction in price) so the DV is part of the optimal solution. Rate at which the optimal objective function value deteriorates when a nonoptimal DV is required. Reduced cost of a DV happens to be equivalent to the shadow price of the nonnegativity constraint for that DV. Why? Because forcing a variable into a solution is the same as increasing the RHS of its nonnegativity constraint (e.g. from 0 to 1). RC = 0 implies DV is part of the optimal solution. RC 0 implies DV does not contribute to the optimal objective function value, and that forcing that DV into the solution would worsen the optimal solution
25 Reduced Costs Example Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$11 Production Boxers 500 $ $C$11 Production Padded 0 ($1.00) 6 1 1E+30 $D$11 Production Briefs 200 $ New line: padded briefs, 1 yard of material and 2 hours of labor. Contribution is $6.00 per padded brief. Forced to produce one unit of padded brief per week, what would be cost? Reduced cost is $1.00 per padded brief. 49 Amended Model To Demonstrate Reduced Cost Force the model to construct ONLY boxers: Objective function: Maximize ( $10.00 x Boxers ) + ( $1 x Briefs ) Constraints: Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) 300 yards Logos: ( 1 x Boxers ) + ( 0 x Briefs ) 600 logos Labor: ( 1 x Boxers ) + ( 2 x Briefs ) 900 hrs Demand: ( 0 x Boxers ) + ( 1 x Briefs ) 375 units Boxers 0 Briefs
26 Reduced Cost Example The optimal solution to the amended model is: 300 Boxers, 0 Briefs for Optimal Value: $3,000 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$3 Decision Variables Boxers E $D$3 Decision Variables Briefs E+30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$6 Material $E$7 Logos E $E$8 Labor E $E$9 Demand E Now, the President comes to visit Penn, but he has forgotten his briefs so we must manufacture at least one pair. What is the penalty? Think carefully 51 Reduced Cost Example (Amended Model) Maximize: ( $10.00 x Boxers ) + ( $1 x Briefs ) Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) 300 yards Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$3 Decision Variables Boxers E $D$3 Decision Variables Briefs E+30 Constraints Final Shadow Constraint Allowable Allowable Only Binding Constraint Cell Name Value Price R.H. Side Increase Decrease $E$6 Material $E$7 Logos E $E$8 Labor E $E$9 Demand E Every boxer contributes $10.00 Every brief contributes $1.00 If we make 1 brief we get $1 more profit, but we lose 0.5 yards of material, which costs us 0.5 boxers (i.e. $5 of boxer profits) Thus the marginal loss is $4 (= $1  $5). Notice that the labor, logo, and demand constraints weren t binding so producing an extra pair of briefs doesn t cost us anything there! 52 26
27 Multiple Optima We saw earlier that one indicator of multiple optima was a zero allowable increase and decrease, since that implied that the objective function overlapped with a line segment (rather than merely being tangent to a single point), and thus could not pivot within a range around a single point. As second indicator of multiple optima is finding a decision variable with a Final Value (at the optimum) of zero, and a Reduced Cost of zero. This means that the variable can be forced into the optimal solution at no cost, and therefore an alternative optimum is available. 53 Reduced Cost in Maximization vs Minimization Problems Reduced cost tells you the effect on the objective function value of forcing a variable into the optimal solution. Forcing a variable into the optimal solution always worsens the solution, irrespective of whether its a max or min problem. However: a worse solution in a max problem involves a lower optimal value (i.e. negative reduced cost) a worse solution in a min problem involves a higher optimal value (i.e. positive reduced cost). This is why: max problems have negative reduced costs min problems have positive reduced costs 54 27
28 Changing Multiple Parameters Simultaneously: The 100% Rule We've only changed on parameter at a time. What happens if we change more than one? Use the 100% rule for simultaneous changes in constraint RHSs and Decision Variable coefficients. Calculate each change as a percentage (%) of its respective allowable increase/decrease. If the accumulated (absolute value) % changes are less than 100%, then you sum their shadow price / reduced cost impacts. Otherwise, you must recalculate the LP 55 The 100% Rule 100% also holds for objective coefficients. Can also be combined for changes in constraint functions. If a single value is outside of range, or if the sum of ratios > 1, then you need to recompute a solution (resolve the LP) for new constraints. Remember: Change constraint functions, production mix changes Change objective function (within allowable increase / decrease), production mix does not
