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Example 2 Ingredients Mixing
Fauji Foundation produces a cereal SUNFLOWER, which they advertise as meeting the minimum daily requirements for vitamins A and D. The mixing department of the company uses three main ingredients in making the cereal-wheat, oats, and rice, all three of which contain amounts of vitamin A and D. Given that each box of cereal must contain minimum amounts of vitamin A and D, the company has instructed the mixing department determine how many ounces of each ingredient should go into each box of cereal in order to minimize total cost.
This problem differs from the previous one in that its objective is to minimize cost, rather than Maximize profit.
Each ingredient has the following vitamin contribution and requirement per box.
VITAMIN CONTRIBUTION
Vitami n
Wheat (mg./oz.)
Oats (mg./oz)
Rice (mg./oz.)
Milligrams Required/Box A 10 20 08 100 D 07 14 12 70
The cost of one ounce of wheat is Rs. 0.4, the cost of an ounce of oats is Rs. 0.6, and the cost of one ounce of rice is Rs. 0.2.
Decision Variables
This problem contains three decision variables for the number of ounces of each ingredient in a box of cereal:
X 1 = ounces of wheat
X 2 = ounces of oats
X 3 = ounces of rice
The Objective Function
The objective of the mixing department of the Fauji Foundation is to minimize the cost of each box of cereal. The total cost is the sum of the individual costs resulting from each ingredient. Thus, the objective function that is to minimize total cos, Z, is expressed as
Minimize Z = Rs. 0.4X 1 + 0.6X 2 + 0.2X 3
where Z = total cost per box
Rs. 0.4 X 1 = cost of wheat per box
0.6 X 2 = cost of rice per box
0.2 X 3 = cost of rice per box
Model Constraints
In this problem the constraints reflect the requirements for vitamin consistency of the cereal. Each ingredient contributes a number of milligrams of the vitamin to the cereal. The constraint for vitamin A is
10 X 1 + 20 X 2 + 8 X 3 > 100 milligrams
where 10 X 1 = vitamin A contribution (in mg.) for wheat
20 X 2 = vitamin A contribution (in mg.) for oats
8X 3 = vitamin A contribution (in mg.) for rice
Notice that rather than an (<) inequality, as used in the previous example, this constraint requires a > (greater than or minimum requirement) specifying that at least 100 mg of vitamin A must be in a box. If a minimum cost solution results so that more than 100 mg is in the cereal mix, which is acceptable, however, the amount cannot be less than 100 mg.
The constraint for vitamin D is constructed like the constraint for vitamin A.
7X 1 + 14 X 2 + 12X 3 > 70 milligrams
As in the previous problem there are also nonnegative constraints indicating that negative amounts of each ingredient cannot be in the cereal.
X 1 , X 2 , X 3 > 0
The L.P. model for this problem can be summarized as
Minimize Z = Rs. 0.4 X 1 + 0.6 X 2 + 0.2 X 3
Subject to 10X 1 + 20 X 2 + 8X 3 > 100
7 X 1 + 14 X 2 + 12 X 3 > 70
X 1 , X 2 , X 3 > 0
Example 3 Investment Planning
Mr. Majid Khan has Rs. 70, 000 to investment in several alternatives. The alternative investments are national certificates with an 8.5% return, Defence Savings Certificates with a 10% return, NIT with a 6.5% return, and khas deposit with a return of 13%. Each alternative has the same time until maturity. In addition, each
In this problem the constraints are the guidelines established by the investor for diversifying the total investment. Each guideline will be transformed into a mathematical constraint separately.
