Elasticsearch nested sort based on minimum values of child of child arrays

Assuming that each order is a separate document, you could create an order-focused index where both shipments and products are nested fields to prevent array flattening.

The minimal index mapping could then look like:

PUT orders
{
  "mappings": {
    "properties": {
      "shipments": {
        "type": "nested",
        "properties": {
          "products": {
            "type": "nested"
          }
        }
      }
    }
  }
}

The next step is to ensure the quantity is always numeric -- not a string. When that's done, insert said docs:

POST orders/_doc
{"orderNo":"1","shipments":[{"products":[{"productName":"AAA","quantity":2},{"productName":"AAA","quantity":2}]},{"products":[{"productName":"AAA","quantity":3},{"productName":"AAA","quantity":6}]}]}

POST orders/_doc
{"orderNo":"2","shipments":[{"products":[{"productName":"AAA","quantity":1},{"productName":"AAA","quantity":6}]},{"products":[{"productName":"AAA","quantity":4},{"productName":"AAA","quantity":5}]}]}

Finally, you can use nested sorting:

POST orders/_search
{
  "sort": [
    {
      "shipments.products.quantity": {
        "nested": {
          "path": "shipments.products"
        },
        "order": "asc"
      }
    }
  ]
}

Tip: To make the query even more useful, you could introduce sorted inner_hits to not only sort the top-level orders but also the individual products enclosed in a given order. These inner hits need a nested query so you could simply add a non-negative condition on shipments.products.quantity.

When you combine this query with the above sort and restrict the response to only relevant attributes with filter_path:

POST orders/_search?filter_path=hits.hits._id,hits.hits._source.orderNo,hits.hits.inner_hits.*.hits.hits._source
{
  "_source": ["orderNo", "non_negative_quantities"], 
  "query": {
    "nested": {
      "path": "shipments.products",
      "inner_hits": {
        "name": "non_negative_quantities",
        "sort": {
          "shipments.products.quantity": "asc"
        }
      },
      "query": {
        "range": {
          "shipments.products.quantity": {
            "gte": 0
          }
        }
      }
    }
  },
  "sort": [
    {
      "shipments.products.quantity": {
        "nested": {
          "path": "shipments.products"
        },
        "order": "asc"
      }
    }
  ]
}

you'll end up with both sorted orders AND sorted products:

{
  "hits" : {
    "hits" : [
      {
        "_id" : "gVc0BHgBly0XYOUcZ4vd",
        "_source" : {
          "orderNo" : "2"                   <---
        },
        "inner_hits" : {
          "non_negative_quantities" : {
            "hits" : {
              "hits" : [
                {
                  "_source" : {
                    "quantity" : 1,         <---
                    "productName" : "AAA"
                  }
                },
                {
                  "_source" : {
                    "quantity" : 4,         <---
                    "productName" : "AAA"
                  }
                },
                {
                  "_source" : {
                    "quantity" : 5,         <---
                    "productName" : "AAA"
                  }
                }
              ]
            }
          }
        }
      },
      {
        "_id" : "gFc0BHgBly0XYOUcYosz",
        "_source" : {
          "orderNo" : "1"
        },
        "inner_hits" : {
          "non_negative_quantities" : {
            "hits" : {
              "hits" : [
                {
                  "_source" : {
                    "quantity" : 2,
                    "productName" : "AAA"
                  }
                },
                {
                  "_source" : {
                    "quantity" : 2,
                    "productName" : "AAA"
                  }
                },
                {
                  "_source" : {
                    "quantity" : 3,
                    "productName" : "AAA"
                  }
                }
              ]
            }
          }
        }
      }
    ]
  }
}