29 The 100% Rule Example If material decreases from 300 to 290 yards and labor increases from 900 to 1000 hours, what is the change in weekly contribution? 100 / labor hours + 10 / 52.5 material = = < So, change in Objective Function Value = ($1/hr x 100 hrs)  ($5/yard x 10 yards) = $50 So, new Objective Function Value = $2,450 (= $2,400 + $50) NOTE: The solution changes to boxers and briefs (need to resolve to get this information) % Rule Example: Pricing out a New Product Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$4 Yards_Material $D$5 Logos E $D$6 Hours_Labor $D$7 Demand_Briefs E Product designers offer a new line of padded briefs that require 1 yard of material and 2 hours of labor. Contribution would be $6.00 per brief. Should management introduce this line? Percentage changes: 1 / 52.5 material 2 / 60 labor hours = = < Loss from one padded brief = reducing relevant constraints ($5.00/yard x 1 yard) + ($1.00/hour x 2 hours) = $7.00 Do not produce! $7.00 cost exceeds the $6.00 contribution 58 29
30 SolverTable SolverTable is an Excel AddIn that allows you to produce 2way data tables which look at the sensitivity of the optimum solution to changes in any 2 parameters. Question: How does SolverTable differ from a regular 2way data table? Answer: They re pretty much the same, except: SolverTable reruns Excel Solver for each combination of parameter values, whereas a regular 2way data table could not do that. SolverTable does not autoupdate it merely pastes values. So SolverTable would need to be rerun if other model parameters (besides the 2 you are testing) change. In contrast, a regular 2way data table uses formulas, and the values of these formulas automatically update as model parameters change. 59 SolverTable Getting it: Download it from solver_table.html Installing It: Follow the instruction on the webpage above. Then open Excel and go to Tools AddIns Solver Table Addin Using it: Lay out your 2way table, putting the formula to evaluate under the different scenarios in the top left hand corner, like you would in a regular 2way data table. Go to Tools SolverTable Warning: Because of the complexity of the 2way data tables and SolverTables in the Boxers and Briefs example spreadsheet, it could take you up to 20 minutes to run the scenario analysis. Press Escape repeatedly at any time, or Ctrl+Break, if you wish to terminate the analysis
31 SolverTable Example Changing Multiple Decision Variables Coefficients in a Constraint The example below shows the results of running SolverTable to investigate the effects of changing perunit labor requirements for boxers and briefs on the optimal solution. In other words, it shows the effect of changing the coefficients of the decision variables in the labor constraint. The top table shows the effect on the optimal solution value (i.e. optimal profit in dollars). The bottom table shows the effect on the optimal product mix. You can see that increasing the number of hours required per product type (boxers or brief) decreases the amount of that product type in the optimal mix. Effect on Optimal Solution Value Effect on Optimal Product Mix 61 Key Points Constraint Right Hand Sides: Slack and Shadow Price Changes within the allowable / never alter whether a constraint is (non) binding. Change constraint RHS: value of decision variables (product mix) changes. Objective Function Coefficients: Reduced Cost Changes within the allowable / never alter the optimal vertex. Changes within the allowable / : value of decision variables (product mix) does not change
32 Optimal product mix (Optimal decision variable values) Sensitivity Analysis Report Summary Nonzero if the variable is not in the solution. Zero if it is. Allowable objective function coefficient range: Solution vertex stays the same. Usage of resource (Left Hand Side of constraint) Allowable constraint RHS range: Shadow price is valid. Increase in optimal objective function value per unit increase in right hand side (RHS) of constraint. Z = (shadow price) ( RHS) 63 Sensitivity Analysis Why Should You Care? Uncertainty welcome to the real world. Slack what to do with unused resources? Iteration constraints are rarely fixed e.g. begin with a budget allocation and then evaluate alternatives 64 32
Quiz Sensitivity Analysis
1 Quiz Sensitivity Analysis 1. The difference between the righthand side (RHS) values of the constraints and the final (optimal) value assumed by the lefthand side (LHS) formula for each constraint is
More informationLinear Programming Models: Graphical and Computer Methods
Linear Programming Models: Graphical and Computer Methods Learning Objectives Students will be able to: 1. Understand the basic assumptions and properties of linear programming (LP). 2. Graphically solve
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationSUPPLEMENT TO CHAPTER
SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationChapter 6: Sensitivity Analysis
Chapter 6: Sensitivity Analysis Suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production
More informationLinear Programming with PostOptimality Analyses
Linear Programming with PostOptimality Analyses Wilson Problem: Wilson Manufacturing produces both baseballs and softballs, which it wholesales to vendors around the country. Its facilities permit the
More informationChap 4 The Simplex Method
The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.
