Skip to main content

Engineering Computer Science Operations Research : Applications and Algorithms

Calculating the extra payment: Let x 1 be the number of pairs of shoes that were produced during the month “t” with non-overtime labor, o t be the number of pairs of shoes that were produced during month “t” with the overtime labor, i t be the inventory of shoes at the end of month “t”, h t is the number of workers that were hired at the beginning of month “t”, f t be the number of workers who were fired at the beginning of month “t” and w t be the number of workers who were available for month “t”. For the objective function, the following cost must be considered as worker’s salaries, hiring costs, firing costs, holding costs, overtime costs and raw-material costs. Hence, the derived objective function is: min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 25 × 4 o t + 5 ( x t + o t ) min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. The constraints are described where the first constraint for inventory is, i 1 = 50 + x 1 + o 1 − 300 x 1 + o 1 − i 1 = 250 For the second month, i 2 = i 1 + x 2 + o 2 − 500 i 1 + x 2 + o 2 − i 2 = 500 In the same way, for next two months, x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 The second constraint for relation available workers to hiring and firing the workers, for the first month, w 1 = 3 + h 1 − f 1 w 1 + f − h 1 = 3 For the second month, w 2 = w 1 + h 2 − f 2 w 2 + f 2 − w 1 − h 2 = 0 In the same way, for next two months, w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 The third constraint for each month, the amount of shoes made with non-overtime labors is limited by the number of workers, here for the first month, 4 x 1 ≤ 160 w 1 4 x 1 − 160 w 1 ≤ 0 In the same way, for next three months, 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 The fourth constraint for each month, the amount of shoes made with overtime labor is limited by the number of workers, for the first month, 4 o 1 ≤ 20 w 1 4 o 1 − 20 w 1 ≤ 0 In the same way, for next three months, 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 Hence, the obtained Linear Programming min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. Subjected to, x 1 + o 1 − i 1 = 250 i 1 + x 2 + o 2 − i 2 = 500 x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 w 1 + f − h 1 = 3 w 2 + f 2 − w 1 − h 2 = 0 w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 4 x 1 − 160 w 1 ≤ 0 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 4 o 1 − 20 w 1 ≤ 0 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 The following is the LINDO output for this Linear Programming: min z=

Calculating the extra payment: Let x 1 be the number of pairs of shoes that were produced during the month “t” with non-overtime labor, o t be the number of pairs of shoes that were produced during month “t” with the overtime labor, i t be the inventory of shoes at the end of month “t”, h t is the number of workers that were hired at the beginning of month “t”, f t be the number of workers who were fired at the beginning of month “t” and w t be the number of workers who were available for month “t”. For the objective function, the following cost must be considered as worker’s salaries, hiring costs, firing costs, holding costs, overtime costs and raw-material costs. Hence, the derived objective function is: min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 25 × 4 o t + 5 ( x t + o t ) min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. The constraints are described where the first constraint for inventory is, i 1 = 50 + x 1 + o 1 − 300 x 1 + o 1 − i 1 = 250 For the second month, i 2 = i 1 + x 2 + o 2 − 500 i 1 + x 2 + o 2 − i 2 = 500 In the same way, for next two months, x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 The second constraint for relation available workers to hiring and firing the workers, for the first month, w 1 = 3 + h 1 − f 1 w 1 + f − h 1 = 3 For the second month, w 2 = w 1 + h 2 − f 2 w 2 + f 2 − w 1 − h 2 = 0 In the same way, for next two months, w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 The third constraint for each month, the amount of shoes made with non-overtime labors is limited by the number of workers, here for the first month, 4 x 1 ≤ 160 w 1 4 x 1 − 160 w 1 ≤ 0 In the same way, for next three months, 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 The fourth constraint for each month, the amount of shoes made with overtime labor is limited by the number of workers, for the first month, 4 o 1 ≤ 20 w 1 4 o 1 − 20 w 1 ≤ 0 In the same way, for next three months, 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 Hence, the obtained Linear Programming min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. Subjected to, x 1 + o 1 − i 1 = 250 i 1 + x 2 + o 2 − i 2 = 500 x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 w 1 + f − h 1 = 3 w 2 + f 2 − w 1 − h 2 = 0 w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 4 x 1 − 160 w 1 ≤ 0 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 4 o 1 − 20 w 1 ≤ 0 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 The following is the LINDO output for this Linear Programming: min z=

Solution Summary: The author explains how the objective function calculates the extra payment based on the following constraints: the first constraint for inventory is i_1=50+

Operations Research : Applications and Algorithms

Operations Research : Applications and Algorithms

4th Edition

ISBN: 9780534380588

Author: Wayne L. Winston

Publisher: Brooks Cole

See similar textbooks

Expert Solution & Answer

Chapter 5.3, Problem 10P

Explanation of Solution

Calculating the extra payment:

Let x₁ be the number of pairs of shoes that were produced during the month “t” with non-overtime labor, o_t be the number of pairs of shoes that were produced during month “t” with the overtime labor, i_t be the inventory of shoes at the end of month “t”, h_t is the number of workers that were hired at the beginning of month “t”, f_t be the number of workers who were fired at the beginning of month “t” and w_t be the number of workers who were available for month “t”.

