Mg0c10 - Situation for the Firm to Maximum the Total Profits

MG0C10 Group Assignment

Problem 1:

Executive Summary:

        The following detailed report summaries the situation for the firm to maximum the total profits. Several questions have been addressed in it. We have provided analyses about the machine costs and operating hour costs, whereas the report also reveals how the revenues have been generated. Moreover, the explanations for the objective function and all constraints are been given clearly. Based on the result, the maximum profits can be generated is $1343965.731 with total number of hours that all Type 1, 2 and 3 machines will be operating per week are 274.038hrs, 163.462hrs and 129.808hrs respectively. For obtaining the maximum profits, the number of Type 1,2 and 3 machines rented would be 7,5 and 4.Furthermore, we analyze the answer report comprehensively through different dimensions for providing the recommendations. It evaluated whether the maximum profits would be influenced or not based on our recommendation regarding the number of machine hours should use. It also interpreted the situations when the maximum widgets can be sold per week has been increased. Under this situation, we believed that there is an upper limit for the growing profits. Once the disposing costs occurred, the profits would decrease simultaneously. As a result, there will be advantages and disadvantages in terms of the extensions of capacities, and it will be discussed in the following detailed report.

Detailed Report:

a)

All Variables:

T1--the total (combined) number of hours that all Type 1 machines will be operating per week;

T2--the total (combined) number of hours that all Type 2 machines will be operating per week;

T3--the total (combined) number of hours that all Type 3 machines will be operating per week;

M1--the number of Type 1 machines rented;

M2--the number of Type 2 machines rented;

M3--the number of Type 3 machines rented;

Objective Function:

Maximum Profits: 200(60T2 +40T3)

-{8000M1+25T1+6000M2+35T2+5000M3+22T3+20(100T1)+30(120T1)

+3.5[100T1-(120T2+60T3)]+3.5[120T1-(90T2+140T3)]}

Explanation:

We generate the total revenues and total costs to generate the objective function for maximizing total profits.

200(60T2+40T3) are the total revenue can be generated through the production of widgets.

8000M1+25T1+6000M2+35T2+5000M3+22T3 is the total renting costs and operating hours’ costs for all machines.

20(100T1)+30(120T1) represents the total cost of purchasing elements A and B.

3.5[100T1-(120T2+60T3)]+3.5[120T1-(90T2+140T3)] represent the total costs of disposing the remaining purified A and B.

Constraints:

The maximum units of widgets can be sold are 15000 widgets per week. Based on the information, Type 2 machine produces 1 widget per minute and Type 3 machine produces 1 widget per 1.5 minutes. On the hour-based, we have:

60T2 + 40T3 <= 15000

Each machine can be operated up to 40 hours per week, T1, T2, T3 represent the total number of hours all machines used. Therefore, the total capacity for the operation hours will be 40hr/machine * the number of machines, we generate the following constraints:

T1<=40M1      T2<=40M2      T3<=40M3

According to the information, purified elements A and B that can be used in Type 2 and Type 3 machines must pass Type 1 machine at first for purifying. Therefore, the number of elements A and B in Type 2 and 3 machines must be less or equal to the number of elements A and B that Type 1 machine purified. We generate:

120T2 +60T3<=100T1      90T2 +140T3 <= 120T1

The number of machines of each type rented must be an integer, and all variables cannot be negative, so:

All variables >=0     T1, T2, T3=integer

The linear optimization model will be:

Maximum Profits: 200(60T2 +40T3)

-{8000M1+25T1+6000M2+35T2+5000M3+22T3+20(100T1)+30(120T1)

+3.5[100T1-(120T2+60T3)]+3.5[120T1-(90T2+140T3)]}

Constraints:

T1<=40M1    (1)

T2<=40M2    (2)

T3<=40M3    (3)

60T2 + 40T3 <= 15000   (4)

120T2 +60T3<=100T1   (5)

90T2 +140T3 <= 120T1  (6)

All variables >=0        (7)

T1, T2, T3=integer    

b)

The answer sheet shows that: maximum profits =1343956.731, T1 =274.038, T2 =163.462, T3=129.808, M1=7, M2=5, M3=4

Based on the result, the optimal weekly profits will be $1343956.731, the total number of hours that all Type 1 machines will used is 274.038 per week, the total number of hours that all Type 2 machines will used is 163.462 per week, the total number of hours that all Type 3 machines will used is 129.808 per week. The total number of Type 1 machines rented is 7, the total number of Type 2 machines rented is 5, and the total number of Type 3 machines rented is 4.

c)

Additional Analysis

From the answer report, the first three constraints (1)(2)(3) have slack value. It represents that amounts of resources are not being used. To be more specific, the unused hour for Type 1 machine is 5.962hrs, the unused hour for Type 2 machine is 36.538hrs, and the unused hour for Type 3 machine is 30.192hrs. Since the maximum widgets can be sold per week is 15000 (fixed), the slack values demonstrated that the optimal value would not change even though there are more available hours can be used for the three machines. According to that the constraint (4)(5)(6) is binding, then the corresponding slack values are equal to zero. It means that all the purified elements A and B are been used to produce the widgets and there is no disposing cost.

Based on the answer sheet, the widgets will be sold per week is 15000 for obtaining the maximum profits, which is exactly the maximum amounts can be sold. Assume Type 1 machine used all the operating hours (280hrs), the disposed cost would increase since there will have extra unused purified elements A and B. As a result, the optimal value would decrease. For Type 2 & 3 machine, they will not use more hours to produce widgets because widgets cannot be stored.

In-depth perspective, the optimal value might change if some of the constraints changed. For instance, if the maximum widgets can be sold per week increase, the optimal value would increase because they will use more operating hours to produce more widgets, which means that the total hours used for Type 1,2,3 machines would increase and the total purified elements A and B been used would increase. On the other hand, the profits would decrease once the disposing costs occurred so that there must be an upper limit for the increasing-profits. However, since we cannot get the Sensitivity Report and the Limit Report, it hard to know the dual value, objective coefficient range and right-hand-side range. Therefore, we cannot determine how the optimal value would change.