IE 416: Operations Research I Fall 2010
Source of the problem statement: W. L. Winston, " Operations Research, Application and Algorithms"
Solution by student groups (formulation was given)
Dr. Parisay’s comments are in red.
Brute Production Process, Example 13, Pg 95
Problem Statement
Rylon Corporation manufactures Brute and Chanelle perfumes. The raw material needed to manufacture each type of perfume can be purchased for $3 per pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed raw material yields 3 oz of Regular Brute Perfume and 4 oz of Regular Chanelle Perfume. Regular Brute can be sold for $7/oz and Regular Chanelle for $6/oz. Rylon also has the option of further processing Regular Brute and Regular Chanelle to produce Luxury Brute, sold at $18/oz, and Luxury Chanelle, sold at $14/oz. Each ounce of Regular Brute processed further requires an additional 3 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Brute. Each ounce of Regular Chanelle processed further requires an additional 2 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Chanelle. Each year, Rylon has 6,000 hours of laboratory time available and can purchase up to 4,000 lb of raw material. Formulate an LP that can be used to determine how Rylon can maximize profits. Assume that the cost of the laboratory hours is a fixed cost.
Summary Table
This is an example that a summary graph (as in you book and copied above) is more useful than a summary table. This graph shows the interrelationship between Raw materail, Brute, and Chanelle in a more clear manner. Notice that you cannot process Raw materail to obtain just Brute (or Chanelle). You have no choice. As soon as you process Raw material you will get both Brute and chanelle, and with the amount mentioned in the problem.
The summary table is created for the convenience of handling the information given in the problem statement:
Item / Revenue / Costs (Processing) / Costs (Materials) / Lab TimeRegular Brute / 7 $/oz / 0 $/oz / $3 = 1 $
3oz oz
*This is misleading / 1 hr/lb
This is misleading
Luxury Brute / 18 $/oz / 4 $/oz / $3 = 3 $
1oz oz / (1+3) = 4 hr/lb
Regular Chanelle / 6 $/oz / 0 $/oz / $3 = 0.75 $
4oz oz / 1 hr/lb
Luxury Chanelle / 14 $/oz / 4 $/oz / $3 = 3 $
1oz oz / (2+1) = 3 hr/lb
Max = 4,000 lb/year / Max = 6,000 hrs/year
*This is misleading, as it gives impression that it is possible to process Raw material and obtain just Brute.
This is misleading, as gives the impression that one unit Brute requires one hour of lab and does not show the sharing of that one hour with production of Chanelle.
Having said above, if it helps you to formulate correctly and communicate with others, it is fine.
By Malley, can add more info in there:
X1 + X2 = 3 X5 X3 + X4 = 4 X5
Decision Variables
RB sold/yr = Oz of regular Brute sold per year (It is better to change this variable name as it will create confusion in formula. A variable name that has “/” and space is not common. You may define it as RBsoldPerYr. You may use paranthesis around the variable name in the formula to prevent confusion.)
LB sold/yr = Oz of luxury Brute sold per year
RC sold/yr = Oz of regular Chanelle sold per year
LC sold/yr = Oz of luxury Chanelle sold per year
Raw mat bought/yr = Lb of raw materials bought per year (It is better to change this variable name as it will create confusion in formula.)
Objective Function
The objective function of this problem is to maximize profits, and this can be obtain by subtracting various costs by the revenues earned from sales.
Notice the paranthesis added here > O.F max z= 7 (RB sold/yr) + 18 (LB sold/yr) + 6 (RC sold/yr) + 14 (LC sold/yr) – (4 (LB sold/yr) + 4 (LC sold/yr) ) – 3 (Raw mat bought/yr)
= 7 RB sold/yr + 14 LB sold/yr + 6 RC sold/yr + 10 LC sold/yr - 3 Raw mat bought/yr
Constraints
•No more than 4,000 lb of raw material purchased annually
•No more than 6,000 hours of laboratory time can be used each year
•Total ounces of Regular and Luxury Brute sold is equal to number of Brute produced per pound of raw material
•Total ounces of Regular and Luxury Chanelle sold is equal to the number of Chanelle produced per pound of raw material
S.T. Raw mat bought/yr ≤4000
3 LB sold/yr + 2 LC sold/yr + Raw mat bought/yr ≤6000
RB sold/yr + LB sold/yr - 3 Raw mat bought/yr = 0
RC sold/yr + LC sold/yr – 4 Raw mat bought/yr = 0
WinQSB Problem Specification
WinQSB Input
If you change name of the constraints it will assist with analysis and communication.
