11/15/2011
Operations Research I IE 416
Extra Credit Assignment
Problem #2 P.199
Group 9

Fruit Computer Company

Jin woo Choi
Philip Liu
Nallely Tagle
Table of Contents

Problem Statement……………………………………………………………Pg. 3

Problem Summary…………………………………………………………….Pg. 3

Problem Formulation…………………………………………………………Pg. 4

Initial Constraints………………………………………………………….Pg. 5

Constraints with Deviational Variables………………………………….Pg. 6

Assumptions & Example…………………………………………………..Pg. 6

Objective Function………………………………………………………...Pg. 7

WinQSB for Linear Programming…………………………………………...Pg. 8

Report to Manager for Linear Programming……………………………….Pg. 9

WinQSB for Preemptive Goal Programming…….………………………....Pg. 10

Report to Manager for Preemptive Goal Programming…………………....Pg. 13

Sensitivity Analysis…………………………………………………………..…Pg. 14

Scenario 2………………………………………………………………..….Pg. 15

Report to Manager 2…………………………………………………..…..Pg. 17

Scenario 3…………………………………………………………….….....Pg. 19

Report to Manager 3………………………………………………..……..Pg. 20

Scenario 4…………………………………….…………………….……....Pg. 21

Report to Manager 4……………………………………………………….Pg. 22

Scenario 5…………………………………………………….……………..Pg. 23

Report to Manger 5…………………………………………………….Pg. 26

Sensitivity Analysis Summery for Goal Programming……………...………Pg. 27
Problem Statement:
Fruit Computer Company is ready to make its annual purchase of computer chips. Fruit can purchase chips (in lots of 100) from three suppliers. Each chip is rated as being of excellent, good, or mediocre quality. During the coming year, Fruit will need 5,000 excellent chips, 3,000 good chips, and 1,000 mediocre chips. The characteristics of the chips purchased from each supplier are shown in Table 1 (see Problem Summary).
Each year, Fruit has budgeted $28,000 to spend on chips. If Fruit does not obtain enough chips of a given quality, then the company may specialorder additional chips at $10 per excellent chip, $6 per good chip, and $4 per mediocre chip. Fruit assesses a penalty of $1 for each dollar by which the amount paid to supplier 13 exceeds the annual budget. Formulate and solve a linear program to help Fruit minimize the penalty associated with meeting the annual chip requirements. Also use preemptive goal programming to determine a purchasing strategy. Let the budget constraint have the highest priority, followed in order by the restrictions on excellent, good, and mediocre chips. Table 2 (see Problem Summary) will summarize the problem.
Problem Summary:
As stated earlier, Table 1 is a compilation of the given data about how many types of chips are in one lot from each supplier, as well as how much each supplier charges for a single lot. Additionally, Table 2 includes the rest of the information that was given to solve the problem (Fruit’s desired amount of each chip, penalty cost, etc.)

Table 1



Characteristics of a Lot of 100 Chips


Supplier

Excellent

Good

Mediocre

Price Per 100 Chips ($)

1

60

20

20

400

2

50

35

15

300

3

40

20

40

250


Table 2




Characteristics of a Lot of 100 Chips



Supplier

Excellent

Good

Mediocre

Price Per 100 Chips ($)

Price Per One Chip ($)

Supplier 1

60

20

20

400

4.00

Supplier 2

50

35

15

300

3.00

Supplier 3

40

20

40

250

2.50

Total

≥ 5,000

≥ 3,000

≥ 1,000

≤ $28,000


Price Per Special Order Per One Chip (Penalty Cost)

