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Linear Programming We can make Product1 and Product2. There are 3 resources; Resource1, Resource2, Resource3. Product1 needs one hour of Resource1, nothing of Resource2, and three hours of resource3. Product2 needs nothing from Resource1, two hours of Resource2, and two hours of resource3. Available hours of resources 1, 2, 3 are 4, 12, 18, respectively. Contribution Margin of product 1 and Product2 are $300 and $500, respectively. Formulate the Problem Solve the problem using solver in excel Problem Formulation Objective Function Z = 3 x1 +5 x2 Constraints Resource 1 x1 4 Resource 2 2x2 12 Resource 3 3 x1 + 2 x2 18 Nonnegativity x1 0, x2 0 Wyndor Example; Enter data Organize the data for the model on the spreadsheet. Type in the coefficients of the constraints and the objective function Noncomputational Entries and Changing Cells Changing cells. Assign a set of cells to represent the decision variable in the model. The problem starts with assuming a value of 0 in each decision variable cell, and find the optimal solution. SUMPRODUCT Constraint cells. For each constraint, create a formula in a separate cell that corresponds to the left-hand side (LHS) of the constraint. SUMPRODUCT function is used to multiply element by element of two tables and addup all values. SUMPRODUCT: sums the products of individual cells in two ranges. SUMPRODUCT(C6:E6, C4:E4) = C6*C4 + D6*D4+ E6*E4 The two specified ranges must be of the same size ( the same number of rows and columns). For linear programming you should try to always use the SUMPRODUCT function (or SUM) for the objective function and constraints. This is to remember that the equations are all linear. In LP we do not have square, square-root, log, exponential, cos, etc. SUMPRODUCT F4 Copy Down; LHS, and the Objective Function Target Cell: The Objective Function Target cell. A cell containing a formula that corresponds to the objective function. Tools/Add-Ins/Solver Defining the Target Cell ( The Objective Function) The target cell contains an equation that defines the objective and depends on the decision variables. You can only have one objective function. Target cell must be a single cell. In the Solver dialogue box select the “Set Target Cell” window, then click on the cell that you have already defined it as the objective function. This is the cell you wish to optimize. Then lick on the radio button of either Max or Min. Identifying the Changing Cells ( Decision Variables) You next tell Excel which cells are decision variables, i.e., which cells excel is allowed to change when trying to optimize. Move the cursor to the “By Changing Cells” window, and drag the cursor across all cells you wish to treat as decision variables Adding Constraints Click on the “Add” button to the right of the constraints window. A new dialogue box will appear. The cursor will be in the “Cell Reference” window within this dialogue box. Click on the cell that contains the quantity you want to constrain. The default inequality that first appears for a constraint is “<= ”. To change this, click on the arrow beside the “<= ” sign. After setting the inequality, move the cursor to the “Constraint” window. Click on the cell you want to use as the constraining value for that constraint. Adding Constraints Adding Constraints You may define a set of similar constraints (e.g., all <= constraints, or all >= constraints) in one step if they are in adjacent rows. Simply select the range of cells for the set of constraints in both the “Cell Reference” and “Constraint” window. After you are satisfied with the constraint(s), click the “Add” button if you want to add another constraint, or click the “OK” button if you want to go back to the original dialogue box. Notice that you may also force a decision variable to be an integer or binary (i.e., either 0 or 1) using this window. Important Options The Solver dialogue box now contains the optimization model, including the target cell (objective function), changing cells (decision variables), and constraints. Important Options Click on the “Options” button in the Solver dialogue box, and click in both the “Assume Linear Model” and the “Assume Non-Negative” box. The “Assume Linear Model” option tells Excel that it is a linear program. This speeds the solution process, makes it more accurate, and enables the more informative sensitivity report. The “Assume