More informationLinear Programming Supplement E
Linear Programming Supplement E Linear Programming Linear programming: A technique that is useful for allocating scarce resources among competing demands. Objective function: An expression in linear programming
More informationChapter 5: Solving General Linear Programs
Chapter 5: Solving General Linear Programs So far we have concentrated on linear programs that are in standard form, which have a maximization objective, all constraints of type, all of the right hand
More informationLINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad
LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources
More informationLINEAR PROGRAMMING THE SIMPLEX METHOD
LINEAR PROGRAMMING THE SIMPLE METHOD () Problems involving both slack and surplus variables A linear programming model has to be extended to comply with the requirements of the simplex procedure, that
More informationLEARNING OBJECTIVES. Introduction 1/4/2013 CHAPTER. Linear Programming Models: Graphical and Computer Methods
CHAPTER 2 Linear Programming Models: Graphical and Computer Methods PowerPoint presentation to accompany Balakrishnan/Render/Stair Managerial Decision Modeling with Spreadsheets, 3/e 21 LEARNING OBJECTIVES
More informationLinear Programming Notes VII Sensitivity Analysis
Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization
More informationThe Simplex Method. yyye
Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationLecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2
Lecture 1: Linear Programming Models Readings: Chapter 1; Chapter 2, Sections 1&2 1 Optimization Problems Managers, planners, scientists, etc., are repeatedly faced with complex and dynamic systems which
More informationTutorial Resource Allocation
MARKETING ENGINEERING FOR EXCEL TUTORIAL VERSION 130528 Tutorial Resource Allocation Marketing Engineering for Excel is a Microsoft Excel addin. The software runs from within Microsoft Excel and only
More informationSolving Linear Programs using Microsoft EXCEL Solver
Solving Linear Programs using Microsoft EXCEL Solver By Andrew J. Mason, University of Auckland To illustrate how we can use Microsoft EXCEL to solve linear programming problems, consider the following
More informationSensitivity Report in Excel
The Answer Report contains the original guess for the solution and the final value of the solution as well as the objective function values for the original guess and final value. The report also indicates
More informationQuiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs
Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a realvalued
More informationChapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach
Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we
More informationChapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY
Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationLinear Programming: Using the Excel Solver
Outline: Linear Programming: Using the Excel Solver We will use Microsoft Excel Solver to solve the four LP examples discussed in last class. 1. The Product Mix Example The Outdoor Furniture Corporation
More informationLinear Programming Sensitivity Analysis
Linear Programming Sensitivity Analysis Massachusetts Institute of Technology LP Sensitivity Analysis Slide 1 of 22 Sensitivity Analysis Rationale Shadow Prices Definition Use Sign Range of Validity Opportunity
More informationLinear Programming Sensitivity Analysis
Linear Programming Sensitivity Analysis Massachusetts Institute of Technology LP Sensitivity Analysis Slide 1 of 19 Sensitivity Analysis Rationale Shadow Prices Definition Use Sign Range of Validity Opportunity
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationSolving Linear Programs in Excel
Notes for AGEC 622 Bruce McCarl Regents Professor of Agricultural Economics Texas A&M University Thanks to Michael Lau for his efforts to prepare the earlier copies of this. 1 http://ageco.tamu.edu/faculty/mccarl/622class/
More informationCHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS
Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a realworld problem can be represented accurately
More informationLecture 4 Linear Programming Models: Standard Form. August 31, 2009
Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods
More informationObjective Function Coefficients and Range of Optimality If c k changes to values outside the range of optimality, a new c j  z j row may be generated
Chapter 5 SimplexBased Sensitivity Analysis and Duality Sensitivity Analysis with the Simplex Tableau Duality 1 Objective Function Coefficients and Range of Optimality The range of optimality for an objective
More informationGraphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method
The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible
More informationUsing EXCEL Solver October, 2000
Using EXCEL Solver October, 2000 2 The Solver option in EXCEL may be used to solve linear and nonlinear optimization problems. Integer restrictions may be placed on the decision variables. Solver may be
More informationThe Simplex Solution Method
Module A The Simplex Solution Method A1 A2 Module A The Simplex Solution Method The simplex method is a general mathematical solution technique for solving linear programming problems. In the simplex
More informationSolving Linear Programs
Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,
More information2. Efficiency and Perfect Competition
General Equilibrium Theory 1 Overview 1. General Equilibrium Analysis I Partial Equilibrium Bias 2. Efficiency and Perfect Competition 3. General Equilibrium Analysis II The Efficiency if Competition The
More informationSection Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011
Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More informationLINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH
59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving
More informationCHAPTER 17. Linear Programming: Simplex Method
CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2
More informationAlgebraic Simplex Method  Introduction Previous
Algebraic Simplex Method  Introduction To demonstrate the simplex method, consider the following linear programming model: This is the model for Leo Coco's problem presented in the demo, Graphical Method.
