Linear Programming Break-Even Analysis, Sensitivity Analysis and Transportation Problem

Problem 1: Break-Even Analysis

A small organization builds TV satellite dishes. The investment in plant and equipment is $200,000. The variable cost per dish is $500. The price of the TV dish is $1000.
(Please show your work used to come up with this solution)

1. How many TV satellite dishes would be needed to be sold for the firm to break even?



2. What profit or loss can be anticipated with a demand of 500 dishes?


3. If the demand is 300 dishes, what is the minimum price that the firm must charge to breakeven?



Problem 2: Linear Programming Model Development

A manufacturer makes two products, doors and windows. Each must be processed through two work areas. Work area #1 has 60 hours of available production time. Work area #2 has 48 hours of available production time. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window. As you respond to the following questions, please note that you are only setting the problem up. A solution to determine the number of doors and windows to be manufactured and the resulting profit is NOT necessary.


1. Define the decision variables that will tell how many units to build (doors and windows)


2. Develop an objective function that will maximize profits.


3. Develop production constraints for work areas #1 and #2.  


Problem 3: Model Development, Sensitivity Analysis, Interpretation of Output

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer.  It is assumed that every bike ordered will be sold, and the profits for each, respectively, are $30, $25, $22, and $20. The linear programming model should maximize profit. There are several conditions that the store needs to be concerned about. One of these is the space to hold the inventory. The adult bikes need two feet each, but each children's bike needs only one foot. The store has 500 feet of space available for the bicycles. There are 1200 hours of assembly time available. The children's bikes need 4 hours each to assemble; the Open Trail needs 5 hours each; and the Cityscape needs 6 hours each. The store would like to place an order for 275 bicycles.

Requirements:

1. Formulate a model for the problem, state the objective function and the subject to constraints.


2. Refer to Appendix 1 to answer the following:

A. Based on the optimal solution, how many of each kind of bicycle should be ordered and what will the total profit be?


B. What would the profit be if the store had 100 more feet of storage space?


C. Over what range of assembly hours is the dual price available?


D. Which resource should the company work to increase? - inventory space or assembly time?


Appendix 1

Objective Function Value = 6850.000

Variable Value Reduced Costs
Open Trail 100.000                                              0.000
City Scape 0.000        13.000
Sea Sprite 175.000           0.000
Trail Blazer   0.000           2.000


Constraint                               Slack/Surplus                           Dual Price
1 (Space Available) 125.000 0.000
2 (Assembly Hours Available)         0.000                                     8.000
3 (Order Desired)                      0.000              - 10.000


Right Hand Side Ranges:

Constraint                     Lower Limit                    Current Value            Upper Limit
1 (Space Available)                       375.000           500.000                No Upper Limit
2 (Assembly Hours Available)        1100.000          1200.000               1325.000
3 (Order Desired)                           240.000         275.000                    300.000


Problem 4: Transportation Problem

Chicago Transport is to move goods from three factories to three distribution centers. Information about the move is as follows:

Factory            Units Supplied                       Distribution Center             Units Demanded  

A              200           X            100
B              100           Y            150
C              150           Z                               200


Costs per unit to ship from each factory to each distribution center are:
                                               Distribution Center
Factory               X         Y          Z                                      
A $3       $2       $5
B $9       $10      $7
C                                              $5       $6       $4


Requirements:


1. Develop a Network representative of this problem.



2. Formulate the problem as a linear program showing the objective function and the constraints



3. Using Appendix 2, what is the total shipping cost of the optimal solution?




4. Based on the optimal solution of Appendix 2, how many units will be shipped from each factory to each distribution center?   Note Variable X1 represents units shipped from Factory A to Distribution Center X; X2 - A to Y; X3 - A to Z; X4 - B to X; X5 - B to Y; X6 - B to Z; X7 - C to X; X8 - C to Y; and X9 - C to Z.



Appendix 2

Objective Function Value = 1800.000

Variable Value                  Reduced Costs
X1 50.000                                              0.000
X2 150.000        0.000
X3 0.000        3.000
X4 0.000        1.000
X5                            0.000                                               3.000
X6                            100.000                                            0.000
X7                             50.000                                             0.000    
X8                               0.000                                             2.000
X9                             100.000                                           0.000