Guideline one states that no more than 20% of the total investment should be in khas deposit. Since the total investment will be Rs. 70, 000 (i.e., the investor desires to invest the entire amount), then 20% of Rs. 70, 000 is Rs. 14, 000. Thus, this constraint is
X 4 < Rs. 14, 000
The second guideline indicates that the amount invested in Defence Savings Cert. should not exceed the amount invested in the other three alternatives. Since the investment in Defence Savings Cert. is X 2 and the amount invested in the other alternatives is X 1 + X 3 + X 4 the constraint is
X 2 < X 1 + X 3 + X 4
However, the solution technique for linear programming problems will require that constraints be in a standard form so that all decision variables are on the left side of the inequality (i.e., < ) and all numerical values are on the right side. Thus, by subtracting, X 1 + X 3 + X 4 from both sides of the sign, this constraint in proper from becomes
X 2 - X 1 - X 3 - X 4 < 0
Thus third guideline specifies that at least 30% of the investment should be in NIT and Defence Savings Certificates. Given that 30% of the Rs. 70, 000 totals is Rs. 21, 000 and the amount invested in Defence Savings Certificates and NIT is represented by X 2 + X 3 , the constraint is,
X 2 + X 3 > Rs. 21, 000
The fourth guideline states that the ratio of the amount invested in national certificates to the amount invested in NIT should not exceed one to three. This constraint is expressed as
(X 1 ) / (X 3 ) < 1/
This constraint is not in standard linear programming form because of the fractional relationship of the decision variables, X 1 /X 3. It is converted as follows;
X 1 < 1 X 3 /
3X 1 - X 3 < 0
Finally, Majid Khan wants to invest all of the Rs. 70, 000 in the four alternatives. Thus, the sum of all the investments in the four alternatives must equal Rs. 70, 000,
X 1 + X 2 + X 3 + X 4 = Rs. 70, 000
This last constraint differs from the < and > inequalities previously developed, in that a specific requirement exists to invest an exact amount. Thus, the possibility of investing more than Rs. 70, 000 or less than Rs. 70, 000 is not considered.
This problem contains all three of the types of constraints that are possible in a linear programming problem: <, = and >. Further, note that there is no restriction on a model containing any mix of these types of constraints as demonstrated in this problem.
The complete LP model for this problem can be summarized as
Maximize Z = .085X 1 + .100X 2 + .065X 3 + .130X 4
Subject to X 4 < 14, 000
X 2 - X 1 - X 3 - X 4 < 0
X 2 + X 3 > 21, 000
3X 1 - X 3 < 0
X 1 + X 2 + X 3 + X 4 = 70, 000
X 1 , X 2 , X 3 , X 4 , > 0
Example 4 Chemical Mixture
United Chemical Company produces a chemical mixture for a customer in 1, 000 - pound batches. The mixture contains three ingredients - zinc, mercury, and potassium. The mixture must conform to formula specifications (i.e., a recipe) supplied by the customer. The company wants to know the amount of each ingredient to put in the mixture that will meet all the requirements of the mix and minimize total cost.
The formula for each batch of the mixture consists of the following specifications:
- The mixture must contain at least 200 lbs. of mercury.
- The mixture must contain at least 300 lbs. of zinc.
- The mixture must contain at least 100 lbs. of potassium.
The cost per pound for mercury is Rs. 4; for zinc, Rs. 8; and for potassium, Rs. 9.
Decision Variables
The model for this problem contains three decision variables representing the amount of each ingredient in the mixture:
X 2 > 300
X 3 > 100
X 1 + X 2 + X 3 = 1, 000
X 1 , X 2 , X 3 > 0
Example 5 Marketing
The Bata Shoe Company has contracted with an advertising firm to determine the types and amount of advertising it should have for its stores. The three types of advertising available are radio and television commercials and newspaper ads. The retail store desires to know the number of each type of advertisement it should purchase in order to Maximize exposure. It is estimated that each ad and commercial will reach the following potential audience and cost the following amount.
Type of Advertisement Exposure (people/ad or commercial)
Cost
Television commercial 20, 000 Rs. 15, 000 Radio commercial 12, 000 6, 000 Newspaper ad 9, 000 4, 000
The following resource constraints exist:
- There is a budget limit of Rs. 100,000 available for advertising.
- The television station has enough time available for four commercials.
- The radio station has enough time available for ten radio commercials.
- The newspaper has enough space available for seven ads.
- The advertising agency has time and staff to produce at most a total of fifteen commercials ads.
Decision Variables
This model consists of three decision variables representing the number of each type of advertising produced:
X 1 = the number of television commercials
X 2 = the number of radio commercials
X 3 = the number of newspaper ads