More informationUsing the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood
PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720E October 1998 $3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced
More informationIso profit or Iso cost method for solving LPP graphically
Unit Lesson 4: Graphical solution to a LPP Learning Outcomes How to get an optimal solution to a linear programming model using Iso profit (or Iso cost method) Iso profit or Iso cost method for solving
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More informationComponents of LP Problem. Chapter 13. Linear Programming (LP): Model Formulation & Graphical Solution. Introduction. Components of LP Problem (Cont.
Chapter 13 Linear Programming (LP): Model Formulation & Graphical Solution Components of LP Problem Decision Variables Denoted by mathematical symbols that does not have a specific value Examples How much
More informationNotes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation. Jon Bakija
Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More information1 2 3 Total /25 /45 /30 /100. Economics 172A Midterm Exam Vincent Crawford Winter 2008
1 2 3 Total /25 /45 /30 /100 Economics 172A Midterm Exam NAME Vincent Crawford Winter 2008 Your grade from this exam is 35% of your course grade. The exam ends promptly at 9:20, so you have 80 minutes.
More informationMathematics Notes for Class 12 chapter 12. Linear Programming
1 P a g e Mathematics Notes for Class 12 chapter 12. Linear Programming Linear Programming It is an important optimization (maximization or minimization) technique used in decision making is business and
More informationLinear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16.
LINEAR PROGRAMMING II 1 Linear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16. Introduction A minimization problem minimizes the value of the objective function rather than
More informationUSING EXCEL 2010 TO SOLVE LINEAR PROGRAMMING PROBLEMS MTH 125 Chapter 4
ONETIME ONLY SET UP INSTRUCTIONS Begin by verifying that the computer you are using has the Solver AddIn enabled. Click on Data in the menu across the top of the window. On the far right side, you should
More informationLINEAR PROGRAMMING: MODEL FORMULATION AND 21
LINEAR PROGRAMMING: MODEL FORMULATION AND GRAPHICAL SOLUTION 21 Chapter Topics Model Formulation A Maximization Model Example Graphical Solutions of Linear Programming Models A Minimization Model Example
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More informationQuestion 2: How will changes in the objective function s coefficients change the optimal solution?
Question 2: How will changes in the objective function s coefficients change the optimal solution? In the previous question, we examined how changing the constants in the constraints changed the optimal
More informationSENSITIVITY ANALYSIS AS A MANAGERIAL DECISION
SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL SENSITIVITY ANALYSIS AS A MANAGERIAL DECISION MAKING TOOL SUMMARY Martina Briš, B.Sc.(Econ) Faculty of Economics in Osijek 87 Decision making is
More information3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
More informationAirport Planning and Design. Excel Solver
Airport Planning and Design Excel Solver Dr. Antonio A. Trani Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia Spring 2012 1 of
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationFundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras
Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras Linear Programming solutions  Graphical and Algebraic Methods
More informationMath 407A: Linear Optimization
Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More informationChapter 2 Solving Linear Programs
Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (AddisonWesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A
More informationProject and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi
Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture  15 Limited Resource Allocation Today we are going to be talking about
More informationQuestion 2: How do you solve a linear programming problem with a graph?
Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.
More informationOur development of economic theory has two main parts, consumers and producers. We will start with the consumers.
Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies,
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadowprice interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationThe Graphical Simplex Method: An Example
The Graphical Simplex Method: An Example Consider the following linear program: Max 4x 1 +3x Subject to: x 1 +3x 6 (1) 3x 1 +x 3 () x 5 (3) x 1 +x 4 (4) x 1, x 0. Goal: produce a pair of x 1 and x that
More informationExcel Modeling Practice. The Svelte Glove Problem StepbyStep With Instructions
Excel Modeling Practice The Svelte Glove Problem StepbyStep With Instructions EXCEL REVIEW 20012002 Contents Page Number Overview...1 Features...1 The Svelte Glove Problem...1 Outputs...2 Approaching
More informationOutline. Linear Programming (LP): Simplex Search. Simplex: An ExtremePoint Search Algorithm. Basic Solutions
Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions
More information(a) Let x and y be the number of pounds of seed and corn that the chicken rancher must buy. Give the inequalities that x and y must satisfy.
