This method is called the Simplex Method. The Simplex Method , developed by George Dantzig in incorporates both optimality and feasibility tests to find the optimal solution s to a linear program if one exists. An optimality test shows whether or not an intersection point corresponds to a value of the objective function better than the best value found so far.
A feasibility test determines whether the proposed intersection point is feasible. It does not violate any of the constraints. The simplex method starts with the selection of a corner point usually the origin if it is a feasible point and then, in a systematic method, moves to adjacent corner points of the feasible region until the optimal solution is found or it can be shown that no solution exists.
Tableau Format : Place the linear program in Tableau Format, as explained below. This is accomplished by adding a unique, non-negative variable, called a slack variable, to each constraint.
Adding slack variables makes the constraint set a system of linear equations. We write these with all variables on the left side of the equation and all constants on the right hand side.
We will even rewrite the objective function by moving all variables to the left-hand side. Initial Extreme Point : The Simplex Method begins with a known extreme point, usually the origin 0, 0 for many of our examples. The requirement for a basic feasible solution gives rises to special Simplex methods such as Big M and Two-Phase Simplex, which can be studied in a linear programming course.
The Tableau previously shown contains the corner point 0, 0 is our initial solution. We can have at most 3 solutions. Z will always be a solution by convention of our tableau. These non-zero variables are called the basic variables. The remaining variables are called the non-basic variables.
The corresponding solutions are called the basic feasible solutions FBS and correspond to corner points. The complete step of the simplex method produces a solution that corresponds to a corner point of the feasible region. These solutions are read directly from the tableau matrix. We also note the basic variables are variables that have a column consisting of one 1 and the rest zeros in their column.
We will add a column to label these as shown below:. Optimality Test : We need to determine if an adjacent intersection point improves the value of the objective function. If not, the current extreme point is optimal. If an improvement is possible, the optimality test determines which variable currently in the independent set having value zero should enter the dependent set as a basic variable and become nonzero.
For our maximization problem, we look at the Z-Row The row marked by the basic variable Z. If any coefficients in that row are negative then we select the variable whose coefficient is the most negative as the entering variable.
The variable with the most negative coefficient is x 2 with value — Thus, x 2 wants to become a basic variable. Feasibility Test : To find a new intersection point, one of the variables in the basic variable set must exit to allow the entering variable from Step 3 to become basic. The feasibility test determines which current dependent variable to choose for exiting, ensuring we stay inside the feasible region.
We will use the minimum positive ratio test as our feasibility test. Make a quotient of the r h s j a j. Note that we will always disregard all quotients with either 0 or negative values in the denominator. This gives the location of the matrix pivot that we will perform. Pivot: We can form a new equivalent system by using row operations to change the pivot element to a 1 and all other numbers in the pivot column to zero.
We do the row operations by adding a suitable multiple of the pivot row to a multiple of each row in the tableau, thus eliminating the new basic variable. Then set the new non-basic variables to zero in the new system to find the values of the new basic variables, thereby determining an intersection point. We note that x 1 has a coefficient of —5 in the Z-Row therefore, we are not optimal.
The minimum non-negative quotient is This indicates that to remain in the feasible region that x 1 enters as a basic variable and S 2 leaves being a basic variable. Step 5. We make the highlighted position a 1 and all other column entries 0 for the column of x 1.
The set of points satisfying the constraint of this linear programming problem the convex set as a shaded region. Each solution found in the tableau corresponds to a corner point. We went from corner 0, 0 to corner 0, 23 to corner 12, The Tableau Format with slack variables y 1 , y 2 :. Extreme Point 0, 0 Corresponding to the values of x 1 , x 2. Optimality Test: The entering variable is x 1 corresponding to -3 in the Z- row. Choose y 1 to leave since it corresponds to the minimum positive ratio test value of 3.
Pivot: Divide the row containing the exiting variable the first row in this case by the coefficient of the entering variable in that row the coefficient of x 1 in this case , giving a coefficient of 1 for the entering variable in this row.
Then eliminate the entering variable x 1 from the remaining rows which do not contain the exiting variable y 1 and have a zero coefficient for it. The results are summarized in the next tableau. Extreme Point 3, 0 Corresponding to the values of x 1 , x 2. Optimality Test: There are no negative coefficients in the Z- row.
Remarks : We have assumed that the origin is a feasible extreme point. If it is not, then an extreme point must be found before the Simplex Method, as presented, can be used.
These and other topics are studied in more advanced optimization courses. Technology is critical to solving, analyzing, and performing sensitivity analysis on linear programming problems. Technology provides a suite of powerful, robust routines for solving optimization problems, including linear programs LPs. We tested all these software packages and found them useful. From the standpoint of sensitivity analysis Excel is satisfactory in that it provides shadow prices.
In our case study we present linear programming for supply chain design. We consider producing a new mixture of gasoline. We desire to minimize the total cost of manufacturing and distributing the new mixture. There is a supply chain involved with a product that must be modeled. The product is made up of components that are produced separately. Depending on whether we want to additionally minimize delivery across different locations or maximize sharing by having more distribution point involved then we have choices.
