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Available on the Companion
List of Aha! Moments
What’s New in the Tenth Edition
Acknowledgments
About the Author
Trademarks
- IntroductIon
- operatIons research Models
- What are the decision alternatives?
- Under what restrictions is the decision made?
- What is an appropriate objective criterion for evaluating the alternatives?
- Buy five regular FYV-DEN-FYV for departure on Monday and return on Wednesday of the same week
- Buy one FYV-DEN, four DEN-FYV-DEN that span weekends, and one DEN-FYV
- cost = 5 * $400 = $2000
- is the cheapest
- solvIng the or Model
Buy five regular FYV-DEN-FYV for departure on Monday and return on Wednesday of the same week. An obvious objective criterion for assessing the proposed alternatives is the price of the tickets.
- art oF ModelIng
- Production Department: Production capacity expressed in terms of available machine and labor hours, in-process inventory, and quality control standards
- Materials Department: Available stock of raw materials, delivery schedules from outside sources, and storage limitations
- Sales Department: Sales forecast, capacity of distribution facilities, effectiveness of the advertising campaign, and effect of competition
- Production rate
- Consumption rate
- More than Just MatheMatIcs
- The stakes were high in 2004 when United Parcel Service (UPS) unrolled its ORION software (based on the sophisticated Traveling Salesman Algorithm—see
- In a steel mill in India, ingots were first produced from iron ore and then used in the manufacture of steel bars and beams. The manager noticed a long delay between
- In response to complaints of slow elevator service in a large office building, the OR team initially perceived the situation as a waiting-line problem that might
- Before jumping to the use of sophisticated mathematical modeling, a bird’s eye view of the situation should be adopted to uncover possible nontechnical reasons
- phases oF an or study
- Definition of the problem
- Construction of the model
- Solution of the model
- Validation of the model
- Implementation of the solution
- about thIs book
The model acceptably expresses the mathematical functions that represent the behavior of the assumed real world. What is the shortest time to move all four people to the other side of the river?
Real-Life Application—Frontier Airlines Purchases Fuel Economically
Two-VARiAbLE LP ModEL
The daily demand for interior paint cannot exceed that for exterior paint by more than 1 ton. The first constraint on product demand states that the daily production of interior paint cannot exceed that of exterior paint by more than 1 ton, which translates to:
GRAPhiCAL LP SoLuTion
Determination of the feasible solution space
Determination of the optimum solution from among all the points in the solution space
- Solution of a Maximization Model Example 2.2-1
- Solution of a Minimization Model Example 2.2-2 (diet Problem)
- CoMPuTER SoLuTion wiTh SoLVER And AMPL
- LP Solution with Excel Solver
- LP Solution with AMPL 5
- LinEAR PRoGRAMMinG APPLiCATionS
The conclusion is that one should not 'predict' the solution by imposing the additional equality constraint. If the optimal objective value is unbounded (not finite), Solver will issue the somewhat ambiguous message: "The set cell values do not converge".
Investment
The top line defines subscript j, and subscript i is entered at the beginning of each line as. Solution command; calls the solution algorithm and the command display z, x; gives the solution.
Production planning and inventory control
Actually, AMPL has formatting capabilities that improve the readability of the output results (see Section C.5.2). This section presents realistic LP models in which the definition of the variables and the construction of the objective function and the constraints are not as simple as in the case of the two-variable model.
Workforce planning
Urban development planning
Oil refining and blending
- investment
- Production Planning and inventory Control
- workforce Planning
But there are two more reasons why you shouldn't use If other constraints in the model are such that all $12 million cannot be used (eg, the bank may set limits on different loans), then the choice .4 * 12 may lead to an impossible or incorrect solution. 2) If you want to experiment with the effect of changing the available funds (say from $12 to $13 million) on the optimal solution, there is a real chance that you might forget to change 0.4 * 12 to .4 * 13, in which in case the solution will not be correct. The substitution Si = Si- - Si+ is the basis for developing the hiring and firing cost.
Real-Life Application—Telephone Sales workforce Planning at Qantas Airways
In the new development, the triple and quadruple units make up at least 25% of the. In addition, distillation capacities and demand limits can directly affect the level of production of the various grades of gasoline. One goal of the model is to determine the optimal mix of final products that will maximize an appropriate profit function.
82 Feed-
- Additional LP Applications
The objective of the model is to maximize the total profit resulting from the sale of all three grades of gasoline. The octane number of a gasoline product is the weighted average of the octane numbers of the input streams used in the blending process and can be calculated as Supplies ON * BBl/day supply + Cracker Unit ON * Cracker Unit bbl/day Total bbl/day of regular gasoline.
