How to assign a known number of different size of outbound packages, to a known (different) number of inbound deliveries?

Question

I am trying to solve a problem that looks like the multiple knapsack problem, the multiple bin packing problem and the subset-sum problem, but isn't exactly one of them.

Imagine a warehouse where 4 deliveries of different quantities (2208, 1920, 400, 1120) arrived at different times. From those boxes 39 "subpackages" of different quantities were taken (these quantities are smaller, see table below) and sent out.

How can we link each outbound package, to one of the inbound deliveries? And/or how is this type of problem called (so I can research further)? There is at least one solution I found manually for the example below. However, I want some strategies that I can put into code, so I can apply this to other similar cases. So I am looking to do this programatically with heuristics and/or with analytical optimization with a certain objective function (eg. minimal time difference between outbound 'groups').

Constraints:

• The total quantity delivered equals the total quantity sent out.
• The outbound packages cannot be divided
• A departure cannot be linked to an arrival that occurs at a later timestamp. Meaning "we can not send out what has not arrived" (time-constraint)

Compared with other known problems:

• This is the multiple knapsacks problem but with all items values = 0 and different size knapsacks
• This is the multiple bin problem but with fixed number of bins of different sizes and the bin must be filled completely
• This is the subset sum problem, but with multiple non-overlapping subsets to be found
• This is the number partition problem, but with fixed number of partitions and the sum of the partitions should not be equal to each other.

Here's the example table (the 4 deliveries are at index 1,3,6,9; the rest are the outbound packages):

index	TIMESTAMP	DELIVERY	QUANTITY
1	1614355831	Inbound	2208
2	1614355831	Departure	400
3	1614355833	Inbound	1920
4	1617200107	Departure	720
5	1617804783	Departure	520
6	1618562403	Inbound	400
7	1618926163	Departure	118
8	1618930180	Departure	160
9	1619169080	Inbound	1120
10	1619170760	Departure	84
11	1619450233	Departure	120
12	1619525526	Departure	240
13	1619768362	Departure	80
14	1619770192	Departure	40
15	1620058521	Departure	160
16	1620061855	Departure	3
17	1620135685	Departure	120
18	1620139836	Departure	200
19	1620144816	Departure	2
20	1620232806	Departure	1
21	1620373216	Departure	240
22	1620375018	Departure	40
23	1620663042	Departure	40
24	1620742517	Departure	120
25	1620744519	Departure	280
26	1620744525	Departure	50
27	1620835864	Departure	1
28	1620922242	Departure	1
29	1620977846	Departure	280
30	1621268120	Departure	80
31	1621347504	Departure	280
32	1621347512	Departure	80
33	1621582717	Departure	120
34	1621584500	Departure	80
35	1621874565	Departure	240
36	1621952070	Departure	80
37	1621953896	Departure	240
38	1621961031	Departure	1
39	1621961035	Departure	1
40	1621961046	Departure	1
41	1622187496	Departure	120
42	1622189393	Departure	120
43	1622533239	Departure	185

Answer 1

You can solve this as a Generalized Assignment Problem, where the departures are the items and the arrivals are the bins. Here, $w_{ij}=\text{QUANTITY}_j$ for items, and $t_i=\text{QUANTITY}_i$ for bins. Your third constraint corresponds to fixing $x_{ij}=0$ (or just omitting the variable) when the timestamp of bin $i$ is later than the timestamp of item $j$.

One approach is to take $p_{ij}=\text{QUANTITY}_j$ and accept the resulting optimal solution only if the objective value is $\sum_j \text{QUANTITY}_j$.

Alternatively, use a constant $0$ objective function and replace the $\le$ constraints with $=$.