Non-Negative” box adds non-negativity constraints to all of the decision variables. Check the Options then Solve The Solution The Solution When it is done, you will receive one of four messages: Solver found a solution. All constraints and optimality conditions are satisfied. This means that Solver has found the optimal solution. Cell values did not converge. This means that the objective function can be improved to infinity. You may have forgotten a constraint (perhaps the nonnegativity constraints) or made a mistake in a formula. Solver could not find a feasible solution. This means that Solver could not find a feasible solution to the constraints you entered. You may have made a mistake in typing the constraints or in entering a formula in your spreadsheet. Conditions for Assume Linear Model not satisfied. You may have included a formula in your model that is nonlinear. There is also a slim chance that Solver has made an error. (This bug shows up occasionally.) The Solution If Solver finds an optimal solution, you have some options. First, you must choose whether you want Solver to keep the optimal values in the spreadsheet (you usually want this one) or go back to the original numbers you typed in. Click the appropriate box to make you selection. you also get to choose what kind of reports you want. Once you have made your selections, click on “OK”. You will often want to also have the “Sensitivity Report”. To view the sensitivity report, click on the “Sensitivity Report” tab in the lower-left-hand corner of the window. Questions Answered by Excel What is the Optimal Solution? What is the value of the Objective Function for the Optimal Solution? If the net profit for a product changes, will the optimal solution change? If more (or less) of a resource is available, how it affect our profit? For example: How much do you pay to have one extra unit of Resource1 How much do you pay to have one extra unit of Resource3 How much do you pay to have one extra unit of Resource2 Sensitivity Output from Computer Solution : Changing Cells Adjustable Cells Cell Name $B$9 Solution Product1 $C$9 Solution Product2 Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 2 0 3 4.5 3 6 0 5 1E+30 3 Final Value The value of the variable in the optimal solution Reduced Cost Increase in the objective function value per unit increase in the value of a zero-valued variable (a product that the model has decided not to produce). Allowable Increase/ Decrease Defines the range of the cost coefficients in the objective function for which the current solution (value of the variables in the optimal solution) will not change. Output from Computer Solution : Constraints Constraints Cell Name $D$5 Resource1 LHS $D$6 Resource2 LHS $D$7 Resource3 LHS Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 2 0 4 1E+30 2 12 1.5 12 6 6 18 1 18 6 6 Final Value The usage of the resource in the optimal solution. Shadow price The change in the value of the objective function per unit increase in the right hand side of the constraint: Z = (Shadow Price)(RHS) (Only for change is within the allowable range) Output from Computer Solution : Constraints Constraints Cell Name $D$5 Resource1 LHS $D$6 Resource2 LHS $D$7 Resource3 LHS Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 2 0 4 1E+30 2 12 1.5 12 6 6 18 1 18 6 6 Constraint R.H. Side The current value of the right hand side of the constraint (the amount of the resource that is available). Allowable Increase/ Decrease The range of values of the RHS for which the shadow price is valid and hence for which the new objective function value can be calculated. (NOT the range for which the current solution will not change.) Net Profit product 1 = $7 Change the profit to 7, Solver again, Solve again Adjustable Cells Cell Name $B$9 Solution Product1 $C$9 Solution Product2 Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 2 0 3 4.5 3 6 0 5 1E+30 3 Net Profit product 1 = $8 Change the profit to 8, Solver again, Solve again Adjustable Cells Cell Name $B$9 Solution Product1 $C$9 Solution Product2 Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 2 0 3 4.5 3 6 0 5 1E+30 3 Wyndor Optimal Solution What is the optimal Objective function value for this problem? What is the allowable range for changes in the objective coefficient for Product 2 What is the allowable range for changes in the RHS for Resource 3. If the coefficient of Product 2 in the objective