The table displays the production requirements of four products (A, B, C, and D) across four machines. The values represent the time (in hours) required by each product on each machine. Additionally, a "Maximum Demand" column lists the maximum units required for each product. ### Table Details: - Columns: - Machines 1 to 4: Represents four different machines needed for production. - Maximum Demand: Indicates the highest quantity demanded for each product. - Rows: - Product A: - Machine 1: 0.083 hours - Machine 2: 0.167 hours - Machine 3: 0.1 hours - Machine 4: 0.05 hours - Maximum Demand: 400 units - Product B: - Machine 1: 0.05 hours - Machine 2: 0.1 hours - Machine 3: 0.067 hours - Machine 4: 0.133 hours - Maximum Demand: 100 units - Product C: - Machine 1: 0.067 hours - Machine 2: 0.083 hours - Machine 3: 0.05 hours - Machine 4: 0.05 hours - Maximum Demand: 150 units - Product D: - Machine 1: 0.067 hours - Machine 2: 0.033 hours - Machine 3: 0.017 hours - Machine 4: 0.033 hours - Maximum Demand: 500 units This data is useful for understanding how to allocate machine time effectively to meet product demands.

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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Related questions

Question

A plant has sufficient capacity to manufacture any combination of four different products (A, B, C, D).
For each product, time is required to be invested in four different machines, which is expressed in hours per kilogram of product, as shown in the following table as shown in the following table:

(attached image)

Each machine has an availability of 60 hours per week. Products A, B, C and D can be sold at $9, $7, $6 and $5
per kilo, respectively. Variable labor costs are $2 per hour for machines 1 and 2, and $3 per hour for machines 3 and 4. The material costs for each kilogram of product A are $4. The material costs for each kilogram of products B, C, D and D are $4 each kilogram of products B, C and D are $1.

What needs to be done:
Formulate a profit-maximizing PL model given the maximum demand per product shown in the table (there are 16 variables).

Note:

do the exercise by hand, or do it digitally, but not in excel.

The table displays the production requirements of four products (A, B, C, and D) across four machines. The values represent the time (in hours) required by each product on each machine. Additionally, a "Maximum Demand" column lists the maximum units required for each product.

### Table Details:

- **Columns:**
- **Machines 1 to 4:** Represents four different machines needed for production.
- **Maximum Demand:** Indicates the highest quantity demanded for each product.

- **Rows:**
- **Product A:**
- Machine 1: 0.083 hours
- Machine 2: 0.167 hours
- Machine 3: 0.1 hours
- Machine 4: 0.05 hours
- Maximum Demand: 400 units
- **Product B:**
- Machine 1: 0.05 hours
- Machine 2: 0.1 hours
- Machine 3: 0.067 hours
- Machine 4: 0.133 hours
- Maximum Demand: 100 units
- **Product C:**
- Machine 1: 0.067 hours
- Machine 2: 0.083 hours
- Machine 3: 0.05 hours
- Machine 4: 0.05 hours
- Maximum Demand: 150 units
- **Product D:**
- Machine 1: 0.067 hours
- Machine 2: 0.033 hours
- Machine 3: 0.017 hours
- Machine 4: 0.033 hours
- Maximum Demand: 500 units

This data is useful for understanding how to allocate machine time effectively to meet product demands.

Expert Solution

Step 1

Decision variable:

Suppose-
$x i j = t h e u n i t s o f p r o d u c t i t o b e m a n u f a c t u r e d b y t h e p l a n t u \sin g t h e m a c h i n e j w h e r e i = (1, 2, 3, 4) a n d j = (1, 2, 3, 4)$