Consider the following project which is operational research problem. gimme the full mathematical model of this problem defining:
1.the decision variables,
2.the constraints to be satisfied for a solution to be considered as valid.
3.the objective functions to be minimized or maximized.



Here is the project:
VEHICLE LOADING PROBLEM
ABSTRACT
Supply chain demands are reaching new heights, pushing companies to find new ways to boost efficiency. Loading and unloading are some of the most routine processes in logistics and are an excellent place to start. Tackling truck loading efficiency presents many opportunities to streamline
operations and reduce costs.
1 Introduction
The problem objective is to pack a set of items into stacks and pack the stacks into trucks to minimize the number of
trucks used.
In terms of data volume, a large instance can contain up to 260000 items and 5000 planned trucks, over a horizon of 7
weeks. Hence, 37142 items per day.
2 Problem Description
Three-dimensional trucks contain stacks packed on the two-dimensional floor. Each stack contains items that are packed one above another. There are three correlated dimensions (length, width, height) and an additional weight dimension. We call horizontal dimensions the length and the width. The position of any element (item or stack) is described in a solution by its position in the three axis X, Y, and Z. There is one origin per truck, corresponding with the leftmost downward position (cf Figure 1)
Figure 1: Views from top to bottom: right view, above view, and left view of a truck, seen from the rear of the truck.
The truck contains 96 items (A1 to AD3) packed into 30 stacks (A to AD).
2.1 Items
An item i is a packaging containing a given product. It is characterized by:
• length li, width wi, and height hi (by convention, the length is larger than the width).
• weight ωi
• stackability code pi
• maximal stackability Pi
• forced orientation Oi
• nesting height h¯i
The stackability code pi is used to define which items can be packed into the same stack: only items with the same stackability code can be packed into the same stack. Items with the same stackability code share the same length and width. But the opposite is false: items with identical length/width may not share the same stackability code. An item i is associated with maximal stackability Pi. It represents the maximal number of items i in a stack. The items may be rotated only in one dimension (horizontal), i.e. the bottom of an item remains the bottom. An item
may be oriented lengthwise (the item’s length is parallel with the length of the truck), or widthwise (the item’s length is parallel with the width of the truck). In Figure 1, stack X is oriented widthwise, while stack Y is oriented lengthwise. Item i may have a forced orientation Oi, in which case the stack s containing item i must have the same orientation.
An item i has a nesting height h¯i which is used for the calculation of the height of stacks: if item i1 is packed above
item i2, then the total height is hi1 + hi2 − h¯i1(cf Figure 2). For the majority of items, the nesting height is equal to
zero.
2.2 Trucks
A vehicle v is characterized by its dimensions length/width/height (Lv, Wv, Hv), its maximum authorized loading
weight Wv, and its cost Cv. The truck’s cost represents a fixed cost that does not depend on the number of items loaded
into the truck.
For structural reasons, there is a maximal total weight TMMv for all the items packed above the bottom item in any stack loaded into truck v.
2.3 Stacks
While items and planned trucks are parameters for the problem, stacks represent the results of the optimization. A stack contains items, packed one above the other, and is loaded into a truck. A stack s is characterized by its dimensions length/width/height (ls, ws, hs) and its weight ωs. The length and width of a stack are the length and width of its items (all the items of a stack share the same length/width). A stack’s orientation is the orientation of all its items. There is a unique orientation (lengthwise or widthwise) for all the items of a stack. The stack is characterized by the coordinates of its origin (xo s, yso, zso) and extremity points (xe s, yse, zse).
As shown in Figure 3, the origin is the leftmost bottom point of the stack (the point nearest the origin of the truck), while the extremity point is the rightmost top point of the stack (the farthest point from the origin of the truck). The coordinates of origin and extremity points of a stack must satisfy the following assumptions:
• xe s - xo s = ls and yse - yso = ws if stack s is oriented lengthwise
• xe s - xo s = ws and yse - yso = ls if stack s is oriented widthwise
• zo s = 0 and zse = hs
A stack’s weight ωs is the sum of the weights of the items it contains:
ωs = X
i∈Is
ωi.
The stack’s height hs is the sum of the heights of its items, minus their nesting heights (see Fig 2):
hs = X
i∈Is
hi − X
i∈Is,i̸=bottomitem
h¯i
3 Problem
The objective function is the minimization of the total cost, subject to the constraints described above and that no free space between objects: Any stack must be adjacent to another stack on its left on the X axis, or if there is a single stack in the truck, the unique stack must be placed at the front of the truck (adjacent to the truck driver):
Figure 4: Example of non-feasible packing.
4 Data
4.1 Input Data
Each instance has two files: the file of items and the file of vehicles. An example of file of item is:
id_item length width height weight nesting_height stackability_code forced_orientation max_stackability
I0000 570 478 81 720 47 0 w 4
I0001 570 478 96 256 47 0 w 4
I0002 1000 975 577 800 45 1 n 100
NB: the dimension of the items is set lengthwise. The file of vehicles:
id_truck length width height max_weight max_weight_stack cost max_density
V0 14500 2400 2800 24000 100000 1410.6 1500
V1 14940 2500 2950 24000 100000 1500 1500