WinQSB Output
After solving using WinQSB, we have obtained the optimal solution of z= 172,666.7, number of regular Brute to be sold per year = 11,333.33, number of luxury Brute to be sold per year = 666.6667, number of regular Chanelle to be sold per year = 16,000, number of luxury Chanelle to be sold per year = 0, and raw materials to be bought per year = 4000.
Sensitivity Analysis (O.F.)
We perform sensitivity analysis on the objective function, changing the cost of the raw materials per pound. (Mention the motivation for this selection.)
Sensitivity Analysis (O.F.) Output Summary
Sensitivity Analysis (O.F) Graphical Summary
Sensitivity Analysis (RHS)
We also perform sensitivity analysis on the right hand side of constraints for the number of raw materials to be bought per year. (Mention the motivation for this selection.)
Sensitivity Analysis (RHS) Output Summary
Sensitivity Analysis (RHS) Graphical Summary
Objective Function Coefficient Sensitivity Analysis: This is repeated!!
Sensitivity Analysis for selling price of one ounce of Regular Brute (OFC):
(Mention the motivation for this selection.)
Right Hand Side Sensitivity Analysis: This is repeated!!
Sensitivity Analysis for available Lab Time (RHS):
Mention the motivation for this selection.
Report to the Manager
Dear Manager,
After performing calculations on the data, Rylon Corporation should be able to maximize profits to $172,666.67. [ what is this?! Expected value!! This is an expected value, which means that the actual value will not be exactly the same as the value that we have calculated.] In order to obtain the maximized profit, Rylon will need to purchase 4,000 lb of raw materials per year while producing 11,333.33 oz of regular Brute, 666.667 oz of luxury Brute and 16,000 oz of regular Chanelle. . [ what is this?! Again, these are only expected values, meaning that they are averages of a number of possible values. ]
It is better to rewrite the report as: Rylon Corporation should be able to maximize profits to $172,666.67. To maximize profit, Rylon will need to purchase 4,000 lb of raw materials per year and use all 6000 hours of available labor. Then they will produce 12000 oz of Regular Brute and 16000 oz of Regular Chanelle. It is suggested that further process 666.7 oz of Regular Brute to Luxury Brute, which will leave 11333.3 oz of Regular Brute for sale. Do not produce any Luxury Chanelle.
Perfume Type / Amount to Produce (Oz)Regular Brute / 11,333.33
Luxury Brute / 666.67
Regular Chanelle / 16,000
Luxury Chanelle / 0
Profit: $172,666.70
If you would like to increase our profits, we need to look at our constraints and adjust from there. We have performed sensitivity analysis on the case, where we want to find out the changes that the profits will have if the amount of raw materials purchased per year varies. From the sensitivity analysis output summary and the graphical summary, we can see that the profit immediately [This is wrong! gets lesser to a value of $14,000 once Rylon buys 42,667lb of raw material per year. There is a pattern showing the decreasing profits with increasing number of raw materials purchased per year. ]
Therefore, we performed another sensitivity analysis on the [Do not use these terms in report to the manager. right hand side of the constraint ] which limits the number (This is the same RHS as the previous paragraph.) of raw materials to be bought per year. The result shows that when Rylon slightly increases the number of raw materials purchased to 6000lb, they will gain an increase in profit of $252,000. However, if they reduces the number of raw materials purchased per year, the profit will correspond and reduce as well.
We could either increase the amount of lab time available, which would increase the profits by $2.33 for every extra hour added to the laboratory time that we can use (up to …). Or, an alternative that would add even more profit than adding lab time, is finding a way to get more raw material (this was discussed before). For every extra pound that we buy, we could create a profit of $ 39.67 per pound. (You can discuss managerial use of shadow price here.) If you decide that you would like to go this route in order to increase profits, further analysis would need to be done, since this would change our optimal solution. This solution would depend on which constraint you would like to change in order to gain more profit. However, without changing any of the constraints, there is no flexibility to change this optimal solution (not true, you can change coefficient of OF) (or change the amount we produce of each perfume) because all of our constraints [Do not use these terms in report to the manager. our are binding.]
From all the sensitivity analysis conducted, we can conclude that Rylon Corporation should not make drastic changes (not clear what drastic change is, be specific) to the number of raw materials purchased per year (not a correct conclusion). By keeping the number in the range of 4000lb to 6000lb, Rylon Corp will be able to maximize their profits (this is not a correct claim, for any value they can maximize, the point is under what condition).
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