The SAVE-IT COMPANY operates a reclamation center that collects four types of solid waste materials and treats them so that they can be amalgamated into a salable product. (Treating and amalgamating are separate processes.) Three different grades of this product can be made (see the first column of Table 1), depending upon the mix of the materials used. Although there is some flexibility in the mix for each grade, quality standards may specify the minimum or maximum amount allowed for the proportion of a material in the product grade. (This proportion is the weight of the material expressed as a percentage of the total weight for the product grade.) For each of the two higher grades, a fixed percentage Table 1 Grade Specifications Amalgamation cost per pound $ Selling price per pound $ A Material 1: Not more than 30% of total 3 8.5 Material 2: Not less than 40% of total Material 3: Not more than 50% of total Material 4: Exactly 20% of total B Material 1: Not more than 50% of total 2.50 7.00 Material 2: Not less than 10% of total Material 4: Exactly 10% of total C Material 1: Not more than 70% of total 2.00 5.50 is specified for one of the materials. These specifications are given in Table 1 along with the cost of amalgamation and the selling price for each grade. The reclamation center collects its solid waste materials from regular sources and so is normally able to maintain a steady rate for treating them. Table 2 gives the quantities available for collection and treatment each week, as well as the cost of treatment, for each type of material. The Save-It Co. is solely owned by Green Earth, an organization devoted to dealing with environmental issues, so Save- Its profits are used to help support Green Earth’s activities. Green Earth has raised contributions and grants, amounting to $30,000 per week, to be used exclusively to cover the entire treatment cost for the solid waste materials. The board of directors of Green Earth has instructed the management of Save-It to divide this money among the materials in such a way that at least half of the amount available of each material is actually collected and treated. These additional restrictions are listed in Table 2 Table 2 Materials Pounds per week available Treatment cost per pound ($) Additional Restrictions 1 3,000 3 1.For each material, at least half of the pound per week available should be collected and treated 2 2,000 6 2. $30,000 per week should be used to treat these materials 3 4,000 4 4 1,000 5 Within the restrictions specified in Tables 1 and 2, management wants to determine the amount of each product grade to produce and the exact mix of materials to be used for each grade. The objective is to maximize the net weekly profit (total sales income minus total amalgamation, cost), exclusive of the fixed treatment cost of $30,000 per week that is being covered by gifts and grants. Formulate the blended LP for this problem showing 1) The Objective function 2) The mixture specifications from Table 1 3) The availability of materials (2nd column Table 2) 4) Restrictions on the amount treated(Note 1 in 4th column Table 2 5) Restriction on treatment cost(Note 2 in 4th column table 2 6) Non negativity constraints
The SAVE-IT COMPANY operates a reclamation center that collects four types of solid waste materials and treats them so that they can be amalgamated into a salable product. (Treating and amalgamating are separate processes.) Three different grades of this product can be made (see the first column of Table 1), depending upon the mix of the materials used. Although there is some flexibility in the mix for each grade, quality standards may specify the minimum or maximum amount allowed for the proportion of a material in the
product grade. (This proportion is the weight of the material expressed as a percentage of the total weight for the product grade.) For each of the two higher grades, a fixed percentage
Table 1
|
Grade |
Specifications |
Amalgamation cost per pound $ |
Selling price per pound $ |
|
A |
Material 1: Not more than 30% of total |
3 |
8.5 |
|
|
Material 2: Not less than 40% of total |
|
|
|
|
Material 3: Not more than 50% of total |
|
|
|
|
Material 4: Exactly 20% of total |
|
|
|
|
|
|
|
|
B |
Material 1: Not more than 50% of total |
2.50 |
7.00 |
|
|
Material 2: Not less than 10% of total |
|
|
|
|
Material 4: Exactly 10% of total |
|
|
|
|
|
|
|
|
C |
Material 1: Not more than 70% of total |
2.00 |
5.50 |
is specified for one of the materials. These specifications are given in Table 1 along with
the cost of amalgamation and the selling price for each grade.
The reclamation center collects its solid waste materials from regular sources and so is normally able to maintain a steady rate for treating them. Table 2 gives the quantities available for collection and treatment each week, as well as the cost of treatment, for each type of material.
The Save-It Co. is solely owned by Green Earth, an organization devoted to dealing with environmental issues, so Save- Its profits are used to help support Green Earth’s activities. Green Earth has raised contributions and grants, amounting to $30,000 per week, to be used exclusively to cover the entire treatment cost for the solid waste materials.
The board of directors of Green Earth has instructed the management of Save-It to divide this money among the materials in such a way that at least half of the amount available of each material is actually collected and treated. These additional restrictions are listed in Table 2
Table 2
|
Materials |
Pounds per week available |
Treatment cost per pound ($) |
Additional Restrictions |
|
1 |
3,000 |
3 |
1.For each material, at least half of the pound per week available should be collected and treated |
|
2 |
2,000 |
6 |
2. $30,000 per week should be used to treat these materials |
|
3 |
4,000 |
4 |
|
|
4 |
1,000 |
5 |
|
|
|
|
|
|
|
|
|
|
|
Within the restrictions specified in Tables 1 and 2, management wants to determine the amount of each product grade to produce and the exact mix of materials to be used for each grade. The objective is to maximize the net weekly profit (total sales income minus total amalgamation, cost), exclusive of the fixed treatment cost of $30,000 per week that is being covered by gifts and grants.
Formulate the blended LP for this problem showing
1) The Objective function
2) The mixture specifications from Table 1
3) The availability of materials (2nd column Table 2)
4) Restrictions on the amount treated(Note 1 in 4th column Table 2
5) Restriction on treatment cost(Note 2 in 4th column table 2
6) Non negativity constraints
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