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=

Student Suite Cd-rom For Winston's Operations Research: Applications And Algorithms

4th Edition

ISBN: 9780534423551

Author: Wayne L. Winston

Publisher: Cengage Learning

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

Trending nowThis is a popular solution!

See solution

Check out sample textbook solution

Chapter 5.3, Problem 9P

Chapter 5.3, Problem 11P

Students have asked these similar questions

What should the next three steps be in my machine based home security system after deployment and after the following current steps: Enhancing Security & Privacy Measures User Interface (UI) and Experience (UX) Improvement Machine Learning Model Refinement & Continuous Improvement

I am creating a machine learning home based security system, have completed initial deployment and in the following phases of the project: Expanding device compatibility and integration, preparing for cloud integration, and implementing system reduncancy and disaster recovery. What should the next three phases be?

Hands-On Assignments Part II Assignment 1-5: Querying the DoGood Donor Database Review the DoGood Donor data by writing and running SQL statements to perform the following tasks: 1. List each donor who has made a pledge and indicated a single lump sum payment. Include first name, last name, pledge date, and pledge amount. 2. List each donor who has made a pledge and indicated monthly payments over one year. Include first name, last name, pledge date, and pledge amount. Also, display the monthly payment amount. (Equal monthly payments are made for all pledges paid in monthly payments.) 3. Display an unduplicated list of projects (ID and name) that have pledges committed. Don't display all projects defined; list only those that have pledges assigned. 4. Display the number of pledges made by each donor. Include the donor ID, first name, last name, and number of pledges. 5. Display all pledges made before March 8, 2012. Include all column data from the DD PLEDGE table.

Knowledge Booster

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+

Explanation of Solution

Chapter 5 Solutions