Operation Management Research/ Linear ProgrammingPlease refer to the attached file for all the questions.There are some true or false questions but I mainly need help on the linear programming problems/ problems that don’t involve definitions. Thanks!
Problem 1. Solving Linear Programming Problems —- The Graphical Method （20 points）
Read chapter 3 (pp. 25–38), and answer the following questions.
er the following statements about linear programming and the simplex method. Label each statement as true or false:
Given that an optimal solution to a linear programming problem exists, it must occur at a vertex of the feasible set. ( )
If the optimal solution occurs at two adjacent vertices of the feasible set, then the linear programming problem has infinitely many solutions. Any point on the line segment joining the two vertices is also a solution. ( )
Outline the steps of the graphical method for solving linear programming problems.
Step 1: Graph the system of constraints. This will give the feasible set.
Step 2:
……
The last step: State the solution to the problem.
Use the graphical method to solve the following linear programming problem.
Maximize Q = 3x + 5y
Problem 2. Solving Linear Programming Problems —- The Simplex Method（40 points）
Read chapter 4 and answer the following questions.
Suppose that, in a maximization problem, every nonbasic variable has a nonpositive coefficient in the objective function of a canonical form. Then the basic feasible solution given by the canonical form maximizes the objective function over the feasible region.
Suppose that, in a maximization problem, some nonbasic variable has a positive coefficient in the objective function of a canonical form. If that variable has a positive coefficient in some constraint, then a new basic feasible solution may be obtained by pivoting.
determine the entering and leaving basic variables.
Initialization: Set up to start iterations, including finding an initial CPF solution.
Optimality test: Is the current CPF solution optimal?
Iteration: Perform an iteration to find a better CPF solution.
Problem 3. Solving Transportation and Assignment Problems（20 points）
Read chapter 9 and answer the following questions.
transportation simplex algorithm.
Step 1. Determine a initial basic feasible solution, and go to step 2.
Step 2. ….
Step 3. ….
Transportation Problem: Wheat is harvested in the Midwest and stored in grain elevators in three different cities: Kansas City, Omaha, and Des Moines. These grain elevators supply three flour mills, located in Chicago, St. Louis, and Cincinnati. Grain is shipped to the mills in railroad cars, each of which is capable of holding one ton of wheat. Each grain elevator is able to supply and mill demands the following number of tons (i.e., railroad cars) of wheat per month:
Grain Elevator Supply 1. Kansas City 150 2. Omaha 175 3. Des Moines 275 Total 600 tons
Mill Demand 1. Chicago 200 2. St. Louis 100 3. Cincinnati 300 Total 600 tons
The cost of transporting one ton of wheat from each grain elevator (source) to each mill (destination) differs according to the distance and rail system. These costs are shown in the following table.
Grain Elevator
Mill
1. Chicago
2. St. Louis
3. Cincinnati
1. Kansas City
$ 6
$ 8
$ 10
2. Omaha
7
11
11
3. Des Moines
4
5
12
Use the information provided above, setting up the transportation simplex tableau.
Use the transportation simplex to solve the transportation problem.
Problem 4. Network Optimization Problems（20 points）
Read chapter 9 and answer the following questions.
Use the algorithm described in Sec. 10.3 to find the shortest path through each of the following network, where the numbers represent actual distances between the corresponding nodes.
Reconsider the above network. Use the algorithm described in Sec. 10.4 to find the minimum spanning tree for the networks.
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