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Linear Programming:VITAMIN CONTRIBUTION, Decision Variables

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Linear Programming:LINEAR PROGRAMMING PROBLEM >>
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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
Wheat
Oats
Rice
Milligrams
n
(mg./oz.)
(mg./oz)
(mg./oz.)
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:
X1 = ounces of wheat
X2 = ounces of oats
X3 = 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. 0.4X1 + 0.6X2 + 0.2X3
where
Z = total cost per box
Rs.
0.4 X1 = cost of wheat per box
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0.6 X2 = cost of rice per box
0.2 X3 = 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 X1 + 20 X2 + 8 X3 > 100 milligrams
where 10 X1 = vitamin A contribution (in mg.) for wheat
20 X2 = vitamin A contribution (in mg.) for oats
8X3 = 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.
7X1 + 14 X2 + 12X3 > 70 milligrams
As in the previous problem there are also nonnegative constraints indicating that negative amounts of each
ingredient cannot be in the cereal.
X1, X2, X3 > 0
The L.P. model for this problem can be summarized as
Minimize
Z = Rs. 0.4 X1 + 0.6 X2 + 0.2 X3
Subject to
10X1 + 20 X2 + 8X3 > 100
7 X1 + 14 X2 + 12 X3 > 70
X1, X2, X3 > 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
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investment alternative has a different perceived risk thus creating a desire to diversify. Majid Khan wants to know
how much to invest in each alternative in order to maximize the return.
The following guidelines have been established for diversifying the investments and lessening the risk;
1.
No more than 20% of the total investment should be in khas deposit.
2.
The amount invested in Defence Savings Certificates should not exceed the amount invested in the other
three alternatives.
3.
At least 30% of the investment should be in NIT and Defence Savings Certificates.
4.
The ration of the amount invested in national certificates to the amount invested in NIT should not exceed
one to three.
Decision Variables
There are four decision variables in this model representing the monetary amount invested in each
investment alternative.
X1 = the amount (Rs. ) invested in national certificates
X2 = the amount (Rs. ) invested in Defence Savings Cert.
X3 = the amount (Rs. ) invested in NIT.
X4 = the amount (Rs. ) invested in khas deposit.
The Objective Function
The objective of the investor is to maximize the return from the investment in the four alternatives. The
total return is the sum of the individual returns from each separate alternative.
Thus, the objective function is expressed as
Maximize
Z = Rs. .085 X1 + .100 X2 + .65 X3 + .130 X4
Where
Z = the total return from all investments
Rs. .085 X1 = the return from the investment in nat. Cer.
.100 X2 = the return from the investment in certificates of deposit.
.065 X3 = the return from the investment in NIT.
.130 X4 = the return from the investment in khas deposit.
Model Constraints
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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
X4 < 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 X2 and the amount
invested in the other alternatives is X1 + X3 + X4 the constraint is
X2 < X1 + X3 + X4
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, X1 + X3 + X4 from both sides of the sign, this constraint in proper from
becomes
X2 - X1 - X3 - X4 < 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 total is Rs. 21, 000 and the amount invested in Defence Savings
Certificates and NIT is represented by X2 + X3, the constraint is,
X2 + X3 > 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
(X1) / (X3 )< 1/3
This constraint is not in standard linear programming form because of the fractional relationship of the
decision variables, X1/X3. It is converted as follows;
X1 < 1 X3/3
3 X1 - X3 < 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,
X1 + X2 + X3 + X4 = Rs. 70, 000
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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 = .085X1 + .100X2 + .065X3 + .130X4
Subject to
X4 < 14, 000
X2 - X1 - X3 - X4 < 0
X2 + X3 > 21, 000
3X1 - X3 < 0
X1 + X2 + X3 + X4 = 70, 000
X1, X2, X3, X4, > 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:
1.
The mixture must contain at least 200 lbs. of mercury.
2.
The mixture must contain at least 300 lbs. of zinc.
3.
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:
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X1 = the number of lbs. of mercury in a batch.
X2 = the number of lbs. of zinc in a batch.
X3 = the number of lbs. of potassium in a batch.
The Objective Function
The objective of the company is to minimize the cost of producing a batch of the chemical mixture. The
total cost is the sum of the individual costs of each ingredient:
Minimize Z = Rs. 4X1 + 8 X2 + 9 X3
where
Z = the total cost of all ingredients
Rs.
4X1 = the cost of mercury in each batch
8X2 = the cost of zinc in each batch
9X3 = the cost of potassium in each batch.
Model Constraints
In this problem the constraints are derived for the chemical formula.
