1. A company has three operational departments (weaving, processing, and packing) with the capacity to produce three different types of clothes namely suiting, shirting, and woolens yielding a profit of $2,$4, and $3 respectively. One meter of suiting requires 3 minutes in weaving, 2 minutes in processing, and 1 minute in packing. Similarly, one meter of shirting requires 4 minutes in weaving, 1 minute in processing, and 3 minutes in packing. One meter of woolen requires 3 minutes in each department. In a week, the total run time of each department is 60, 40, and 80 hours for weaving, processing, and packing respectively.

Question

A. Formulate the linear programming problem (LPP)

Let x1, x2, and x3 represent the number of meters of suiting, shirting, and woolen produced respectively. The objective function can be formulated as:

Maximize: P = 2x1 + 4x2 + 3x3

Subject to the constraints:

3x1 + 4x2 + 3x3 ≤ 60 (Weaving constraint)
2x1 + x2 + x3 ≤ 40 (Processing constraint)
x1 + x2 + x3 ≤ 80 (Packing constraint)

2. A company has four warehouses W1, W2, W3, and W4. It is required to deliver a product from these warehouses to three customers A, B, and C. The warehouses have the following amounts in stock:

Warehouse: W1 W2 W3 W4
No of units: 15 16 12 13

And the customer's requirements are:

Customer: A B C
No of units: 18 20 18

The table below shows the costs of transporting one unit from warehouse to customer is given below.

Minimize z = 8x11 + 9x12 + 6x13 + 3x14 + 6x21 + 11x22 + 5x23 + 10x24 + 3x31 + 8x32 + 7x33 + 9x34

A. Find the initial solution using each of the three methods.

To find the initial solution, we can use the simplex method, the dual simplex method, or the interior-point method. These methods are commonly used to solve linear programming problems. The choice of method depends on the specific problem and the available software or computational resources.

B. Test optimality using the Stepping stone method.

To test the optimality of the initial solution, we can use the Stepping stone method. This method involves finding a feasible direction in which the objective function value can be improved. If no such direction is found, then the current solution is optimal. Otherwise, the solution can be further improved by moving in the direction of the improving feasible direction.

3. A person requires 10, 12, and 12 units of chemicals A, B, and C, respectively for his garden.

A liquid product contains 5, 2, and 1 unit of A, B, and C, respectively per jar. A dry product contains 1, 2, and 4 units of A, B, and C per unit. If the liquid product sells for Br. 3 per jar and the dry product for Br. 2 per carton, how many of each should be purchased in order to minimize the cost and meet the requirement?

A. Formulate the linear programming problem (LPP)

Let x1, x2, and x3 represent the number of jars of the liquid product, the number of units of the dry product, and the number of units of the dry product, respectively. The objective function can be formulated as:

Minimize: C = 3x1 + 2x2

Subject to the constraints:

1, A person requires 10,12, and 12, units of chemicals A, B, and C ,respectivitily for his garden a liquid product contains 5, 2 , and 1 unit of A, B, and ,C respectivily per jar a dry product and the dry product contains 1,2, and 4 units of A,B, and C per unit . if the liquid product sells for birr 3 per jar and the dry product for birr 2 per Carton , How many of each should be perchased in order to minimize the cost and meet the requirement ?
A, formulate tht linear programming problem (lpp)
B, Solve the problems using a graphical method ?

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Answer 1

A. Formulate the linear programming problem (LPP)

Let x1, x2, and x3 represent the number of jars of the liquid product, the number of units of the dry product, and the number of units of chemical C purchased, respectively. The objective function can be formulated as:

Minimize: C = 3x1 + 2x2

Subject to the constraints:
5x1 + x2 ≥ 10 (Requirement for chemical A)
2x1 + 2x2 ≥ 12 (Requirement for chemical B)
x1 + 4x2 ≥ 12 (Requirement for chemical C)

B. Solve the problem using a graphical method:

To solve the problem using a graphical method, we can plot the constraints on a graph and find the feasible region. By plotting the objective function line (C = 3x1 + 2x2) on the graph, we can determine the optimal solution that minimizes the cost while meeting the requirements for chemicals A, B, and C.