MA 44 Practice Exam Justify your answers and show all relevant work. The exam paper will not be graded, put all your work in the blue book provided. Problem A chicken rancher concludes that his flock
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationStandard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationvalue is determined is called constraints. It can be either a linear equation or inequality.
Contents 1. Index 2. Introduction 3. Syntax 4. Explanation of Simplex Algorithm Index Objective function the function to be maximized or minimized is called objective function. E.g. max z = 2x + 3y Constraints
More informationLinear Programming: Basic Concepts
Linear Programming: Basic Concepts Table of Contents Three Classic Applications of LP The Wyndor Glass Company Product Mix Problem Formulating the Wyndor Problem on a Spreadsheet The Algebraic Model for
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationProblem Set #3 Answer Key
Problem Set #3 Answer Key Economics 305: Macroeconomic Theory Spring 2007 1 Chapter 4, Problem #2 a) To specify an indifference curve, we hold utility constant at ū. Next, rearrange in the form: C = ū
More informationLinear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables
Linear Inequalities and Linear Programming 5.1 Systems of Linear Inequalities 5.2 Linear Programming Geometric Approach 5.3 Geometric Introduction to Simplex Method 5.4 Maximization with constraints 5.5
More informationLINEAR PROGRAMMING WITH THE EXCEL SOLVER
cha06369_supa.qxd 2/28/03 10:18 AM Page 702 702 S U P P L E M E N T A LINEAR PROGRAMMING WITH THE EXCEL SOLVER Linear programming (or simply LP) refers to several related mathematical techniques that are
More informationLinear Programming I: Maximization 2009 Samuel L. Baker Assignment 10 is on the last page.
LINEAR PROGRAMMING I 1 Learning objectives: Linear Programming I: Maximization 2009 Samuel L. Baker Assignment 10 is on the last page. 1. Recognize problems that linear programming can handle. 2. Know
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More informationLinear Programming IS 601
Linear Programming IS 01 eoptimization.com A Resource Page on Optimization Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and redesign their North American distribution
More informationSpreadsheets have become the principal software application for teaching decision models in most business
Vol. 8, No. 2, January 2008, pp. 89 95 issn 15320545 08 0802 0089 informs I N F O R M S Transactions on Education Teaching Note Some Practical Issues with Excel Solver: Lessons for Students and Instructors
More information56:171. Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.
56:171 Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationNotes on Excel Forecasting Tools. Data Table, Scenario Manager, Goal Seek, & Solver
Notes on Excel Forecasting Tools Data Table, Scenario Manager, Goal Seek, & Solver 20012002 1 Contents Overview...1 Data Table Scenario Manager Goal Seek Solver Examples Data Table...2 Scenario Manager...8
More informationInsurance. Michael Peters. December 27, 2013
Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall
More informationCONSUMER PREFERENCES THE THEORY OF THE CONSUMER
CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the
More informationChapter 5.1 Systems of linear inequalities in two variables.
Chapter 5. Systems of linear inequalities in two variables. In this section, we will learn how to graph linear inequalities in two variables and then apply this procedure to practical application problems.
More information36106 Managerial Decision Modeling Revenue Management
36106 Managerial Decision Modeling Revenue Management Kipp Martin University of Chicago Booth School of Business October 5, 2015 Reading and Excel Files 2 Reading (Powell and Baker): Section 9.5 Appendix
More informationBONUS REPORT#5. The SellWrite Strategy
BONUS REPORT#5 The SellWrite Strategy 1 The SellWrite or Covered Put Strategy Many investors and traders would assume that the covered put or sellwrite strategy is the opposite strategy of the covered
More information1. Setting up a small sample model to be solved by LINDO. 5. How do solve LP with Unrestricted variables (i.e. nonstandard form)?
Contents: 1. Setting up a small sample model to be solved by LINDO 2. How to prepare the model to be solved by LINDO? 3. How to Enter the (input) Model to LINDO [Windows]? 4. How do I Print my work or
More informationIn this section, we will consider techniques for solving problems of this type.
Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving
More information