$10/chip

$6/chip

$4/chip


$1 for every dollar over budget

Problem Formulation:
With a budget of $28,000 Fruit must determine how many lots should be ordered from each Supplier in order to meet their desired goal for each type of chip, while at the same time attempting to avoid going over budget and being forced to specialorder additional chips to meet their demand.
Decision Variables
Let X_{i} : number of lots provided by Supplier_{i} (where i = 1, 2, and 3)
Deviational Variables:
The deviational variables were added because it is unknown whether the costminimizing solution will undersatisfy or oversatisfy the given goals. Thus, the following deviational variables will be formed:
S_{i}^{} : amount that goal_{i} is under (goal is not achieved and where i = 1, 2, and 3)
S_{i}^{+} : amount that goal_{i} is over (goal has been passed and where i = 1, 2, and 3)
Initial Constraints:
Using the amount of chips of each type that must be ordered and the budget, the constraints were formed. The first constraint was formed using the total amount of excellent chips that were ordered from all three suppliers and how many were desired; in this case it was 60 from Supplier 1, 50 from Supplier 2, 40 from Supplier 3, and a desired amount of 5,000. Similarly, the second and third constraints were formed the same way; the second constraint had 20 from Supplier 1, 35 from Supplier 2, and 20 from Supplier 3 with a desired amount of 3,000. The third constraint was formed the same way, with 20 from Supplier 1, 15 from Supplier 2, 40 from Supplier 3, and a minimum order of 1,000.
1) The total number of excellent chips ordered from all three suppliers, with a desired quantity of 5,000 excellent chips:
60X_{1} + 50X_{2} + 40X_{3} ≥ 5,000
2) The total number of good chips ordered from all three suppliers, with a desired quantity of 3,000 good chips:
20X_{1} + 35X_{2} + 20X_{3} ≥ 3,000
3) The total number of mediocre chips ordered from all three suppliers, with a desired quantity of 1,000 mediocre chips:
20X_{1} + 15X_{2} + 40X_{3} ≥ 1,000
Due to the fact that Fruit is operating on a budget, a fourth constraint was created by taking the sum of the product of how much each supplier sold a lot for. Also, due to the fact that a supplier could not demand Fruit to ship chips to them, a constraint had to be formed to keep Fruit as a receiver.
4) Fruit has a budget of $28,000 with Supplier 1, Supplier 2, and Supplier 3 charging $400, $300, and $250 per lot respectively:
400X_{1} + 300X_{2} + 250X_{3} ≤ 28,000
5) Fruit does not ship any chips, it only receives:
X_{i}, S_{i}^{}, S_{i}^{+} ≥ 0
In summary, the following five constraints were formed:
1) 60X_{1} + 50X_{2} + 40X_{3} ≥ 5,000
2) 20X_{1} + 35X_{2} + 20X_{3} ≥ 3,000
3) 20X_{1} + 15X_{2} + 40X_{3} ≥ 1,000
4) 400X_{1} + 300X_{2} + 250X_{3} ≤ 28,000
5) Xj, S_{i}^{}, S_{i}^{+} ≥ 0
Constraints with Deviational Variables:
When the constraints were formed, it was noticed that with the first four constraints were flexible. Flexible refers to the fact that the costminimizing solution may either undersatisfy or oversatisfy the goal and it will still be used. Thus, the deviational variables S_{i}^{ }and S_{i}^{+} (where i = 1, 2, 3, and 4) will be used.
1) 60X_{1} + 50X_{2} + 40X_{3} +S_{1}^{} – S_{1}^{+} = 5,000
2) 20X_{1} + 35X_{2} + 20X_{3} +S_{2}^{} – S_{2}^{+} = 3,000
3) 20X_{1} + 15X_{2} + 40X_{3} + S_{3}^{} – S_{3}^{+} = 1,000
4) 400X_{1} + 300X_{2} +250X_{3} + S_{4}^{} –S_{4}^{+} = 28,000
Assumption and Explanation
In reality most of the time a companies’ budget is fixed; that is to say, that there is no additional money to be used. If that is the case and the budget is a constraint, it is not flexible. Because there is no possible way to go above the budget, deviational variables cannot be added.
For this problem, the budget is not fixed. We can assume that Fruit Computer Company’s annual budget is not fixed at $28,000 and there is additional money available to be used if penalty costs are incurred and if specialorders need to be placed. Thus, deviational variables can be added to the budget constraint.
To further explain the concepts of deviational variables in this problem, refer to the following example:
Example:
After inputting the objective function (refer to Objective Function below) into the computer program WinQSB, several values will be generated. Using the priorities given in the Problem Statement and the deviational variables that we defined will mean that S_{3}^{} and S_{3}^{+ }are the deviational variables for the third goal. Thus, if the output defines a value for S_{3}^{ }to be 5, then it means that the goal has not been reached and it is below by 5 units. On the other hand, if the output had defined a value for S_{3}^{+ }to be 5, it would mean that the goal had been exceeded by 5 units. The former option is undesirable because the goal needs to be met, while the latter option is desirable because not only has the goal been met, but it has been exceeded.
It should be noted that the assignment of the terms “undesirable” and “desirable” are not always affiliated with S_{3}^{} and S_{3}^{+} respectively. The assignment is based off of the constraint.
Objective Function:
Recall that the objective is to minimize the penalty cost, and that the penalty cost for specialordering additional chips cost $10, $6, and $4 for excellent, good, and mediocre. There is also a penalty cost of $1 for every dollar that the budget is exceeded by. The costs are then treated as a penalty cost, which would result in the following objective function:
Objective Function: Min Z = 10S_{1}^{ }+6S_{2}^{ }+ 4 S_{3}^{} + 1 S_{4}^{+}
Refer to Table 3 (below) for a summary of the equations that were formed.
Table 3