For the Reddy Mikks model, construct each of the following constraints and express it. with the left side linear and the right side constant:. a) The daily demand for interior paint exceeds that of exterior paint by at least 1 ton. Determine the solution space and optimal solution of the Reddy Mikks model for each of the following independent variables:. a) The maximum daily requirement for interior paint is 1.9 tons and that for exterior paint is a maximum of 2.5 tons. A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of A and B. However, the company cannot sell more than 110 units of A per day.
Real-Life Application—Optimization of Heart Valve Production
LP mOdeL in equAtiOn FORm
All the constraints are equations with nonnegative right-hand side
All the variables are nonnegative. 1
- tHe simPLex metHOd
- iterative nature of the simplex method
2 If the number of (independent) equations m is equal to the number of variables n (and the equations are consistent), the system has exactly one solution. But this does not prevent enumeration of all the corner points of the solution space. Instead of enumerating all the basic solutions (corner points) of the LP problem (as we did in Section 3.2), the simplex method investigates only a "select few" of these solutions.
Setting up a simple table automatically provides a solution on the initial iteration. How the calculated ratios determine the output variable and the value of the input variable. The following table is a restatement of the initial table with the pivot row and column highlighted.
Sensitivity analysis, which deals with determining the conditions that will keep the current solution unchanged. Post-optimal analysis, which deals with finding a new optimal solution when the data of the model is changed.
Pivot row
- M-method 6
- two-Phase method
So, we add the artificial variables R1 and R2 in the first two equations and penalize them in the objective function with MR1 + MR2 (because we are minimizing). The result is that R1 and R2 are now replaced (have zero coefficients) in the z-row with z = 900 as desired. Using the penalty M will not force an artificial variable to zero in the final simplex iteration if the LP does not have a feasible solution (i.e., the constraints cannot be satisfied simultaneously).
In this case, the last simplex iteration will contain at least one artificial variable with a positive value. As the name suggests, the method solves the LP in two phases: Phase I attempts to find an initially feasible basic solution, and if found, Phase II is invoked to solve the original problem.
Put the problem in equation form, and add the necessary artificial vari- ables to the constraints (exactly as in the M-method) to secure a start-
Use the feasible solution from Phase I as a starting basic feasible solution for the original problem
Select a zero artificial variable to leave the basic solution and designate its row as the pivot row. The entering variable can be any nonbasic nonartificial vari-
- sPeciAL cAses in tHe simPLex metHOd
- Degeneracy 2. Alternative optima
- Nonexisting (or infeasible) solutions
- degeneracy
- Alternative Optima
- unbounded solution
- infeasible solution
- sensitiVity AnALysis
- Sensitivity of the optimum solution to changes in the availability of the resources (right-hand side of the constraints)
- Sensitivity of the optimum solution to changes in unit profit or unit cost (coefficients of the objective function)
- Algebraic sensitivity Analysis—changes in the Right-Hand side
- A unit change in operation 1 capacity 1 D 1 = { 1 min 2 changes z by $
- A unit change in operation 2 capacity 1 D 2 = { 1 min 2 changes z by $
- A unit change in operation 3 capacity 1 D 3 = { 1 min 2 changes z by $0
- Algebraic sensitivity Analysis—Objective Function
- By increasing the unit revenue
- By decreasing the unit cost of consumed resources
- sensitivity Analysis with tORA, solver, and AmPL
- cOmPutAtiOnAL issues in LineAR PROGRAmminG 13
- Speed
- Accuracy
In the TOYCO model (Example 3.6-2), the objective z equation in the optimal tableau can be written as. Classic (section 3.3.2) The incoming variable is the one with the most favorable lower cost of all non-basic variables. The aim is to 'simplify' the model in two important ways:17. a) Reducing model size (rows and columns) by identifying and removing redundant constraints and potentially capturing and replacing variables.
Case Study: Optimization of heart Valves production 18 tool: LP
Definition of the Dual Problem
As such, it can be computationally advantageous in some cases to determine the primary solution by solving the dual. Our definition of the dual problem requires us to express the primal problem in the equation form presented in Section 3.1, a format consistent with the initial simplex tableau (all the constraints are equations with nonnegative right-hand sides, and all the variables are nonnegative ). Therefore, any result obtained from the primary optimal solution applies unambiguously to the associated dual problem.
