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# QMB Assessment: Case Study 2

Introduction

The report is determined to give non quantitative analysis of the relationship between variables of interest which is cost of advertisement, price of listing per stay and the best location for the listing. In addition to that, the different types of listing considered are luxury type, designer type, and classic type. Also, the report aimed at maximizing the revenue for the stay.

A linear programming model has been developed which has been used to give a sensitivity report on the optimization of revenue using set constraints such as cleaning budget and number of listings per stay. From my sensitivity report, the maximum revenue is \$40,000 for the total number of listings on the website given the optimum number of listings for classic is zero, for designer is zero anf for luxuy listing is 500.

The CLO’s decision to sell at least twice of luxury stays should not be adhered to because the optimal number of designer stays for an optimal revenue should be zero and also the shadow price is zero, hence it will have no any effect on the main function. The original solution should remain since it gives the optimal solution and the other parts rely on the shadow points which do not give any positive difference.

The statistic on the goodness of the model is 0.8413. This is a higher value which shows that the model is a good fit for the data. 84.13% of the data dispersion is being explained by the model and therefore the closer the model will be able to fit the data and predict the average cost of the advertisement. The variable type of price per listing is a dependent variable as the variable of interest since price per listing depends on cost of advertisement which causes a change in the prices of listing. The listing type is discrete since the distribution takes values which are not continuous.

The shape of the distribution is uniformity in all listing classes. This is because the observation is even distributed at the midpoint value as shown in the box and whisker plot hence there is no change of distribution position and the data is normally distributed in each type.For determining the relationship between location and listing type we use a p value in statistics in order to determine if there is any relationship. The value of the test statistic is 35 and the p value is 0.243. Since the p-value is greater than the significant level, (α = 0.05), we do not reject the H0 and therefore there is no association between location and luxury listing.

Report Body

The main aim of the analysis is to provide a descriptive statistical analysis of the listing price. The variable type of price per listing is a dependent variable as the variable of interest since price per listing depends on cost of advertisement which causes a change in the prices of listing. The listing type is discrete since the distribution takes values which are not continuous. The price limits for the price data to be inside the observation:

• The lower bound for classic price is\$73 and the upper bound is \$200.
• The lower bound for designer is price is \$100 and the upper bound is \$200.
• The lower bound for luxury price is \$132 and the upper bound for luxury is \$316.

The shape of the distribution is uniformity in all listing classes. This is because the observation is even distributed at the midpoint value as shown in the box and whisker plot hence there is no change of distribution position and the data is normally distributed in each type.

For classic type, the midpoint value is 138. The upper whisker length of the data distribution is 200 and the lower whisker length is 73.

For designer type, the midpoint value is 150. The upper whisker length of the data distribution is 200 and the lower whisker length is 100.

For luxury type, the midpoint value is 220. The upper whisker length of the data distribution is 315 and the lower whisker length is 125.

The midpoint valueis located at the center of the box. The impact is that it leads to a distribution that does not change position and hence the data is normally distributed.

Revenue Optimization

The main aim of the analysis is to optimize the revenue determining number of stays per listing price listings given that the total budget of cleaning is \$30,000, and there are at most 84 designer listings per stay and at most 50 classic per stay.

The maximum revenue is \$40,000 for the total number of listings on the website given the optimum number of listings for classic is zero, for designer is zero anf for luxuy listing is 500. If the revenue made from luxury listing is increased by \$20 the solution will still be optimal since \$20 is within the allowable increase of \$100 and the shadow price is \$ 80. New revenue 80*20 + 40,000= \$ 41,600.

The CLO’s decision to sell at least twice of luxury stays should not be adhered to because the optimal number of designer stays for an optimal revenue should be zero and also the shadow price is zero, hence it will have no any effect on the main function. The original solution should remain since it gives the optimal solution and the other parts rely on the shadow points which do not give any positive difference.

The aim of the analysis is to come up with a scatter plot that will help to determine the relationship between cost of advertising and price of listing and the data distribution. In addition to that, it guides on observing the trend in the data which is upward moving. An increase in the price of listing leads to an increase in the cost of advertisement.

From the above scatter plot, there is a positive gradient for the data whereby the trend is upward moving. The gradient of the statistical relationship is 0.3049 which shows that the gradient moves upwards as we move from left to right. For each dollar in the price of a listing leads to a 0.3049 average increase per unit in the cost of advertising.

