Baó cáo nghiên cứu và áp dụng kiến thức của QUY HOẠCH TUYẾN TÍNH dựa trên lý thuyết được học của môn Phương pháp định lượng và phần mềm QM for Windows. selecting an effective media mix. Sometimes the technique is used for allocating a fixed or limited budget across various media including radio or television commercials, newspaper ads, direct mailings, magazine ads, and so on. In other applications, the objective of the technique is the maximization of reach of audience. Therefore, we decided to create a case study about media selection to acknowledge about LP in marketing application. Obito Mandara is a manager of a hoodie store in Suwon city. Recently, he and marketing team have been planning a campaign to attract customers as well as gain competitive advantages. There are four selected ways to advertise: television ads, facebook ads, google ads, poster ads. The cost of ads, the reach of audience following each type of ad and the maximum number of each ads are shown in the below table: Type of ad Cost per ad Audience reachedad Maximum number TV 1000 35,000 10 Facebook 650 25,000 10 Google 720 30,000 10 Poster 120 9,000 10 Furthermore, they decided that TV or Facebook or come combination of those two should be at least 8ads. The amount spent on Google and Poster must not exceed that on TV ads. Addition, meanwhile looking for funding, 17,000 has been monthly budget for ads. How many ads of each type to maximize the total number of people reached?
Trang 1HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY SCHOOL OF INDUSTRIAL MANAGEMENT
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GROUP ASSIGNMENT COURSE: QUANTITATIVE METHODS
CLASS: CC17QKD INSTRUCTOR: PHAM QUOC TRUNG
Group members :
Ho Chi Minh city - 2019
Trang 2OBJECTIVES OF THE STUDY
The objective of the study is to focus on analyzing and applying the knowledge of linear programming based on the theoretical principles and the practical soft-wares In this study, we decide to choose the case about Obito Mandara (a manager)- to help him select
an effective media channel; and Chase Manhattan Bank- to arrange labor forces to
minimize the labor cost
Trang 3CASE STUDY 1
Linear programming models have been used in the advertising field as a decision aid in selecting an effective media mix Sometimes the technique is used for allocating a fixed
or limited budget across various media including radio or television commercials,
newspaper ads, direct mailings, magazine ads, and so on In other applications, the
objective of the technique is the maximization of reach of audience Therefore, we
decided to create a case study about media selection to acknowledge about LP in
marketing application
Obito Mandara is a manager of a hoodie store in Suwon city Recently, he and marketing team have been planning a campaign to attract customers as well as gain competitive advantages There are four selected ways to advertise: television ads, facebook ads, google ads, poster ads The cost of ads, the reach of audience following each type of ad and the maximum number of each ads are shown in the below table:
Type of ad Cost per ad Audience
reached/ad
Maximum number
Furthermore, they decided that TV or Facebook or come combination of those two should
be at least 8ads The amount spent on Google and Poster must not exceed that on TV ads Addition, meanwhile looking for funding, $17,000 has been monthly budget for ads How many ads of each type to maximize the total number of people reached?
