Practical Management Science
Practical Management Science
6th Edition
ISBN: 9781337406659
Author: WINSTON, Wayne L.
Publisher: Cengage,
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I need help with the constraint and how to solve the probelm on excel

### Problem 2 – Network Model

You've just started a job at a company that produces and distributes products around Ohio. Your boss is struggling with the logistics of shipping your products in the most cost-effective way possible. She has tasked you with creating a network model in Excel to help her out. 

Your logistics network consists of three factories, two distribution centers, and two retail stores. The factories are located in Toledo, Cleveland, and Steubenville, OH. The distribution centers are located in Marion and Newark, OH. The retail locations are located in Columbus and Dayton, OH. A map of these nodes can be seen below:

![Ohio Network Map](https://www.example.com/path_to_image.jpg)

**Map Explanation:**

The map illustrates the logistics network across Ohio with the following key points:
- **Factories**:
  - **Toledo** (marked by a red dot in the northwest)
  - **Cleveland** (marked by a red dot in the northeast)
  - **Steubenville** (marked by a red dot in the east)
  
- **Distribution Centers**:
  - **Marion** (marked by a red dot in the center-north)
  - **Newark** (marked by a red dot in the center-east)
  
- **Retail Stores**:
  - **Columbus** (marked by a red dot in the center)
  - **Dayton** (marked by a red dot in the southwest)

These locations will serve as the points in your network model to facilitate the efficient transportation and distribution of products within Ohio.
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Transcribed Image Text:### Problem 2 – Network Model You've just started a job at a company that produces and distributes products around Ohio. Your boss is struggling with the logistics of shipping your products in the most cost-effective way possible. She has tasked you with creating a network model in Excel to help her out. Your logistics network consists of three factories, two distribution centers, and two retail stores. The factories are located in Toledo, Cleveland, and Steubenville, OH. The distribution centers are located in Marion and Newark, OH. The retail locations are located in Columbus and Dayton, OH. A map of these nodes can be seen below: ![Ohio Network Map](https://www.example.com/path_to_image.jpg) **Map Explanation:** The map illustrates the logistics network across Ohio with the following key points: - **Factories**: - **Toledo** (marked by a red dot in the northwest) - **Cleveland** (marked by a red dot in the northeast) - **Steubenville** (marked by a red dot in the east) - **Distribution Centers**: - **Marion** (marked by a red dot in the center-north) - **Newark** (marked by a red dot in the center-east) - **Retail Stores**: - **Columbus** (marked by a red dot in the center) - **Dayton** (marked by a red dot in the southwest) These locations will serve as the points in your network model to facilitate the efficient transportation and distribution of products within Ohio.
### Transportation Cost Optimization Problem

In this problem, we are focusing on optimizing the transportation costs associated with shipping products between different cities. The table below details the transportation costs for a single unit of product between various source nodes and destination nodes.

#### Cost Table

| Source       | Node 4 - Marion | Node 5 - Newark | Node 6 - Columbus | Node 7 - Dayton |
|--------------|------------------|------------------|-------------------|------------------|
| **Node 1 - Toledo**    | $119              | $103              | -                 | -                |
| **Node 2 - Cleveland** | $113              | $110              | -                 | -                |
| **Node 3 - Steubenville** | $120              | $107              | -                 | -                |
| **Node 4 - Marion**    | -                 | -                 | $106              | $132              |
| **Node 5 - Newark**    | -                 | -                 | $78               | $91               |

Note: A dash ("-") indicates that no direct shipping cost is available between those nodes.

### Problem Description
Your task involves optimizing the shipping schedule and costs between these nodes. There are three factories located in:

- Toledo
- Cleveland
- Steubenville

The production capacities for these factories are as follows:

- **Toledo:** 420 units per week
- **Cleveland:** 470 units per week
- **Steubenville:** 455 units per week

The demand at the retail stores in different locations is also provided:

- **Columbus:** 710 units per week
- **Dayton:** 620 units per week

Each distribution center can handle a maximum of 670 units per week.

### Problem-solving Task
Formulate a linear program to determine the optimal number of units your company should ship between these locations to minimize costs. Compute the associated cost and present your answers to two decimal places.

### Instructions for Formulation
1. **Decision Variables:**
   - Represent shipping from node \( i \) to node \( j \) using \( X_{ij} \).
   - Example: Shipping a unit from Node 1 (Toledo) to Node 2 (Cleveland) will be denoted as \( X_{12} \).

2. **Objective Function:**
   - Minimize the total transportation cost.
   
3. **Constraints:**
   - Production
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Transcribed Image Text:### Transportation Cost Optimization Problem In this problem, we are focusing on optimizing the transportation costs associated with shipping products between different cities. The table below details the transportation costs for a single unit of product between various source nodes and destination nodes. #### Cost Table | Source | Node 4 - Marion | Node 5 - Newark | Node 6 - Columbus | Node 7 - Dayton | |--------------|------------------|------------------|-------------------|------------------| | **Node 1 - Toledo** | $119 | $103 | - | - | | **Node 2 - Cleveland** | $113 | $110 | - | - | | **Node 3 - Steubenville** | $120 | $107 | - | - | | **Node 4 - Marion** | - | - | $106 | $132 | | **Node 5 - Newark** | - | - | $78 | $91 | Note: A dash ("-") indicates that no direct shipping cost is available between those nodes. ### Problem Description Your task involves optimizing the shipping schedule and costs between these nodes. There are three factories located in: - Toledo - Cleveland - Steubenville The production capacities for these factories are as follows: - **Toledo:** 420 units per week - **Cleveland:** 470 units per week - **Steubenville:** 455 units per week The demand at the retail stores in different locations is also provided: - **Columbus:** 710 units per week - **Dayton:** 620 units per week Each distribution center can handle a maximum of 670 units per week. ### Problem-solving Task Formulate a linear program to determine the optimal number of units your company should ship between these locations to minimize costs. Compute the associated cost and present your answers to two decimal places. ### Instructions for Formulation 1. **Decision Variables:** - Represent shipping from node \( i \) to node \( j \) using \( X_{ij} \). - Example: Shipping a unit from Node 1 (Toledo) to Node 2 (Cleveland) will be denoted as \( X_{12} \). 2. **Objective Function:** - Minimize the total transportation cost. 3. **Constraints:** - Production
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