Page 199, Problem 2
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 excellent, good, or mediocre quality. During the coming year, Fruit will need 5000 excellent chips, 3000 good chips, and 1000 mediocre chips. The characteristics of the chips purchased from each supplier are shown below:
Characteristics of a Lot of 100 Chips / Prices per 100 Chips.Excellent / Good / Mediocre
Supplier 1 / 60 / 20 / 20 / $400
Supplier 2 / 50 / 35 / 15 / $300
Supplier 3 / 40 / 20 / 40 / $250
Each year, Fruit has budgeted $28,000 to spend on chips. If Fruit does not obtain enough chips of a given quality, the company may special-order additional chips at $10 per excellent chip, $6 per good chips, and $4 for mediocre chip. Fruit assesses a penalty of $1 for each dollar by which the amount paid to Suppliers 1-3 exceeds the annual budget. Formulate and solve an LP 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.
Summary of Problem:
Characteristics of a Lot of 100 Chips / Price per 100 Chips / Cost of 1 ChipExcellent / Good / Mediocre
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 / => 5000 / => 3000 / => 1000 / < = $28,000
Penalty Cost / $10/chip / $6/chip / $4/chip / $1 for every dollar over budget
Decision Variable: Xj = Number of lots provided by Supplier j (where j = 1, 2 and 3).
Deviational Variables: SiN = Quantity by which goal i is not reached.
SiP = Quantity by which goal i is exceeded.
Original constraints:
- 60X1 + 50X2 + 40X3 >= 5000.
- 20X1 + 35X2 + 20X3 >= 3000.
- 20X1 + 15X2 + 40X3 >= 1000.
- 400(X1) + 300(X2) +250(X3) = 28,000.
Sign Restrictions: Xij, SiN and SiP =>0
Constraints modified with deviations:
60X1 + 50X2 + 40X3 + S1N – S1P = 5000.
20X1 + 35X2 + 20X3 + S2N – S2P = 3000.
20X1 + 15X2 + 40X3 + S3N – S3P = 1000.
400(X1) + 300(X2) +250(X3) + S4N – S4P = 28,000.
Objective Function (OF):
If using penalty cost information and LP: (just for your information)
Min Z = 10S1N + 6S2N + 4S3N + 1S4P
If using priority and GP:
- Z1 = P1* S4P
- Z2 = P2 * S1N
- Z3 = P3 * S2N
4. Z4 = P4 * S3N
WinQSB Input for Penalty Cost Analysis (LP) (just for your information)
WinQSB Output for Penalty Cost Analysis (LP) (just for your information)
WinQSB Input for Preemptive GP
WinQSB Output for Preemptive GP
Report to Manager
1. Penalty Cost and LP Solution Analysis (just for your information)
The minimum penalty cost for meeting the chip requirements and maintaining the budget at $28,000 is $2000. The penalty cost is a result of a budget shortfall. This is illustrated in Table 1 below:
TABLE 1
Description / Requirements / Unit of Deviation from Requirements / Penalty Cost ($)Number of excellent chips / => 5000 / 0 / $0.0
Number of good chips / => 3000 / 500 / Above* / $0.0
Number of mediocre chips / => 1000 / 500 / Above / $0.0
Budget / <= $28,000 / $2000 / Above / $2,000
Total / $2,000
Supplier: /
Supplier 2
Quantity Purchased (Lots): / 100 at $300/lotTotal Cost Required: / $30,000
Types of Chips Acquired: / 5000 excellent chips; 3500 good chips; and 1500 mediocre chips.
*Note: Above = above the requirement
As indicated in the above Table, to minimize the penalty cost, the company needs to meet the excellent chip requirement of 5000 chips. The company will have to buy 100 lots of chips from Supplier 2, which charges $300 per lot of chips. At this price ($30,000 for the 100 lots), the company will obtain 5000 excellent chips, 3500 good chips, and 1500 mediocre chips. The latter two requirements will be exceeded, the good chip by 500 and mediocre chip by the same amount as well. The budget will also be exceeded. It will have to increase to $30,000 from its current level of $28,000.
The reason why the company should focus solely on excellent chips is because the penalty cost of not meeting this requirement is far higher than the penalty cost of other requirements. Deviation from the excellent chip restriction will cost the company $10 per chip. With the other restrictions the deviation is less than $10. For example, the penalty cost for not meeting the good chip requirement is $6; for the mediocre chip, it is $4. Thus, to lower the penalty cost, it is best that the company focus on satisfying the excellent chip restriction.