Data envelopment analysis DEA , occasionally called frontier analysis, was first put forward by Charnes, Cooper and Rhodes in It is a performance measurement technique which, as we shall see, can be used for evaluating the relative efficiency of decision-making units DMU's in organizations. Here a DMU is a distinct unit within an organization that has flexibility with respect to some of the decisions it makes, but not necessarily complete freedom with respect to these decisions.
Examples of such units to which DEA has been applied are: banks, police stations, hospitals, tax offices, prisons, defense bases army, navy, air force , schools and university departments. Note here that one advantage of DEA is that it can be applied to non-profit making organizations.
Since the technique was first proposed much theoretical and empirical work has been done. Many studies have been published dealing with applying DEA in real-world situations. Obviously there are many more unpublished studies, e. We will initially illustrate DEA by means of a small example. Note here that much of what you will see below is a graphical pictorial approach to DEA.
This is very useful if you are attempting to explain DEA to those less technically qualified such as many you might meet in the military or management world. There is a mathematical approach to DEA that can be adopted however. We will present the single measure first to demonstrate the idea and then move to multiple measures and use linear programming methodology from our course. Example 1. Ranking Banks. Consider a number of bank branches. For each branch we have a single output measure number of personal transactions completed and a single input measure number of staff.
For example, for the Branch 2 in one year, there were 44, transactions relating to personal accounts and 16 staff members were employed. A commonly used method is ratios. Typically we take some output measure and divide it by some input measure.
Note the terminology here, we view branches as taking inputs and converting them with varying degrees of efficiency, as we shall see below into outputs. For our bank branch example we have a single input measure, the number of staff, and a single output measure, the number of personal transactions.
Hence we have:. Here we can see that Branch1 has the highest ratio of personal transactions per staff member, whereas Branch 4 has the lowest ratio of personal transactions per staff member. As Branch 1 has the highest ratio of 6.
To do this we divide the ratio for any branch by 6. This gives:. The other branches do not compare well with Branch 1, so are presumably performing less well.
That is, they are relatively less efficient at using their given input resource staff members to produce output number of personal transactions. We could, if we wish, use this comparison with Branch 1 to set targets for the other branches. For example we could set a target for Branch 4 of continuing to process the same level of output but with one less member of staff.
This is an example of an input target as it deals with an input measure. Plainly, in practice, we might well set a branch a mix of input and output targets which we want it to achieve. We can use linear programming. Example 2. Ranking banks with linear programming. Typically we have more than one input and one output. For the bank branch example suppose now that we have two output measures number of personal transactions completed and number of business transactions completed and the same single input measure number of staff as before.
We start be scaling via ratios the inputs and outputs to reflect the ratio of 1 unit. Let W1 and W2 be the personal and business transactions at branch. In this example we choose to maximize branch two, E2.
Maximize E2. Now, what did we learn from this. If we ranked ordered the branches on efficiency performance of our inputs and outputs, we find. We know we need to improve on branch 2 and branch 4 performances while not losing our efficiency in branches 1 and 3.
A better interpretation could be that the practices and procedures used by the other branches were to be adopted by Branch 4, they could improve their performance. This invokes issues of highlighting and disseminating examples of best practices. Equally there are issues relating to identification of poor practices.
In DEA the concept of the reference set can be used to identify best performing branches with which to compare poorly performing branches. If you use this procedure, use it wisely. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. We are IntechOpen, the world's leading publisher of Open Access books.
Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. Fox and Fausto P. Downloaded: Introduction Consider planning the shipment of needed items from the warehouses where they are manufactured and stored to the distribution centers where they are needed.
Problem Identification: Maximize the profit of selling these new drinks. Financial planning formulation Note here that as in all formulation exercises we are translating a verbal description of the problem into an equivalent mathematical description.
Variables Essentially we are interested in the amount in dollars the bank has loaned to customers in each of the four different areas not in the actual number of such loans. Hence the constraint relating to policy condition 3 is 0. For integer programming, there exist some difficulties and problems for the direct applications of AFSA due to the variables belonging to the set of integers.
In this paper, a novel AFSA is proposed for integer programming after three behaviors having been designed, which evolves on the set of integer space. Several mathematical functions and cutting stock problem simulation results show that the proposed algorithm is significantly superior to other algorithms. Authors Close. Assign yourself or invite other person as author. It allow to create list of users contirbution. Assignment does not change access privileges to resource content. Wrong email address.
You're going to remove this assignment. Are you sure? Yes No. Share This Paper. Figures and Tables from this paper. One Citation. Citation Type. Has PDF. Publication Type. More Filters. Aceh is a province that is rich in fishery resource potential. Fish resources become one of the leading commodities, therefore we need a supply chain optimization model.
Optimization is one of the … Expand. Fish and its processed products are the most affordable source of animal protein in the diet of most people in Indonesia.
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