### Linear Programming Formulations
#### Problem 1: Clothing Production
Let's denote the meters of suiting, shirting, and woolens produced as \( x_1 \), \( x_2 \), and \( x_3 \) respectively.
**Objective Function:**
Maximize Profit \( Z = 2x_1 + 4x_2 + 3x_3 \)
**Constraints:**
- Weaving time: \( 3x_1 + 4x_2 + 3x_3 \leq 60 \times 60 \) (total minutes available)
- Processing time: \( 2x_1 + x_2 + 3x_3 \leq 40 \times 60 \)
- Packing time: \( x_1 + 3x_2 + 3x_3 \leq 80 \times 60 \)
Convert hours to minutes for constraints:
- Weaving: \( 3x_1 + 4x_2 + 3x_3 \leq 3600 \)
- Processing: \( 2x_1 + x_2 + 3x_3 \leq 2400 \)
- Packing: \( x_1 + 3x_2 + 3x_3 \leq 4800 \)
\[ \begin{cases}
3x_1 + 4x_2 + 3x_3 \leq 3600 \\
2x_1 + x_2 + 3x_3 \leq 2400 \\
x_1 + 3x_2 + 3x_3 \leq 4800 \\
x_1, x_2, x_3 \geq 0
\end{cases} \]
#### Problem 2: Transportation Problem
**Objective Function:**
Minimize Cost \( Z = 8x_{11} + 9x_{12} + 6x_{13} + 3x_{14} + 6x_{21} + 11x_{22} + 5x_{23} + 10x_{24} + 3x_{31} + 8x_{32} + 7x_{33} + 9x_{34} \)
**Constraints:**
- Supply Constraints:
- Warehouse W1: \( x_{11} + x_{12} + x_{13} + x_{14} \leq 15 \)
- Warehouse W2: \( x_{21} + x_{22} + x_{23} + x_{24} \leq 16 \)
- Warehouse W3: \( x_{31} + x_{32} + x_{33} + x_{34} \leq 12 \)
- Warehouse W4: \( x_{41} + x_{42} + x_{43} + x_{44} \leq 13 \)
- Demand Constraints:
- Customer A: \( x_{11} + x_{21} + x_{31} + x_{41} \geq 18 \)
- Customer B: \( x_{12} + x_{22} + x_{32} + x_{42} \geq 20 \)
- Customer C: \( x_{13} + x_{23} + x_{33} + x_{43} \geq 18 \)
\[ \begin{cases}
x_{11} + x_{12} + x_{13} + x_{14} \leq 15 \\
x_{21} + x_{22} + x_{23} + x_{24} \leq 16 \\
x_{31} + x_{32} + x_{33} + x_{34} \leq 12 \\
x_{41} + x_{42} + x_{43} + x_{44} \leq 13 \\
x_{11} + x_{21} + x_{31} + x_{41} \geq 18 \\
x_{12} + x_{22} + x_{32} + x_{42} \geq 20 \\
x_{13} + x_{23} + x_{33} + x_{43} \geq 18 \\
x_{ij} \geq 0 \quad \forall i,j
\end{cases} \]
#### Problem 3: Chemical Purchase for Garden
Let:
- \( x \) = number of jars of liquid product
- \( y \) = number of cartons of dry product
**Objective Function:**
Minimize Cost \( Z = 3x + 2y \)
**Constraints:**
- For chemical A: \( 5x + y \geq 10 \)
- For chemical B: \( 2x + 2y \geq 12 \)
- For chemical C: \( x + 4y \geq 12 \)
\[ \begin{cases}
5x + y \geq 10 \\
2x + 2y \geq 12 \\
x + 4y \geq 12 \\
x, y \geq 0
\end{cases} \]
#### Problem 4: Transportation Using VAM
**Initial Solution using VAM**
- Calculate penalties for each row and column.
- Allocate as much as possible to the lowest cost cell in the row or column with the highest penalty.
- Update supplies and demands, and recalculate penalties, iterating until all supplies or demands are met.
#### Problem 5: Production Hours Optimization
Let:
- \( x_1 \) = units of product A
- \( x_2 \) = units of product B
- \( x_3 \) = units of product C
**Objective Function:**
Maximize Profit \( Z = 30x_1 + 40x_2 + 35x_3 \)
**Constraints:**
- Machine I: \( 3x_1 + 4x_2 + 2x_3 \leq 90 \)
- Machine II: \( 2x_1 + x_2 + 2x_3 \leq 54 \)
- Machine III: \( x_1 + 3x_2 + 2x_3 \leq 93 \)
\[ \begin{cases}
3x_1 + 4x_2 + 2x_3 \leq 90 \\
2x_1 + x_2 + 2x_3 \leq 54 \\
x_1 + 3x_2 + 2x_3 \leq 93 \\
x_1, x_2, x_3 \geq 0
\end{cases} \]
### Summary
1. Use the simplex method for linear programming problems.
2. Use the North-West Corner, Least Cost, or Vogel's Approximation Method (VAM) for transportation problems.
3. Test for optimality using the Stepping Stone Method.
4. Use graphical methods for two-variable LP problems.
Assigning deadlines and validating through optimality methods ensures realistic and efficient operational strategies. Provide detailed solutions and validate the results through different methodologies.
Department of Management
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 Birr 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.
Formulate the linear programming problem (LPP)
A company has four ware houses 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:
Ware house:
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.
Minz=8x11+9x12+6x13+3x14
+6x21+11x22+5x23+10x24
+3x31+8x32+7x33+9x34
Find the initial solution using each of the three methods.
Test optimality using Stepping stone method.
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?
Formulate the linear programming problem (LPP).
Solve the problems using a graphical method.
A manufacturer wants to ship 22 loads of his product as shown below. The matrix gives the kilometers from sources of supply to the destinations.
Destination
D1
D2
D3
D4
D5
SUPPLY
S1
5
8
6
6
3
8
S2
4
7
7
6
5
5
S3
8
4
6
6
4
9
DEMAND
4
4
5
4
8
Use VAM method to find initial solution and test optimality using steeping stone method.
A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, 2 hours on machine II and one hour on machine III. Each unit of product B requires 4 hours on machine I, one hour on machine II and 3 hours on machine III, while each unit of product C requires 2 hours on each of the three machines. The contribution margin of the three products is birr 30, birr 40, and birr 35 per unit respectively. The machine hours available on three machines are 90, 54, and 93 respectively.
Formulate the above problem as LPP
Deadline for Submission: October 16, 2023.
Assignments that do not meet deadline will bear penalties and shall not be considered.
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