function is changed to 7, what will happen to the value of the objective function? If the coefficient of Product 1 in the objective function is changed to 8, what will happen to the value of the objective function? If the RHS of Resource 2 is increased by 2 , what will happen to the objective function. If the RHS of Resource 1 is increased by 2, what will happen to the objective function. If the RHS of Resource 2 is decreased by 10, what will happen to the objective function. Wyndor Optimal Solution Practice: Given the following Sensitivity Analysis Report Adjustable Cells Cell Name $B$9 Solution Product1 $C$9 Solution Product2 $D$9 Solution Product3 Final Value 0 500 375 Reduced Cost -1.25 0 0 Objective Coefficient 15 20 25 Allowable Increase 1.25 5 0.71 Allowable Decrease 1E+30 0.42 5 Final Value 400 350 150 Shadow Price 0 12.5 100 Constraint R.H. Side 500 350 150 Allowable Increase 1E+30 50 25 Allowable Decrease 100 50 18.75 Constraints Cell Name $E$5 Resource1 LHS $E$6 Resource2 LHS $E$7 Resource3 LHS Sensitivity Analysis • • • What is the optimal objective function value for this problem? a. It cannot be determined from the given information. b. $900. c. $987.5. d. $875. e. $19375. What is the allowable range for the objective function coefficient for Product 3? a. 0.71 ≤ P3 ≤ 5. b. 20 ≤ P3 ≤ 25.71. c. 0 .71≤ P3 ≤ 25.71. d. 25 ≤ P3 ≤ 25.71. e. non of the above. What is the allowable range of the right-hand-side for Resource1? a. 500 ≤ RHS1 ≤ ∞. b. 0 ≤ RHS1 ≤ 500. c. 100 ≤ RHS1 ≤ ∞. d. 400 ≤ RHS1 ≤ ∞. e. -∞ ≤ RHS1 ≤ 400. Sensitivity Analysis • • • If the coefficient for Product2 in the objective function changes to $24, then the objective function value: a. will increase by $24. b. will increase by $120,000. c. will increase by $2000. d. will remain the same. e. can only be discovered by resolving the problem. If the coefficient for Product1 in the objective function changes to $5, then the objective function value: a. will increase by $5. b. is $0. c. will increase by $10. d. will remain the same. e. can only be discovered by resolving the problem. If the coefficient of Product2 in the objective function changes to $15, then: a. the original solution remains optimal. b. the problem must be resolved to find the optimal solution. c. the optimal solution will decrease by 2500. d. the shadow price will decrease by 5. e. the optimal solution will increase by 2500. Sensitivity Analysis • If the right-hand side of Resource1 increases, then the objective function value: a. will increase. b. will decrease. c. will decrease then increase. d. will remain the same. e. will increase then decrease. • If the right-hand side of Resource2 changes to 370, then the objective function value: a. will increase by $370. b. will increase by $350. c. will increase by $250. d. will remain the same. e. can only be discovered by resolving the problem. • If the right-hand side of Resource3 changes to 130, then: a. the original solution remains optimal. b. the problem must be resolved to find the optimal solution. c. the objective function will decrease by 130. d. the objective function will decrease by 3000. e. the objective function will increase by 3000 More than one profit OR More than one resource • If the sum of the ratio of (Change)/(Change in the Corresponding Direction) <=1 • Things remain the same. • If we are talking about profit, the production plan remains the same. • If we are talking about RHS, the shadow prices remain the same. Sensitivity Analysis • • • If the objective coefficients of Product1 is increased by 1 and the objective coefficient of product3 is decreased by 2, then: a. the objective function will decrease. b. the objective function will increase. c. the optimal solution will remain the same. d. the shadow prices will remain the same. e. can only be discovered by resolving the problem. If the right-hand side of all three resources, each increases by 10 units: a. the optimal solution remains the same. The objective function value will increase. b. the optimal solution will change. The objective function value will increase. c. the optimal solution and the shadow prices will remain the same. d. the optimal solution and the shadow prices both will change. e. can only be discovered by resolving the problem. If the right-hand side of resouses1 and 2 each decreases by 20 and the right and side of resource 3 increases by 5: a. the objective function value will