Question

Show decision variables, objective function and constraints. Use excel solver to solve the problem. The Ferguson Paper Company produces rolls of paper for use in adding machines, desk calculators, and cash registers. The rolls, which are 200 feet long, are produced in widths of 1.5, 2.5, and 3.5 inches. The production process provides 200-foot rolls in 10-inch widths only. The firm must therefore cut the rolls to the desired final product sizes. The current requirements (e.g. pre-orders) for the three products (1.5 inches, 2.5 inches, 3.5 inches) are 1000, 2000, 4000 units, respectively.

a. How many knife settings (cutting alternatives) you will have? [should not that be 10 in total?]

b. If the company wants to minimize the number of 10-inch rolls that must be manufactured, how many 10-inch rolls will be needed and how to cut those rolls?

c. What is the total waste (inches)? If the company wants to minimize the waste generated, re-answer the previous question.

d. Discuss about the two objectives used in the above two questions. When would be each objective more desirable?

Answer 1

Given: Total no.of units for 1.5 inch products = 1000

No.of units for 2.5 inch products = 2000

No.of units for 3.5 inch products = 4000

Total size of roll = 200 feet

Total size of roll (in inches) = 200*12 =2400

1 foot = 12 inches

Available size of roll = 10 inch width

Ans: The major decision variable is maximize the production an minimize the waste

Let xi (i= 1,2,3) be the number of units in all the three different products

Objective function: x1+x2+x3 = 6000

Cost of 10 inch roll = y

No.of 10 inch rolls =z

Cost of entire roll is cost of each roll*no.of 10 inch rolls = y*z

Objective function is to minimize y*z value

Constraints: x1<= 1000

X2<= 2000

X3 <= 4000

Let sizes be represented as a,b,c

Let wastage be w

A+b+c+w = 10

1. How many knife settings you will have (cutting alternatives)

Ans: Number of knife settings we should have are 3 sizes given like 1.5, 2,5 , 3.5 inches =3

The other settings are combinations of all the three parameters i.e 1.5+2.5, 1.5 +1.5, 1.5+3.5, 2.5+2.5, 2.5+3.5, 3.5+3.5 = 6

The other setting is sum of all the three inches i.e is 1.5+2.5+3.5 = 1

Total number of knife settings (cutting alternatives) required are 10.

2. If the company wants to minimize the number of 10-inch rolls that must be manufactured, how many 10-inch rolls will be needed and how to cut those rolls?

Ans: For each 200 feet roll, no.of cuttings available is240 with each cutting measuring 10 inch

Total number of units = 6000

For 6000 units, number of 10 inch rolls is 6000/240 = 5

Total 25 rolls are required in 10 inch width and each 10 inch width roll can be cut in 10 ways

3. What is the total waste (inches)? If the company wants to minimize the waste generated, re-answer the previous question.

Ans: Total waste is 0 if 10 inch is cut in 2 *2.5 inch, 1 3.5 inch and 1 1.5 inch

Waste is 0.5 inch if it is cut into 2 *3.5 and 1*2.5 inch

Wastage is 0 if it is cut into 1.5*4 inch and 1 2.5 inch

Wastage is 0.5 inch if it is cut into 4*1.5 inch and 1 3.5 inch

So, in all 4 probable ways , the total wastage is 1 inch per 30 inches i.e is 30 inches

However, in this way total production units for 30 inches is 4 3.5 inches, 8 1.5 inches, 4 2.5 inches are manufactured.

Minimize the waste generation is done by optimizing the solution

4. Discuss about the two objectives used in the above two questions. When would be each objective more desirable?

Ans: The major 2 objectives discussed are: Maximize the production and minimize the waste

Waste minimization is desirable if lesser inch products have more demand than higher inch products.

Maximize the production is required if number of units are different for each products.