4. Tom would like 3 pints of home brew today and an additional 6 pints of home brew tomorrow. Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and $2.70 per pint tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint today and $2.80 per pint tomorrow. Tom wishes to know what his purchases should be to minimize his cost while satisfying his thirst requirements. (a) Formulate a linear programming model for this problem. (max or min Z=..., subject to ... constraints...) (Let x₁ be the amount of pints of brew bought from Dick today, x₂ be the amount of pints of brew bought from Dick tomorrow, x3 be the amount of pints of brew bought from Harry today, x4 be the amount of pints of brew bought from Harry tomorrow.) (b) Formulate this problem as a transportation problem by constructing the appropriate parameter table (in the format as below). Source Demand 1 2 1 Destination 2 3..... supply (c) Draw the network representation of this problem. (d) Use Excel Solver to obtain an optimal solution.
4. Tom would like 3 pints of home brew today and an additional 6 pints of home brew tomorrow. Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and $2.70 per pint tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint today and $2.80 per pint tomorrow. Tom wishes to know what his purchases should be to minimize his cost while satisfying his thirst requirements. (a) Formulate a linear programming model for this problem. (max or min Z=..., subject to ... constraints...) (Let x₁ be the amount of pints of brew bought from Dick today, x₂ be the amount of pints of brew bought from Dick tomorrow, x3 be the amount of pints of brew bought from Harry today, x4 be the amount of pints of brew bought from Harry tomorrow.) (b) Formulate this problem as a transportation problem by constructing the appropriate parameter table (in the format as below). Source Demand 1 2 1 Destination 2 3..... supply (c) Draw the network representation of this problem. (d) Use Excel Solver to obtain an optimal solution.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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![### Problem Description
Tom needs to purchase home brew over two days: 3 pints today and 6 pints tomorrow. Two vendors, Dick and Harry, offer different prices and maximum quantity limits.
- **Dick's Offer**:
- Max 5 pints total
- $3.00 per pint today
- $2.70 per pint tomorrow
- **Harry's Offer**:
- Max 4 pints total
- $2.90 per pint today
- $2.80 per pint tomorrow
Tom aims to minimize cost while meeting his requirements.
### Formulating a Linear Programming Model
#### Decision Variables
- \( x_1 \): Pints bought from Dick today
- \( x_2 \): Pints bought from Dick tomorrow
- \( x_3 \): Pints bought from Harry today
- \( x_4 \): Pints bought from Harry tomorrow
#### Objective Function
Minimize Cost:
\[ Z = 3x_1 + 2.7x_2 + 2.9x_3 + 2.8x_4 \]
#### Constraints
1. \( x_1 + x_3 = 3 \) (Requirement for today)
2. \( x_2 + x_4 = 6 \) (Requirement for tomorrow)
3. \( x_1 + x_2 \leq 5 \) (Dick's total supply constraint)
4. \( x_3 + x_4 \leq 4 \) (Harry's total supply constraint)
5. \( x_1, x_2, x_3, x_4 \geq 0 \) (Non-negativity)
### Transportation Problem Formulation
The given setup can be reformulated as a transportation problem:
#### Parameter Table
\[
\begin{array}{c|ccc|c}
& \text{Destination 1} & \text{Destination 2} & \text{Destination 3} & \text{Supply} \\
\hline
\text{Source 1} & & & & \\
\text{Source 2} & & & & \\
\hdashline
\text{Demand} & & & & \\
\end{array}
\]
- **Supply Rows** correspond to the vendors and the day:
- Source 1: Dick today
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2F48fa231c-428c-45ba-92a8-ff11fbb8e998%2F7juklhs_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
Tom needs to purchase home brew over two days: 3 pints today and 6 pints tomorrow. Two vendors, Dick and Harry, offer different prices and maximum quantity limits.
- **Dick's Offer**:
- Max 5 pints total
- $3.00 per pint today
- $2.70 per pint tomorrow
- **Harry's Offer**:
- Max 4 pints total
- $2.90 per pint today
- $2.80 per pint tomorrow
Tom aims to minimize cost while meeting his requirements.
### Formulating a Linear Programming Model
#### Decision Variables
- \( x_1 \): Pints bought from Dick today
- \( x_2 \): Pints bought from Dick tomorrow
- \( x_3 \): Pints bought from Harry today
- \( x_4 \): Pints bought from Harry tomorrow
#### Objective Function
Minimize Cost:
\[ Z = 3x_1 + 2.7x_2 + 2.9x_3 + 2.8x_4 \]
#### Constraints
1. \( x_1 + x_3 = 3 \) (Requirement for today)
2. \( x_2 + x_4 = 6 \) (Requirement for tomorrow)
3. \( x_1 + x_2 \leq 5 \) (Dick's total supply constraint)
4. \( x_3 + x_4 \leq 4 \) (Harry's total supply constraint)
5. \( x_1, x_2, x_3, x_4 \geq 0 \) (Non-negativity)
### Transportation Problem Formulation
The given setup can be reformulated as a transportation problem:
#### Parameter Table
\[
\begin{array}{c|ccc|c}
& \text{Destination 1} & \text{Destination 2} & \text{Destination 3} & \text{Supply} \\
\hline
\text{Source 1} & & & & \\
\text{Source 2} & & & & \\
\hdashline
\text{Demand} & & & & \\
\end{array}
\]
- **Supply Rows** correspond to the vendors and the day:
- Source 1: Dick today
-
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