Solution:
The very first step is to formulate the problem and define the variables Let:
x1: TV ads
x2: Facebook ads
x3: Google ads
x4: Poster ads
Objective function:
Max: Z= 35,000x1 + 25,000x2 + 30,000x3 + 9,000x4
Trang 41000x1 + 650x2 + 720x3 + 120x4 <= 17,000 (monthly advertising budget)
x1 <=10 (maximum TV ad per month)
x2 <=10 (maximum Facebook ads per month)
x3 <=10 (maximum Google ads per month)
x4 <=10 (maximum poster ads per month)
720x3 + 120x4 <= 1000x1 (maximum Google and Poster ads)
X1 + x2 >= 8 (minimum TV & Facebook ads)
The solution can be found using Excel’s Solver (QM) The formula must be written in the cell for the objective function value, and the cells where this formula should be copied The results are shown in the spreadsheet
Trang 5The number of ads of each type to maximize the total number of people reached is
shown:
x1= 5.25 TV ads
x2=10 Facebook ads
x3= 5.63 Google ads
x4=10 Poster ads
This produces the total number of people reached is 692500 people Because x1 and x3 is fractional, the hoodie shop would properly round them to 5ads and 6ads in turn
Trang 6CASE STUDY 2 (CASE STUDY 8.1)
Chase Manhattan Bank
At Chase Manhattan Bank in New York, the number of domestic money transfer requests received from customers, if plotted against time of day, would appear to have the shape
of an inverted U curve with the peak around 1 P.M For efficient use of resources, the personnel available should, therefore, vary correspondingly
Figure below shows a typical workload curve and corresponding personnel requirements
at different hours of the day:
A variable capacity can be achieved effectively by employing part-time personnel
Because part-timers are not entitled to all the fringe benefits, they are often more
economical than fulltime employees
Other considerations, however, may limit the extent to which part-time people can be hired in a given department The problem is to find an optimum workforce schedule that would meet personnel requirements at any given time and also be economical
Some of the factors affecting personnel assignment are listed here:
1 By corporate policy, part-time personnel hours are limited to a maximum of 40% of the day’s total requirement
2 Full-time employees work for 8 hours (1 hour for lunch included) per day Thus, a full-timer’s productive time is 35 hours per week
Trang 73 Part-timers work for at least 4 hours per day but less than 8 hours and are not allowed a lunch break
4 Fifty percent of the full-timers go to lunch between 11 A.M and noon, and the
remaining 50% go between noon and 1 P.M
5 The shift starts at 9 A.M and ends at 7 P.M (i.e., overtime is limited to 2 hours) Any work left over at 7 P.M is considered holdover for the next day
6 A full-time employee is not allowed to work more than 5 hours overtime per week He
or she is paid at the normal rate for overtime hours—not at one-and-a-half times the normal rate applicable to hours in excess of 40 per week Fringe benefits are not applied
to overtime hours
Solution:
The very first step is to formulate the problem and define the variables Now need to determine the number of employees who start their work at different timings Let:
F= the number of full time employees without overtime
F1=the number of full time employees with 1 hour overtime from 5pm to 6pm F2= the number of full time employees with 2 hours overtime from 5pm to 7pm P1, P2, P3, P4=Part time employees who work 4, 5, 6, 7 hours from 9am to 1pm, 2pm, 3pm, 4pm in turn
Trang 8P5, P6, P7, P8= Part time employees who work 4, 5, 6, 7 hours from 10am to 2pm, 3pm, 4pm, 5pm in turn
P9, P10, P11, P12= Part time employees who work 4, 5, 6, 7 hours from 11am to 3pm, 4pm, 5pm, 6pm in turn
P13, P14, P15, P16= Part time employees who work 4, 5, 6, 7 hours from 12am to 4pm, 5pm, 6pm, 7pm in turn
P17, P18, P19= Part time employees who work 4, 5, 6 hours from 1pm to 5pm, 6pm, 7pm in turn
