QUANTITATIVE METHODS

Quantitative Methods

Quantitative Methods

Question 2

A furniture manufacturer has recently noticed a reduction in the profit achieved in their Tables' product line and in order to address the product is conducting a review to understand which model(s) are more profitable and it should concentrate on. It currently makes four different tables using three different materials and labour. Prices, inputs and constraints are summarised in the following table:

Table

Square

Round

Rectangular

Elliptic

Max. Available

Price

£600.00

£700.00

£900.00

£1,050.00

Wood

11 units

12 units

52,000

Metal

7 units

8 units

36,000

Other Materials

6 units

4 units

9 units

8 units

48,500

Labour

1½ hours

2¾ hours

2 hours

3½ hours

20,000

Max. Demand

3,500

3,000

1,500

2,000

The costs for the inputs are £25 per hour for Labour and £15, £20 and £10 per unit for Wood, Metal and Other Materials, respectively. Formulate this problem as a linear program and use Excel's Solver to arrive at a solution. Write a short report describing your procedure, justify your formulation and give a recommendation to the firm on the best daily production mix.

Solution

In this question we will be answering how the furniture manufacturer can maximize its profits and which products he should focus more in order to reach its higher profit point. For this task we have used excel solver (Linear Programming technique). Following are the constraints that will be used in order to determine the order quantity resource placement to achieve high profit.

Max: 3500Square + 3000Round + 1500Rectangular + 2000Elliptic

11 Round + 12Elliptic <= 52000

7 Square + 8 Rectangular <= 36000

6 Square + 4 Round + 9 Rectangular + 8 Elliptic <= 48500

1.5 Square + 2.75 Round + 2 Rectangular + 3.5 Elliptic <= 20000

Following are the results that were achieved when we implemented these constraints:

Square

Round

Rectangular

Elliptic



RHS

Dual

Maximize

3500

3000

1500

2000

Wood

0

11

0

12

<=

52000

0

Metal

7

0

8

0

<=

36000

0

Other Materials

6

4

9

8

<=

48500

488.0952

Labor

1.5

2.75

2

3.5

<=

20000

380.9524

Solution->

5083.333

4500

0

0

31291670

The above table states that the company should focus on the square and round tables as there is more demand for this product. Also these tables are providing more profit to the company. If they will make 5083.333 round table and 4500 of round tables they will earn a profit of $ 31,291,670. They should focus on these products and use their resources for the production of these products. Below is the table in which all the formulations of the strategy is present.

Cj

Basic Variables

3500 Square

3000 Round

1500 Rectangular

2000 Elliptic

0 slack 1

0 slack 2

0 slack 3

0 slack 4

Quantity

Iteration 1

cj-zj

3,500

3,000

1,500

2,000

0

0

0

0

0

slack 1

0

11

0

12

1

0

0

0

52,000

0

slack 2

7

0

8

0

0

1

0

0

36,000

0

slack 3

6

4

9

8

0

0

1

0

48,500

0

slack 4

1.5

2.75

2

3.5

0

0

0

1

20,000

Iteration 2

cj-zj

0

3,000

-2,500

2,000

0

-500

0

0

0

slack 1

0

11

0

12

1

0

0

0

52,000

3500

Square

1

0

1.1429

0

0

0.1429

0

0

5,142.86

0

slack 3

0

4

2.1429

8

0

-0.8571

1

0

17,642.86

0

slack 4

0

2.75

0.2857

3.5

0

-0.2143

0

1

12,285.71

Iteration 3

cj-zj

0

0

-4,107.14

-4,000

0

142.8571

-750

0

0

slack 1

0

0

-5.8929

-10

1

2.3571

-2.75

0

3,482.14

3500

Square

1

0

1.1429

0

0

0.1429

0

0

5,142.86

3000

Round

0

1

0.5357

2

0

-0.2143

0.25

0

4,410.71

0

slack 4

0

0

-1.1875

-2

0

0.375

-0.6875

1

156.25

Iteration 4

cj-zj

0

0

-3,654.76

-3,238.10

0

0

-488.0952

-380.9524

0

slack 1

0

0

1.5714

2.5714

1

0

1.5714

-6.2857

2,500.00

3500

Square

1

0

1.5952

0.7619

0

0

0.2619

-0.381

5,083.33

3000

Round

0

1

-0.1429

0.8571

0

0

-0.1429

0.5714

4,500.00

0

slack 2

0

0

-3.1667

-5.3333

0

1

-1.8333

2.6667

416.6667

Question 3

The majority of banks, when making decisions on mortgage applications, will look at two indicators: salary and borrowing as a percentage of purchase prices. On the first indicator, banks are normally willing to lend 2.5 times one's salary or 3.25 times joint salary in a joint mortgage application, while currently most banks will lend up to 75% of the property price on their best rate with penalties for higher percentages. Alan and Karen are getting married and decided to buy a flat to move into once they do. You have been given the following data:

Alan's current salary is £52,000 p.a. and Karen's is £38,500 p.a. plus a bonus likely to be around £5,000 (based on previous 3 years experience);

Both have jobs where ...