The statistic on the goodness of my model is 0.8413. This is a higher value which shows that the model is a good fit for the data. 84.13% of the data dispersion is being explained by the model and therefore the closer the model will be able to fit the data and predict the average cost of the advertisement.

Fitting equation is y=0.3049*x-6.4173, Average cost of advertising= 0.3049*187.16-6.4173= 50.647784

Where y is the average cost of advertising, x is the price per listing.

The average cost of advertising is 50.647784. This shows an accuracy in prediction of the model since the average cost is positive and this is well shown in the scatter plot diagram. The listing of \$ 300 will not be reliable with the model because the average listing price for the model is 187.16. Hence the prediction may not be reliable.

Conclusions and Recommendation

From the above report, we can conclude that there is a strong linear relationship between cost of advertisement and price of listing per stay and the model fitted is a good fit for the data. The CLO should use the model in order to have a clear prediction of the future expected advertisement cost. Luxury listing seemed to be the only variable which highly determines the profitability of the project, hence it should be highly considered as shown in the sensitive report. Location is not statistically dependent on listings and hence the relationship does not exist.

The CLO should, in future include more data for efficiency of the analysis and also use different variables such as factors that may affect cost of advertisement such as economic climate of the country and use other mathematical models to forecast the unknown price.

The variable type of price per listing is a dependent variable. Since price per listing depends on cost of advertisement which causes a change in the prices of listing.

The listing type is discrete since the distribution takes values which are not continuous.

I would use average as a measure of central tendency since the sample size is large and does not include observations that are distant from other observations and the data is also skewed. In the measure of dispersion, l would use measure of data dispersion to measure the spread since average is the appropriate measure of the central tendency.

Descriptive statistical data analysis

• There are no observations that are distant from other observations present since there is no value that is outside the given ranges from the boxplots.
• The lower bound for classic is 73 and the upper bound is 200.
• The lower bound for designer is 100 and the upper bound is 200.
• The lower bound for luxury is 132 and the upper bound for luxury is 316.

The shape of the distribution is uniformity in all listing classes. This is because the observation is even distributed at the median as shown hence there is no skewness and the data is normally distributed in each type.
For classic type, the median is 138. The upper whisker length of the data distribution is 200 and the lower whisker length is 73.
For designer type, the median is 150. The upper whisker length of the data distribution is 200 and the lower whisker length is 100.
For luxury type, the median is 220. The upper whisker length of the data distribution is 315 and the lower whisker length is 125.

Medians are located at the center of the box and whisker. The impact is that it leads to a distribution that is not changing position and hence the data is normally distributed.

Box and whisker plot showing price per listing

Average and measure of data dispersion would be the single most appropriate measure of central tendency and dispersion because the sample size is large and the data does not contain observations that are distant from other observations.
Appendix 2–Revenue Optimization

Linear Programming Model Template
Decision Variables

A decision variables are variables in a mathematical model that are not known. The decision variables are:
• Optimal number of stays for Classic listing (C)
• Optimal number of stays for Designer listing (D)
• Optimal number of stays for Luxury listing (L)

Objective and Objective Function
Objective is to maximize revenue by determining the maximum number of stays between Luxury, Designer and Classic listings.
Objective function: 80L + 60D + 40C = R
Where L, is luxury listing. D, is designer listing and C, is classic listing, R is the maximum revenue.
Constraints
• Cleaning fees:
50L+30D+30C ≤ 30,000
• Number of listings:
D ≤84
C≤50
D≥2L
Required number of stays per month:
L+D+C ≤ 500

Non negativity Restrictions:
D≥0, L≥0, C≥0

Sensitivity report

• The range of optimality for a designer listing is (0,0) whereby the revenue will be zero without any change of the optimal number of designer stays per month.
• The range of optimality for a classic listing is (0,0) and hence there will be no revenue with no any change of the classic stays in each and every month.
• Range optimality for luxury listing is (0,500), hence the revenue for every stay will be between \$ and \$ 500without any change to the optimal number luxury stays per month.

