the number of salespeople assigned to work during a shift is apportioned based on the average number of…

Explanation:

Step1: Calculate the total number of customers

$130 + 275+430 + 475=1310$

Step2: Calculate the standard divisor

The standard divisor $d=\frac{1310}{15}\approx87.33$

Step3: Calculate the standard quota for each shift

Morning: $q_1=\frac{130}{87.33}\approx1.49$ Midday: $q_2=\frac{275}{87.33}\approx3.15$ Afternoon: $q_3=\frac{430}{87.33}\approx4.92$ Evening: $q_4=\frac{475}{87.33}\approx5.44$

Step4: Calculate the lower - quota for each shift

Lower - quota of Morning: $\lfloor1.49\rfloor = 1$ Lower - quota of Midday: $\lfloor3.15\rfloor = 3$ Lower - quota of Afternoon: $\lfloor4.92\rfloor = 4$ Lower - quota of Evening: $\lfloor5.44\rfloor = 5$ The sum of lower - quotas is $1 + 3+4 + 5=13$ We have $15-13 = 2$ remaining salespeople to distribute.

Step5: Use Lowndes' method

We calculate the modified divisor. Try a modified divisor $d' = 85$ Morning: $q_1'=\frac{130}{85}\approx1.53$ Midday: $q_2'=\frac{275}{85}\approx3.24$ Afternoon: $q_3'=\frac{430}{85}\approx5.06$ Evening: $q_4'=\frac{475}{85}\approx5.59$ The lower - quotas are: Morning: $\lfloor1.53\rfloor = 1$, Midday: $\lfloor3.24\rfloor = 3$, Afternoon: $\lfloor5.06\rfloor = 5$, Evening: $\lfloor5.59\rfloor = 5$ The sum of lower - quotas is $1+3 + 5+5=14$ We still need to assign 1 more salesperson. Try a modified divisor $d'' = 84$ Morning: $q_1''=\frac{130}{84}\approx1.55$ Midday: $q_2''=\frac{275}{84}\approx3.27$ Afternoon: $q_3''=\frac{430}{84}\approx5.12$ Evening: $q_4''=\frac{475}{84}\approx5.65$ The lower - quotas are: Morning: $\lfloor1.55\rfloor = 1$, Midday: $\lfloor3.27\rfloor = 3$, Afternoon: $\lfloor5.12\rfloor = 5$, Evening: $\lfloor5.65\rfloor = 5$ The sum of lower - quotas is $1+3+5 + 5=14$ We still need to assign 1 more salesperson. Try a modified divisor $d''' = 83$ Morning: $q_1'''=\frac{130}{83}\approx1.57$ Midday: $q_2'''=\frac{275}{83}\approx3.31$ Afternoon: $q_3'''=\frac{430}{83}\approx5.18$ Evening: $q_4'''=\frac{475}{83}\approx5.72$ The lower - quotas are: Morning: $\lfloor1.57\rfloor = 1$, Midday: $\lfloor3.31\rfloor = 3$, Afternoon: $\lfloor5.18\rfloor = 5$, Evening: $\lfloor5.72\rfloor = 6$ The sum of lower - quotas is $1+3+5 + 6=15$

Answer:

Morning: 1, Midday: 3, Afternoon: 5, Evening: 6