A dual variable is assigned to each primal (equation) constraint and a dual con- straint is assigned to each primal variable
The two problems are closely related, in the sense that the optimal solution to one problem automatically provides the optimal solution to the other. In all textbooks with which this author is familiar, the dual is defined for different forms of the original, depending on the sense of optimization (maximization or minimization), types of constraints 1…, Ú or =2, and sign of the variables (non- negative or unlimited). Not only are there too many combinations to remember, but their use may require some degree of fine-tuning with the results of the simplex algorithm, mainly because the primary that makes up the dual is not in the standard format used by the simplex algorithm (e.g. the primary that makes up the dual can have negative right sides in the constraints).
The right-hand sides of the primal constraints provide the coefficients of the dual objective function
But that computational advantage may be small when compared to what the rich primordial dual theory has to offer, as we will show throughout the book. Each primary (comparison) constraint is assigned a double variable, and each primary variable is assigned a double constraint. The double constraint corresponding to a primary variable is constructed by transposing the column containing the primary variable into a row containing (i) the primary objective coefficient.
The dual constraint corresponding to a primal variable is constructed by transpos- ing the primal variable column into a row with (i) the primal objective coefficient
The sense of optimization, direction of inequalities, and the signs of the variables in the dual are governed by the rules in Table 1
- Primal–Dual relationshiPs
- review of simple matrix operations
Note also that no provision is made for the inclusion of artificial variables in the primitive, because artificial variables will not change the definition of the double (see Problem 4-5). Changes made to the data of an LP model can affect the optimality and/or the feasibility of the current optimal solution. These relationships form the basis for the economic interpretation of the LP model and for post-optimality analysis.
Note that the column headings in the table do not use the designation primal and dual. This section introduces a series of primal-dual relations that can be used to recompute the elements of the optimal simplex tableau. The section starts with a brief review of matrices, a handy tool for performing simplex tableau calculations.
The simplex tableau can be generated by three elementary matrix operations: (row vector) *1matrix2, 1matrix2 * 1columnvector2 and 1scalar2 * 1matrix2.
- optimal Dual solution
- simplex tableau Computations
The order of the optimal primary basic variables in the Basic column is x2 followed by x1. The elements of the original objective coefficients for the two variables should appear in the same order, viz. It is important that the columns of the basic matrix must coincide with the order of the basic variables in the tableau.
Constraint columns (left-hand and right-hand sides)
Note that the relationship does not specify which problem is primary and which problem is dual. This section shows how to generate each iteration of the simplex tableau from the original data of the problem, the inverse associated with the iteration, and the dual problem. Using the simplex tableau layout in Figure 4.1, we can divide the calculations into two types:
Objective z-row
- eConomiC interPretation of Duality
- economic interpretation of Dual Variables
The simplex tableau format in Chapter 3, which generates the current tableau from the immediately preceding one, is a sure recipe for propagating the rounding error, greatly distorting the quality of the optimal solution. You will note from the discussion in Sections 4.2.2 and 4.2.3 that the inverse matrix of an iteration plays the key role in determining all the elements of the associated simplex tableau (using this inverse and the original data for the problem) . This essentially means that at any iteration all the elements of a tableau (inverse matrix included) can be determined from the model's original data.
- economic interpretation of Dual Constraints
- aDDitional simPlex algorithms
As stated in Section 3.6, the quantity ami=1aijyi - cj 1= imputed cost of activity j - cj2 is known by the standard name reduced cost of activity j. Maximizing optimality. Looking at discounted costs for x1, toy trains become economically attractive only if their attributed unit costs are strictly less than unit revenues. A reduction in the assigned cost per unit means a reduction in the assembly time of a toy train unit in three operations.
- Dual simplex algorithm
- The objective function must satisfy the optimality condition of the regular sim- plex method (Chapter 3)
- All the constraints must be of the type 1 … 2
In the current example, the first two inequalities are multiplied by -1 to convert them into constraints 1…2. The following table shows how the dual optimality condition is used to determine the input variable. In all iterations, optimality is preserved (all reduced costs are … 0) as each new iteration moves the solution toward feasibility.
- generalized simplex algorithm
- Post-oPtimal analysis
- Changes affecting feasibility
- Changes affecting optimality
- Compute the dual values using Method 2, Section 4.2.3
- Substitute the new dual values in Formula 2, Section 4.4, to determine the new reduced costs (z-row coefficients)
In this part, we deal with changing the parameters of the model and finding a new optimal solution. Note that the right-hand side of the first dual constraint is 2, the new coefficient in the modified objective function. If the primary constraint is of type ..., the corresponding double variable will be non-negative (non-positive) if the primary objective is maximization (minimization).