The optimal solution is represented by the final value, whereby designer is 0, classic is 0 and luxury is 500. The objective coefficients are 40 for classic variable, 60 for designer and 80 for luxury variable. Allowable increase and decrease values specifies by how much the objective coefficient can change before the optimal solution will change. Allowable increase for classic listing is 40, and the upper limit will be 80. Allowable increase for designer listing is 20 and hence the upper limit will be 80. Allowable increase for luxury listing is infinity and the upper limit is infinity. Allowable decrease for classic listing is infinity and the lower limit is negative infinity, allowable decrease for designer listing is infinity and the lower limit is negative infinity. The allowable decrease for a luxury listing is 20 and the lower limit is 60.
Shadow price is the amount of change in the optimal objective function value per unit increase in the right hand side of the constraints. The shadow price for number of stays per month is 80 which will be the optimal objective value of change for the stays per month. The shadow price of luxury, designer and classic listings and cleaning budget constraints is zero and hence changes in these constraints will have no effect on the objective value. The shadow price value of 80 applies when the upper limit is from 0 to 600. For every extra stay per month, the optimal revenue will increase by \$ 80.
Allowable increase for the cleaning budget is infinity and the shadow price is zero. So the increase in the cost of cleaning budget will have no any effect on the revenue, which will remain to be 40,000.
Change in revenue= 650*0= 0, new revenue= 0 + 40,000=40,000.
Attach the new Answer Report ONLY, for the scenario in which the cleaning budget is increased by \$650 and state the new maximum revenue.
Luxury listing value= 51940/80= 649.25, cleaning fees= 649.25*50= \$32462.5
Amount of additional cleaning budget= 32462.5-25000=\$7462.5

Average price per listing (\$).

Correlation

The value of statistics relationship is 0.9172. This shows a strong positive linear relationship between cost of advertising and the price of listing and hence a linear model is appropriate for the data. From the scatter plot, the data shows an upward pattern from left to right. This indicates a positive relationship between cost of advertising and price listing.
The dependent variable is the cost of advertising (\$) and the independent variable is the price of listing (\$)
The value where y crosses y is -6.4173. The coefficient helps the relationship values and predictions not to be biased. Hence, it’s meaningful. The gradient of the statistical relationship line is 0.3049 which is positive which shows that the gradient moves upwards as we move from left to right.
The statistical relationship value multiplied by itself is 0.8413. This is a higher value which shows that the model is a good fit for the data. 84.13% of the data dispersion is being explained by the model and therefore the closer the model will be able to fit the data.
y=0.3049*x-6.4173, Average cost of advertising= 0.3049*187.16-6.4173= 50.647784
Where y is the average cost of advertising, x is the price per listing.
The average cost of advertising is 50.647784. This shows an accuracy in prediction of the model since the average cost is positive and this is well shown in the scatter plot diagram.

Average cost of advertising= 0.3049*300-6.4173= 85.0527. The average cost of advertising will be 85.0527 which is also well shown in the scatter plot. The value is positive which shows that the model is very accurate.

Appendix 4- Determining the favorable location

Location Listing Type
Classic Designer Luxury Total
Italy 45 55 38 138
Croatia 60 70 40 170
Greece 39 42 59 140
Czech Republic 30 30 20 80
Hungary 26 28 38 92
Total 200 225 195 620
Table 1: Listing Type by Location
Location represents categorical data while listing type represents numerical data.
Probability that someone stays in Croatia,

60/200+ 70/200+40/200 = 0.85
Given the location is Italy, the probability that someone stayed in a Luxury listing,
38/138=0.2754
If the listing type is Classic then the probability someone stayed in Greece is,
39/200= 0.195

Probability that someone stayed in Croatia and booked a Designer listing,
70/225=0.3111
Probability that someone stayed in Czech Republic or they booked a Luxury listing.
30/200 + 20/195= 0.1176

Chi-Square Tests From SPSS
Value df Asymp. Sig. (2-sided)
Pearson Chi-Square 35.000a 30 .243
Likelihood Ratio 24.470 30 .750
N of Valid Cases 7
Test hypothesis

H0: Location has no significant relationship with luxury listing.
H1: Location has a significant relationship with luxury listing.
The value of the test statistic is 35 and the p value is 0.243.
Since the p-value is greater than the significant level, (α = 0.05), we do not reject the H0 and therefore there is no association