The delivery ratio for a given day is defined as the ratio of the number of orders booked on that day which are delivered on the next day to the number of orders booked on that day which are delivered on the second day after booking. On which of the following days, was the delivery ratio the highest?

The correct option is 14th.

Understanding the Delivery Dynamics:
XYZ organization offers a two-day delivery promise. Therefore, orders booked on any given day d can only be delivered on the next day (day d + 1) or on the second day after booking (day d + 2). If an order is not delivered within these two days, it is declared lost at the end of the second day (day d + 2).
Let:
• $B_d$ be the number of orders booked on day d.
• $D_{d,d+1}$ be the orders booked on day d that are delivered on day d + 1 (next-day delivery).
• $D_{d,d+2}$ be the orders booked on day d that are delivered on day d + 2 (second-day delivery).
• $L_d$ be the orders booked on day d that are lost (declared lost at the end of day d + 2).

Thus, the total orders delivered on any day d is the sum of next-day deliveries from day d - 1 and second-day deliveries from day d - 2:
$${\text{Delivered}}_d=D_{d-1,d}+D_{d-2,d}$$
The number of orders lost on day d is the difference in cumulative lost orders between day d and day d - 1. These lost orders must have been booked on day d - 2:
$$L_{d-2}={\text{Cumulative Lost}}_d-{\text{Cumulative Lost}}_{d-1}$$
The number of orders booked on day d is:
$$B_d={\text{Cumulative Booked}}_d-{\text{Cumulative Booked}}_{d-1}$$
Since every booked order must either be delivered on the next day, delivered on the second day, or lost, we have:
$$B_d=D_{d,d+1}+D_{d,d+2}+L_d$$

We are given the following initial parameters:
• Orders booked on the 11th that took two days to deliver: $D_{11,13}=4$
• Orders booked on the 12th that took two days to deliver: $D_{12,14}=6$
• Orders booked on the 13th that took two days to deliver: $D_{13,15}=8$

Step-by-Step Calculation of Daily Deliveries:

1. Analyzing Day 13:
• Total delivered on 13th = 11. These must be $D_{12,13}+D_{11,13}=11$.
Since $D_{11,13}=4$, we get:
$$D_{12,13}=11-4=7$$

2. Analyzing Day 14:
• Total delivered on 14th = 27. These must be $D_{13,14}+D_{12,14}=27$.
Since $D_{12,14}=6$, we get:
$$D_{13,14}=27-6=21$$

3. Analyzing Day 15:
• Total delivered on 15th = 23. These must be $D_{14,15}+D_{13,15}=23$.
Since $D_{13,15}=8$, we get:
$$D_{14,15}=23-8=15$$

4. Analyzing Day 16:
• Total booked on 14th = Cumulative Booked till 14th - Cumulative Booked till 13th = $249-219=30$.
• Cumulative lost on 16th - Cumulative lost on 15th = $106-94=12$. These are the orders booked on the 14th that were lost:
$$L_{14}=12$$
Using the equation for total bookings on the 14th:
$$B_{14}=D_{14,15}+D_{14,16}+L_{14}$$
Substitute the values:
$$30=15+D_{14,16}+12\Rightarrow D_{14,16}=3$$
• Total delivered on 16th = 11. These must be $D_{15,16}+D_{14,16}=11$.
Since $D_{14,16}=3$, we get:
$$D_{15,16}=11-3=8$$

5. Analyzing Day 17:
• Total booked on 15th = Cumulative Booked till 15th - Cumulative Booked till 14th = $277-249=28$.
• Cumulative lost on 17th - Cumulative lost on 16th = $118-106=12$. These are the orders booked on the 15th that were lost:
$$L_{15}=12$$
Using the equation for total bookings on the 15th:
$$B_{15}=D_{15,16}+D_{15,17}+L_{15}$$
Substitute the values:
$$28=8+D_{15,17}+12\Rightarrow D_{15,17}=8$$
• Total delivered on 17th = 21. These must be $D_{16,17}+D_{15,17}=21$.
Since $D_{15,17}=8$, we get:
$$D_{16,17}=21-8=13$$

6. Analyzing Day 18:
• Total booked on 16th = Cumulative Booked till 16th - Cumulative Booked till 15th = $302-277=25$.
• Cumulative lost on 18th - Cumulative lost on 17th = $120-118=2$. These are the orders booked on the 16th that were lost:
$$L_{16}=2$$
Using the equation for total bookings on the 16th:
$$B_{16}=D_{16,17}+D_{16,18}+L_{16}$$
Substitute the values:
$$25=13+D_{16,18}+2\Rightarrow D_{16,18}=10$$

Comparing the Delivery Ratios:
The delivery ratio for a given day is defined as:
$$\text{Delivery Ratio}=\frac{D_{d,d+1}}{D_{d,d+2}}$$

• For the 13th:
$$\text{Ratio}=\frac{D_{13,14}}{D_{13,15}}=\frac{21}{8}=2.625$$

• For the 14th:
$$\text{Ratio}=\frac{D_{14,15}}{D_{14,16}}=\frac{15}{3}=5$$

• For the 15th:
$$\text{Ratio}=\frac{D_{15,16}}{D_{15,17}}=\frac{8}{8}=1$$

• For the 16th:
$$\text{Ratio}=\frac{D_{16,17}}{D_{16,18}}=\frac{13}{10}=1.3$$

Comparing the ratios, the delivery ratio was the highest on the 14th (with a ratio of 5).