P20, P21= Part time employees who work 4, 5 hours from 2pm to 6pm, 7pm in turn
P22= Part time employees who work 4 hours from 3pm to 7pm
Average cost per full time personnel = $10.11
Average cost per overtime personnel hour for full time= $8.08
Average cost per part time personnel = $7.82
Objective function:
Minimize = ($10.11)(7)F + $8.08F1+ ($8.08)(2)F2
+($7.82)(4)(P1+P5+P9+P13+P17+P20+P22)
+($7.82)(5)(P2+P6+P10+P14+P18+P21)
+($7.82)(6)(P3+P7+P11+P15+P19)
+($7.82)(7)(P4+P8+P12+P16)
Constraints:
Workforce requirements
9-10am: F + P1+P2+P3+P4 >= 14 (1)
10-11am: F+P1+P2+P3+P4+P5+P6+P7+P8 >=25 (2)
11-12am: 0.5F + P1+P2+P3+P4+P5+P6+P7+P8+P9+P10+P11+P12 >=26 (3)
12-1pm:0.5F+P1+P2+P3+P4+P5+P6+P7+P8+P9+P10+P11+P12+P13+P14+P15+P16
>=38 (4)
Trang 91-2pm:
F+P2+P3+P4+P5+P6+P7+P8+P9+P10+P11+P12+P13+P14+P15+P16+P17+P18+P19>=55 (5)
2-3pm:
F+P3+P4+P6+P7+P8+P9+P10+P11+P12+P13+P14+P15+P16+P17+P18+P19+P20+P21>=
60 (6)
3-4pm:
F+P4+P7+P8+P10+P11+P12+P13+P14+P15+P16+P17+P18+P19+P20+P21+P22>=51 (7)
4-5pm: F+P8+P11+P12+P13+P14+P15+P16+P17+P18+P19+P20+P21+P22>=29 (8)
5-6pm: (F1+F2) +P12+P15+P16+P17+P20+P21+P22>=14 (9)
6-7pm: (F2) +P16+P19+P21+P22>=9 (10)
For the factor 1: 1 By corporate policy, part-time personnel hours are limited to a
maximum of 40% of the day’s total requirement
Constraint 11: 4(P1+P5+P9+P13+P17+P20+P22) +5(P2+P6+P10+P14+P18+P21)+
6(P3+P7+P11+P15+P19)+ 7(P4+P8+P12+P16) <= 40%
(14+25+26+38+55+60+51+29+14+9)
4(P1+P5+P9+P13+P17+P20+P22) +5(P2+P6+P10+P14+P18+P21)+
6(P3+P7+P11+P15+P19)+ 7(P4+P8+P12+P16) <=128.4
Now, starting with the QM for Windows to calculate the problem, after filling the numbers
we have the tables below:
Trang 10Then, we have the solution:
Trang 111 What is the minimum-cost schedule for the bank?
Hence, based on the solution from QM for Windows, the bank will need:
29 full-timers without overtime (F)
9 full time employees with 2 hours overtime from 5pm to 7pm (F2)
9 Part time employees who work 4 hours from 11am to 3pm (P9)
3 Part time employees who work 5 hours from 11am to 4pm (P10)
12 Part time employees who work 4 hours from 12am to 4pm (P13)
3 Part time employees who work 4 hours from 1pm to 5pm (P17)
5 Part time employees who work 4 hours from 2pm to 6pm (P20)
The minimum cost for such arrangement is $ 3222
Note: From the factor 6, we have:
F1+3*F2 <= F
the number of full-time employees doing overtime should be less than or equal to number of full time employees F2 is multiplied by 3 because each employee can do only 5 hours OT
in a week so each employee can do only 2 days of 2 hours overtime and so we require 3 sets
of people with 2 hours overtime per week:
Set 1: Full time employees do 2 hour of overtime for 2 days (first day)
Set 2: Full time employees do 2 hour of overtime for 2 days (second day)
Set 3: Full time employees do overtime for 2 hours once a week
Total 18 full time employees do 2 hours of overtime for 2 days and 9 employees
do overtime for 2 hours once a week
2 What are the limitations of the model used to answer question 1?
Limitations:
The model does not allow different loading for different days in a week
Trang 12 The model does not allow flexi timings for full time employees If say full time employees are allowed to work from 11am -7pm then the overtime cost could be reduced
3 Costs might be reduced by relaxing the constraint that no more than 40% of the day’s requirement be met by part-timers Would changing the 40% to a higher value significantly reduce costs?
Yes, changing the 40% to a higher value would altogether lesson costs
Changing to 60%, we have constraint 11:
=>4(P1+P5+P9+P13+P17+P20+P22) +5(P2+P6+P10+P14+P18+P21)+
6(P3+P7+P11+P15+P19)+ 7(P4+P8+P12+P16) <= 60%*321
=> cost =$ 2802
Changing to 80%, we have constraint 11:
=>4(P1+P5+P9+P13+P17+P20+P22) +5(P2+P6+P10+P14+P18+P21)+
6(P3+P7+P11+P15+P19)+ 7(P4+P8+P12+P16) <= 80%*321
=> cost = $ 2621
Taken a toll diminishes as the rate increments