For the objective function, the following cost must be considered as worker’s salaries, hiring costs, firing costs, holding costs, overtime costs and raw-material costs.

Hence, the derived objective function is:

min z= 1500wt+1600ht+2000ft+30it+25×4ot+5(xt+ot)

min z= 1500wt+1600ht+2000ft+30it+105ot+5xt

for t=1,2,3,4.

The constraints are described where the first constraint for inventory is,

i1=50+x1+o1−300

x1+o1−i1=250

For the second month,

i2=i1+x2+o2−500

i1+x2+o2−i2=500

In the same way, for next two months,

x3+o3+i2−i3=100

x4+o4+i3−i4=100

The second constraint for relation available workers to hiring and firing the workers, for the first month,

w1=3+h1−f1

w1+f−h1=3

For the second month,

w2=w1+h2−f2

w2+f2−w1−h2=0

In the same way, for next two months,

w3+f3−w2−h3=0

w4+f4−w3−h4=0

The third constraint for each month, the amount of shoes made with non-overtime labors is limited by the number of workers, here for the first month,

4x1≤160w1

4x1−160w1≤0

In the same way, for next three months,

4x2−160w2≤0

4x3−160w3≤0

4x4−160w4≤0

The fourth constraint for each month, the amount of shoes made with overtime labor is limited by the number of workers, for the first month,

4o1≤20w1

4o1−20w1≤0

In the same way, for next three months,

4o2−20w2≤0

4o3−20w3≤0

4o4−20w4≤0

Hence, the obtained Linear Programming

min z= 1500wt+1600ht+2000ft+30it+105ot+5xt

for t=1,2,3,4.

Subjected to,

x1+o1−i1=250

i1+x2+o2−i2=500

x3+o3+i2−i3=100

x4+o4+i3−i4=100

w1+f−h1=3

w2+f2−w1−h2=0

w3+f3−w2−h3=0

w4+f4−w3−h4=0

4x1−160w1≤0

4x2−160w2≤0

4x3−160w3≤0

4x4−160w4≤0

4o1−20w1≤0

4o2−20w2≤0

4o3−20w3≤0

4o4−20w4≤0

The following is the LINDO output for this Linear Programming:

min z=

Expert Solution & Answer

bartleby

Trending nowThis is a popular solution!

Check out sample textbook solution

Blurred answer

Chapter 5.3, Problem 9P

Chapter 5.3, Problem 11P

Students have asked these similar questions

QUESTION 15 Al Watan supermarket has hire-purchases scheme for locals and expats. The customers will get the credit facility if they satisfy any of the following conditions: 1. The customer must hold the present job for more than 5 years and reside in the same place at least for 3 years. In this case, the customer will get credit upto rupees 300 omr. 2. The monthly salary of the customer must exceed 1000 omr and must hold the present job for more than 5 years. In this case credit will be given upto rupees 500 omr. 3. The monthly salary must exceed 1000 omr and reside at the same place at least for 3 years. In this case credit will be given upto 500 omr. 4. In the case, the customer's monthly salary exceeds 1000 omr, holds the present job for more than 5 years and also reside in the same place at least for the 3 years, the credit facility will be upto 1000 omr 5. The credit facility is rejected for all other customers. Prepare a Decision Table for this hire- purchase scheme. Attach File…

QUESTION 15 Al Watan supermarket has hire-purchases scheme for locals and expats. The customers will get the credit facility if they satisfy any of the following conditions: 1. The customer must hold the present job for more than 5 years and reside in the same place at least for 3 years. In this case, the customer will get credit upto rupees 300 omr. 2. The monthly salary of the customer must exceed 1000 omr and must hold the present job for more than 5 years. In this case credit will be given upto rupees 500 omr. 3. The monthly salary must exceed 1000 omr and reside at the same place at least for 3 years. In this case credit will be given upto 500 omr. 4. In the case, the customer's monthly salary exceeds 1000 omr, holds the present job for more than 5 years and also reside in the same place at least for the 3 years, the credit facility will be upto 1000 omr 5. The credit facility is rejected for all other customers. Prepare a Decision Table for this hire- purchase scheme.

1. for October 19–20, starting with a reception on the evening of October 18. The conference consists of all-day meetings that the entire sales force, numbering 120 sales representatives and their 16 managers, must attend. Each sales representative requires his or her own room, and the company needs two common meeting rooms, one large enough to hold the entire sales force plus a few visitors (200) and the other able to hold half the force. Management has set a budget of $195,000 for the representatives' room rentals. The company would like to hold the conference in either Miami or Marco Island, Florida, at a Hilton- or Marriott-owned hotel. The Foremost Composite Materials Company is planning a two-day sales conference Use the Hilton and Marriott websites to select a hotel in whichever of these cities would enable the company to hold its sales conference within its budget and meet its sales conference requirements. Then locate flights arriving the afternoon prior to the conference.…

Knowledge Booster

Background pattern image

Similar questions

SEE MORE QUESTIONS

Recommended textbooks for you

Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole

Text book image

Operations Research : Applications and Algorithms

Computer Science

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Brooks Cole

SEE MORE TEXTBOOKS