Summary of All Equations

Type

Description

Equation

Objective Function

Minimize Penalty Cost

Min Z = 10S_{1}^{} +6S_{2}^{} + 4 S_{3}^{} + 1 S_{4}^{+}

Constraint

Excellent Chips Ordered

60X_{1} + 50X_{2} + 40X_{3} +S_{1}^{} – S_{1}^{+} = 5,000

Constraint

Good Chips Ordered

20X_{1} + 35X_{2} + 20X_{3} +S_{2}^{} – S_{2}^{+} = 3,000

Constraint

Mediocre Chips Ordered

20X_{1} + 15X_{2} + 40X_{3} + S_{3}^{} – S_{3}^{+} = 1,000

Constraint

Annual Budget

400X_{1} + 300X_{2} +250X_{3} + S_{4}^{} –S_{4}^{+} = 28,000

WinQSB for Linear Programming
WinQSB Input for Linear Programming
WinQSB Output for Linear Programming
It should be noted that when dealing with the problem as a Linear Program, the priority levels that were given were not taken into account.
Report to Manager (LP)
Dear Manager,
The team was tasked with minimizing the penalty cost that Fruit Computer Company would incur while fulfilling their orders; in order to efficiently determine that value, the computer program WinQSB was utilized. By assuming that fulfilling the demands for the three types of chips and staying within the annual budget are NOT all equally important (note the penalty costs) , the results from Figure 2 can be interpreted as such:
This paragraph as worded is not quite right. You need to start with the next paragraph and the solution will result in fulfilling this requirement. In order to achieve the lowest penalty cost, Fruit will have to ensure that the desired quantity of 5,000 excellent chips is met. Fruit should focus on achieving the desired number of excellent chips because if there are not 5,000 excellent chips, specialorders will have to be placed to achieve that quantity. Specialorders for excellent chips will cost Fruit $10 per chip. The other two chips have a lower cost for specialorders; they will cost $6 and $4 for good and mediocre chips, respectively. Thus, to achieve the lowest possible penalty cost, Fruit should place their attention towards satisfying the 5,000 unit requirement for excellent chips.
Thus, the most effective way to do so will be to purchase 100 lots of chips from Supplier 2. Doing so will give the company 5,000 excellent chips, meaning that the desired quantity of excellent chips has been met. However, this method of ordering will give the company 3,500 good chips and 1,500 mediocre chips; both of those values are 500 chips above the desired quantity. Despite being over the desired quantity, there is no penalty cost for exceeding the desired quantity; therefore, this overstocking is acceptable.
Supplier 2 charges $300 for each lot; purchasing 100 lots will cost Fruit $30,000. The annual budget that the company has allocated towards purchasing is only $28,000. Thus, the budget will have to be raised to $30,000 to accommodate all chip demands.
While this paragraph is a good analysis it should not be in report to the manager as it makes too many details that may create confusion. It should be noted that all the lots purchased should be from Supplier 2. The reason is that even though Supplier 3 charges less ($50 less per lot), there are less excellent chips in their lot than Supplier 2’s lot. Thus, in order to satisfy the quantity of excellent chips, 125 lots would have to be purchased from Supplier 3. At $250 per lot, Fruit would have to spend $31,250; that cost is $1,250 more than the cost of purchasing solely from Supplier 2.
While this paragraph is a good analysis it should not be in report to the manager as it makes too many details that may create confusion. Despite the fact that Supplier 1 offers more excellent chips per lot, their lot price of $400 is too expensive. In order to facilitate the company’s demand for excellent chips, 83 lots will have to be purchased. However, that would only give the company 4,980 excellent chips; thus, the remaining 20 chips would have to be specialordered. The specialorders would incur the company a penalty cost of $200 dollars, raising the total price of ordering from Supplier 1 to be $33,400. That cost is $3,400 higher than ordering from just Supplier 2.
Please refer to the following Tables (Tables 4 and 5) for a tabular summary of the results that were gathered from the output.

Table 4


Information

Fruit's Requirements

Deviation from Requirements

Penalty Cost

Quantity of excellent chips

≥ 5,000

Goal is met

$0.00

Quantity of good chips

≥ 3,000

Goal exceeded by 500

$0.00

Quantity of mediocre chips

≥ 1,000

Goal exceeded by 500

$0.00

Annual Budget

≤ $28,000

Goal exceeded by $2,000

$2,000.00

Total Penalty Cost



$2,000.00

Table 5

Information

Results

Purchase from Supplier

Supplier 2

Quantity of Lots Purchased

100 lots at $300 per lot

Quantity of Each Chip Obtained

5,000 excellent chips, 3,500 good chips, and 1,500 mediocre chips

Total Cost

(100 lots) * ($300/lot) = $30,000

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