It is also imperative that the company procures its supply of excellent chips from Supplier 2 alone. Though Supplier 3 charges less per lot of chips (at $250/lot), the content of excellent chips within its lot is only 40%, which is lower than the 50% content of
Supplier 2’s lots. Due to the lower content of excellent chip, the company will need to buy more lots from Supplier 3 – about 125 – to satisfy the excellent chip requirement. At 125 lots, the total price the company will have to pay is $31,250 which is $1250 higher than the cost charged by Supplier 2.
2. Priority and GP Solution Analysis
The company has specified the following priorities (or rank) for the goals listed below:
TABLE 2
Rank
/ Priority / Goals1 / Highest priority / a. The budget constraint should be kept.
2 / Second highest priority / b. 5000 excellent chip is required.
3 / Third highest priority / c. 3000 good chips is required.
4 / Lowest priority / d. 1000 mediocre chips is required.
A preemptive goal programming (GP) run has indicated that the company will achieve 3 of the 4 listed objectives. Specifically, the company will be able to satisfy its good and mediocre chips requirement while maintaining its budget at the desired level of $28,000. The only goal where the company will fall short is the excellent chip requirement. The company will miss the minimum target by 400 chips or 4 lots.
To achieve the above goals and their respective priorities, the company should buy 10 lots of chips from the Supplier 1 at a cost of $4000; and 80 lots of chips from Supplier 2 at a cost of $24,000. This will give the company 46 lots (or 4600) excellent chips, 30 lots (or 3000) good chips and 14 lots (or 1400) mediocre chips. While the company will not be able to satisfy its goal on the excellent chip, it shall exceed the goal on the mediocre chip by 400.
The following table (Table 3) summarizes the GP results:
TABLE 3
Supplier / Quantity of Chips(Lots) / Types of Chips(in 100) / Cost
($)
Excellent / Good / Mediocre
1 / 10 / 6 / 2 / 2 / 4,000
2 / 80 / 40 / 28 / 12 / 24,000
3 / 0 / 0 / 0 / 0 / 0
Total / 46 / 30 / 14 / 28,000
Goal / 50 / 30 / 10 / 0
Deviation / -4 / 0 / +4 / 0
If the company decides to fulfill all the chips requirements and is flexible on the budget, the company should buy 100 lots of chips from Supplier 2. The results provided by the LP run as summarized in Table 1 shall apply to this situation.
The following provides the results for different priorities involving the budget and the chip constraints:
TABLE 4
Rank (or Priority)Of Goals / Supplier
(lots) / Amount of Deviation from Goals
1 / 2 / 3 / 4 / 1 / 2 / 3 / 1 / 2 / 3 / 4
*a / b / c / d / 10 / 80 / 0 / 0 / -4 lots** / 0 / +4 lots
a / d / c / b / 10 / 80 / 0 / 0 / +4 lots / 0 / -4 lots
b / c / d / a / 0 / 100 / 0 / 0 / +0.5 lots / +0.5 lots / +$2000**
c / b / d / a / 10 / 80 / 0 / 0 / -4 lots / +4 lots / 0
d / c / b / a / 10 / 80 / 0 / +4 lots / 0 / -4 lots / 0
*Note 1: a = budget constraint goal; b = excellent chip requirement goal; c = good chips requirement goal; and d = mediocre chips requirement goal. **Note 2: A (-) sign indicates that the company has fallen short of the goal. A (+) sign indicates that the company has exceeded the goal.
Table 4 indicates that in four out of the five possible scenarios where the rank of the priorities are changed, regardless of the priority of the budget and the chips requirement, the result will always be the same. The company will need to purchase 80 lots of chips from Supplier 2 and 10 lots of chips from Supplier 3.
Consequently, the company will always be 4 lots of chips short of meeting the excellent chip requirement. It will always, however, exceed the mediocre chip constraint by the same quantity and satisfy the budget limit of $28,000.
The only exception occurs when the excellent chip (goal b) is given the highest priority and the budget (goal a), the lowest. Only when this happens will the goal of the excellent chip be satisfied. In this case, the budget is exceeded by $2000. And the company will have 500 chips (or 0.5 lots) above its mediocre and good chip requirements.
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