The first specification indicates that the mixture must contain at least 200 lbs. of mercury,
X1 > 200
The second specification is that the mixture must contain at least 300 lbs. of zinc,
X2 > 300
The third specification is that the mixture must contain at least 100 lbs. of potassium,
X3 > 100
Finally, it must not be over looked that the whole mixture relates to a 1, 000-lb. batch. As such, the sum of
all ingredients must exactly equal 1, 000 lbs.,
X1 + X2 + X3 = 1, 000
The complete linear programming model can be summarized as
Minimize
Z = 4X1 + 8 X2 + 9 X3
Subject to
X1 > 200
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X2 > 300
X3 > 100
X1 + X2 + X3 = 1, 000
X1, X2, X3 > 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
Cost
(people/ad or commercial)
Television commercial
20, 000
Rs. 15, 000
Radio commercial
12, 000
8, 000
Newspaper ad
9, 000
4, 000
The following resource constraints exist:
1.
There is a budget limit of Rs. 100,000 available for advertising.
2.
The television station has enough time available for four commercials.
3.
The radio station has enough time available for ten radio commercials.
4.
The newspaper has enough space available for seven ads.
5.
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:
X1 = the number of television commercials
X2 = the number of radio commercials
X3 = the number of newspaper ads
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The Objective Function
The objective of this problem is different from the objectives in the previous examples in which only profit
was Maximized (or cost minimized). In this problem profit is not Maximized, but rather the audience exposure is
Maximized.
This objective function demonstrates that although a linear programming model must either Maximize or
Minimize some objective, the objective itself can be in terms of any type of activity or valuation.
For this problem the objective of audience exposure is determined by summing the audience exposure
gained from each type of advertising
Maximize
Z = 20, 000 X1 + 12, 000 X2 + 9, 000 X3
Where
Z = the total number of audience exposures
20, 000 X1 = the estimated number of exposures from television commercials
12, 000 X2 = the estimated number of exposures from radio commercials
9, 000 X3 = the estimated number of exposures from newspaper ads
Model Constraints
The first constraint in this model reflects the limited budget of Rs. 100, 000 allocated for advertisement,
Rs.
15, 000 X1 + 6, 000 X2 + 4, 000 X3 < 100, 000
where
Rs. 15, 000 X1 = the amount spent for television advertising
6, 000 X2 = the amount spent for radio advertising
4, 000 X3 = the amount spent for newspaper advertising
The next three constraints represent the fact that television and radio commercials are limited to four and
ten, respectively, while newspaper ads are limited to seven.
X1 < 4 commercials
X2 < 10 commercials
X3 < 7 ads
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The final constraint specifies that the total number of commercials and ads cannot exceed fifteen due to
the limitations of the advertising firm:
X1 + X2 + X3 < 15 commercials and ads
The complete linear programming model for this problem is summarized as
Maximize
Z = 20, 000 X1 + 12, 000 X2 + 9, 000 X3
Subject to
Rs. 15, 000 X1 + 6, 000 X2 + 4, 000 X3 < Rs. 100, 000
X1 < 4
X2 < 10
X3 < 7
X1 + X2 + X3 < 15
X1, X2, X3 > 0
Example 6
Transportation
The Philips Television Company produces and ships televisions from three warehouses to three retail stores
on a monthly basis. Each warehouse has a fixed demand per month. The manufacturer wants to know the number of
television sets to ship from each warehouse to each store in order to minimize the total cost of transportation.
Each warehouse has the following supply of televisions available for shipment each month.
Warehouse
Supply (sets)
1. Karachi
300
2. Lahore
100
3. Islamabad
200
------
600
Each retail store has the following monthly demand for television sets:
Store
Demand (sets)
A. Faisalabad
150
B. Peshawar
250
C. Hyderabad
200
-----
600
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The costs for transporting television sets from each warehouse to each retail store are different as a result
of different modes of transportation and distances. The shiping cost per television set for each route are,
From
To store
Warehouse
A
B
C
1
Rs. 6
Rs. 8  Rs. 1
2
4
2
3
3
3
5
7
Decision Variables
The model for this problem consists of nine decision variables representing the number of television sets
transported from each of the three warehouses to each of the three stores,
Xij = the No. of television sets shipped from warehouse "i" to store "j" where i = 1, 2, 3 and j = A, B, C.
Xij is referred to as a double subscripted variable. However, the subscript, whether double or single simply
gives a "name" to the variable (i.e., distinguishes it from other decision variables). As such, the reader should not
view it as more complex than it actually is. For example, the decision variable X3A Islamabad to store A in
Faisalabad.
The Objective Function
The objective function of the television manufacturer is to minimize the total transportation costs for all
shipments. Thus, the objective function is the sum of the individual shipping costs from each warehouse to each
store.