increase by 250. b. the objective function value will decrease by 250. c. the objective function value will remain the same. d. the optimal solution will remain the same. e. can only be discovered by resolving the problem. Assignment • The following 11 Questions refer to the following sensitivity report. Adjustable Cells Cell $B$6 $C$6 $D$6 Name Solution Activity 1 Solution Activity 2 Solution Activity 3 Final Value 0 27.5 0 Reduced Cost 425 0.0 250 Objective Coefficient 500 300 400 Allowable Increase Allowable Decrease 1E+30 500 1E+30 425 300 250 Allowable Increase Allowable Decrease Constraints Cell $E$2 $E$3 $E$4 Name Benefit A Totals Benefit B Totals Benefit C Totals Final Value 110 110 137.5 Shadow Price 0 75 0 Constraint R.H. Side 60 110 80 50 1E+30 57.5 1E+30 46 1E+30 Assignment ( Taken from The management Sciences Hillier and Hillier) • What is the optimal objective function value for this problem? a. it cannot be determined from the given information. b. $1,200. c. $975. d. $8,250. e. $500. • What is the allowable range for the objective function coefficient for Activity 3? a. 150 ≤ A3 ≤ ∞. b. 0 ≤ A3 ≤ 650. c. 0 ≤ A3 ≤ 250. d. 400 ≤ A3 ≤ ∞. e. 300 ≤ A3 ≤ 500. • What is the allowable range of the right-hand-side for Resource A? a. 0 ≤ RHSA ≤ 60. b. 0 ≤ RHSA ≤ 110. c. 60 ≤ RHSA ≤ 110. d. 110 ≤ RHSA ≤ 160. e. 0 ≤ RHSA ≤ 160. Assignment ( Taken from The management Sciences Hillier and Hillier) • If the coefficient for Activity 2 in the objective function changes to $400, then the objective function value: a. will increase by $7,500. b. will increase by $2,750. c. will increase by $100. d. will remain the same. e. can only be discovered by resolving the problem. • If the coefficient for Activity 1 in the objective function changes to $50, then the objective function value: a. will decrease by $450. b. is $0. c. will decrease by $2750. d. will remain the same. e. can only be discovered by resolving the problem. • If the coefficient of Activity 2 in the objective function changes to $100, then: a. the original solution remains optimal. The objective function value decreases. b. the problem must be resolved to find the optimal solution. c. the shadow prices will remain the same. d. the original solution and the objective function value remain the same. e. none of the above. Assignment ( Taken from The management Sciences Hillier and Hillier) • • • If the right-hand side of Resource B changes to 80, then the objective function value: a. will decrease by $750. b. will decrease by $1500. c. will decrease by $2250. d. will remain the same. e. can only be discovered by resolving the problem. If the right-hand side of Resource C changes to 140, then the objective function value: a. will increase by $137.50. b. will increase by $57.50. c. will increase by $80. d. will remain the same. e. can only be discovered by resolving the problem. If the right-hand side of Resource C changes to 130, then: a. the original solution remains optimal. The objective function value will increase. b. the problem must be resolved to find the optimal solution. c. the shadow prices remain the same. The objective function value will decrease. d. the shadow prices remain the same. the objective function value remains the same. e. the shadow prices will change. The objective function value will increase. Assignment ( Taken from The management Sciences Hillier and Hillier) • • • If the objective coefficients of Activity 2 and Activity 3 are both decreased by $100, then: a. the objective function will decrease by 2750. b. the objective function will decrease by less than 2750. c. the objective function will decrease by more than 2750. d. The objective function will remain the same. e. can only be discovered by resolving the problem. If the right-hand side of Resource C is increased by 40, and the right-hand side of Resource B is decreased by 20, then: a. the optimal solution remains the same. b. the objective function value increases by 1500. c. the shadow prices remain the same. d. can only be discovered by resolving the problem. e. the objective function value decreases by 1500. If the right-hand side of Resource A is increased by 25, and the right-hand side of Resource B is decreased by 20, then: a. the optimal solution remains the same. b. the objective function value increases by 1500. c. the objective function value remains the same. d. can only be discovered by resolving the problem.