Minimize
Z = Rs. 6X1A + 8X1B + 1X1C + 4X2A + 2X2B + 3X2C + 3X3A + 5X3B + 7X3C
Model Constraints
The constraints in this model are available television sets at each warehouse and the number of sets
demanded at each store. As such, six constraints exist -- one for each warehouse's supply and one for each store's
demand. For example, warehouse 1 retail stores. Since the amount shipped to the three stores is the sum of X1A, X1B,
and X1C the constraint for warehouse 1 is
X1A + X1B + X1C = 300
This constrain is an equality (=) for two reasons. First, more than 300 television sets cannot be shipped,
because that is cannot be shipped, because all 300 are needed at the three stores, the three warehouses must supply
all that can be supplied. Thus, since the total shipped from warehouse 1 cannot exceed 300 or be less than 300 the
constraint is equality. Similarly, the other two supply constraints for warehouse 2 and 3 are also equalities.
X2A + X2B +X2C = 100
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X3A + X3B + X3C = 200
The three demand constraints are developed in the same way except that television sets can be supplied
from any of the three warehouses. Thus, the amount shipped to one store is the sum of the shipments from the three
warehouses:
X1A + X2A + X3A = 150
X1B + X2B + X3B = 250
X1C + X2C + X3C = 200
The complete linear programming model for this problem is summarized as:
Minimize
Z = Rs. 6X1A+8X1B+1X1C + 4X2A + 2X2B + 3X2C + 3X3A + 5X3B + 7X3C
subject to
X1A + X1B + X1C = 300
X2A + X2B + X2C = 100
X3A + X3B + X3C = 200
X1A + X2A + X3A = 150
X1B + X2B + X3B = 250
X1C + X2C + X3C = 200
Xij > 0
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Table of Contents:
  1. Introduction:OR APPROACH TO PROBLEM SOLVING, Observation
  2. Introduction:Model Solution, Implementation of Results
  3. Introduction:USES OF OPERATIONS RESEARCH, Marketing, Personnel
  4. PERT / CPM:CONCEPT OF NETWORK, RULES FOR CONSTRUCTION OF NETWORK
  5. PERT / CPM:DUMMY ACTIVITIES, TO FIND THE CRITICAL PATH
  6. PERT / CPM:ALGORITHM FOR CRITICAL PATH, Free Slack
  7. PERT / CPM:Expected length of a critical path, Expected time and Critical path
  8. PERT / CPM:Expected time and Critical path
  9. PERT / CPM:RESOURCE SCHEDULING IN NETWORK
  10. PERT / CPM:Exercises
  11. Inventory Control:INVENTORY COSTS, INVENTORY MODELS (E.O.Q. MODELS)
  12. Inventory Control:Purchasing model with shortages
  13. Inventory Control:Manufacturing model with no shortages
  14. Inventory Control:Manufacturing model with shortages
  15. Inventory Control:ORDER QUANTITY WITH PRICE-BREAK
  16. Inventory Control:SOME DEFINITIONS, Computation of Safety Stock
  17. Linear Programming:Formulation of the Linear Programming Problem
  18. Linear Programming:Formulation of the Linear Programming Problem, Decision Variables
  19. Linear Programming:Model Constraints, Ingredients Mixing
  20. Linear Programming:VITAMIN CONTRIBUTION, Decision Variables
  21. Linear Programming:LINEAR PROGRAMMING PROBLEM
  22. Linear Programming:LIMITATIONS OF LINEAR PROGRAMMING
  23. Linear Programming:SOLUTION TO LINEAR PROGRAMMING PROBLEMS
  24. Linear Programming:SIMPLEX METHOD, Simplex Procedure
  25. Linear Programming:PRESENTATION IN TABULAR FORM - (SIMPLEX TABLE)
  26. Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE
  27. Linear Programming:The Two Phase Method, First Iteration
  28. Linear Programming:VARIANTS OF THE SIMPLEX METHOD
  29. Linear Programming:Tie for the Leaving Basic Variable (Degeneracy)
  30. Linear Programming:Multiple or Alternative optimal Solutions
  31. Transportation Problems:TRANSPORTATION MODEL, Distribution centers
  32. Transportation Problems:FINDING AN INITIAL BASIC FEASIBLE SOLUTION
  33. Transportation Problems:MOVING TOWARDS OPTIMALITY
  34. Transportation Problems:DEGENERACY, Destination
  35. Transportation Problems:REVIEW QUESTIONS
  36. Assignment Problems:MATHEMATICAL FORMULATION OF THE PROBLEM
  37. Assignment Problems:SOLUTION OF AN ASSIGNMENT PROBLEM
  38. Queuing Theory:DEFINITION OF TERMS IN QUEUEING MODEL
  39. Queuing Theory:SINGLE-CHANNEL INFINITE-POPULATION MODEL
  40. Replacement Models:REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
  41. Replacement Models:ITEMS DETERIORATING WITH TIME VALUE OF MONEY
  42. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS
  43. Dynamic Programming:Analysis of the Result, One Stage Problem
  44. Miscellaneous:SEQUENCING, PROCESSING n JOBS THROUGH TWO MACHINES
  45. Miscellaneous:METHODS OF INTEGER PROGRAMMING SOLUTION