for Model Maximizing Container Loading in the Airfreight Industry
Joseph E. Beaini, Pascal R. Bedrossian
Abstract -- This paper presents models to maximize the loading of freight containers within the time sensitive criteria of the airfreight industry. This paper addresses personnel interaction with loading queues as well as the optimal procedure for loading a freight container within time limits. Different scenarios for the stochastic arrival of packages and alternative solutions will be discussed. Two algorithms for the loading of containers are introduced as well as recommendations on cost bearing optimization methods.
I. INTRODUCTION In the highly competitive industry of overnight parcel delivery, lowering cost is essential. If a company can not find ways to reduce its expenses and pass the savings on to its customers, the competition will. One of the highest individual expense for airfreight companies is associated with the efficient and prompt loading of airplanes. Airplanes are used to deliver packages to destinations in a short amount of time. These packages are loaded into containers and then the containers are loaded on the airplanes. Therefore, if a air transport can have high utilization of these containers, they will be able to save space, use the fewest number of containers, and reduce aircraft expenses A- Assumptions1 The packages are delivered to the containers via a conveyor system. Therefore, not all the packages are available when employees start to fill the
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containers. This makes this aspect of the problem dynamic. The sort window is two hours. That is, employees have two hours to sort the packages to determine what containers they will be placed in as well as pack the container. FedEx1 is not sure what the impact would be if the sort window was extended to four hours for instance. There probably would be a trade-off. The cost would probably increase while the amount of wasted space per container would decrease. By extending the sort window, it drives this aspect of the problem to become static. If the window is increased, an obvious question arises. How long does FedEx wait to start to fill the container? They are not sure of the answer. In solving most problems, other logical and rational assumptions regarding the packages must be made. The following are the assumptions related to the packages that were made that contributed towards the modeling of the problem. 1) Individual packages do not weigh more than 75 pounds. 2) Conveyor speed is constant for the night of the sorting unless an emergency arises where the conveyor is forced into dead stop. 3) The minimum size box is 2 x 2 x 2. 4) The maximum size box is 36 x 36 x 36. II- Breakdown of the problem To determine the high utilization of the containers, the problem must be decomposed into three smaller problems (figure 1). Each part holds crucial information necessary to the optimization of the next. The third part, however, is a problem that has already been resolved by FedEx and requires very little quantitative importance from the others,
Emulating the Memphis hub operations of Federal Express Corporation, Memphis, Tennessee.
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J. Beaini, P. Bedrossian - Model for Maximizing Container Loading in the Air-Freight Industry
but is very important to understand the reasons why the first two problems are important. Package Arrival
Filling the Container
Load Container
Figure 1: Problem Decomposition
These containers vary in size. When they are placed in the aircraft, the heaviest containers are situated over the wings to stabilize the plane. Therefore, this aspect of the problem is not a simple last container off is the first one on the plane. Once they are packed, the containers are loaded on the plane. The order in which they are loaded depends on the weight distribution, and whether or not the plane will be serving more than one city that night. A- Package Arrival and Container Loading A1- The Ramp As mentioned earlier, the packages arrive to the loading area via a conveyor belt. The belt is controlled by a Programmable Logic Controller (PLC) and has the capability to be driven at variable speeds. The belt speed is determined by the amount of packages that it needs to deliver to the loading ramp. Another reason for having a variable speed belt, is the necessity to stop the belt in case of an overflow of packages in tight corners. In other words, this is done to avoid package jams along the boundaries of the ramp. The packages are placed on this belt according to destination zip code and beltaircraft assignment. Carriers in the past have limited the size and weight of each package for the two reasons. 1) The size of the package presents a restriction on the MAESC'99
volume it takes-up inside the container and 2) there is maximum weight a worker is allowed to lift without violating company safety. However, due to the randomness of shipper demands the size and weight variables are not constant. Fed-Ex records the weight of every packages at their pick-up locations. Thus, with the help of its forecasted data, workers attending this ramp could have a priori knowledge of the weight of the packages through the computer database that is accessible in the hub. As far as the size of a package is concerned, the task becomes more difficult. There will be a system to determine the size of a package as it rolls down the ramp. A neural network based pattern recognition system is currently under development by the hub engineering department at Fed-Ex. This will enable the computer to have prior knowledge of the box sizes available on the ramp. If the ramp is not fully loaded, the arrival rate of the boxes coming down the ramp will not be constant. The presence of sensors to determine sizes and the prior knowledge of weight per package will enhance an algorithm to maximize container packing. However, we have to decide on the type of queue that is necessary for the efficient packing of the container. Three different scenarios can be explored. The first is to do away with the prior knowledge of the variables and treat each package on a first come first serve (FCFS) basis. This scenario is very simple to implement since in several situations the algorithm will have to accommodate the box using single constraints as opposed to optimize its position in the container. A simple example is the mandatory placement of a box with fragile contents on the lower stacking level as opposed to the top, because the size does not fit any of the gaps. The second scenario is to allow the computer to have prior knowledge of size and weight of packages in a finite queue, where a certain number of packages are allowed and considered for the algorithm. No other packages are allowed onto the queue unless a package has left. The third scenario is to allow the computer the knowledge of a larger number of boxes for packing consideration by modifying the physical area of the loading zone.
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This is done by, placing the boxes on a carousel similar to the ones you see at baggage claim areas in airports. Thus, the choice of a queue will depend on whether the change of the physical plan is economically feasible. A2- The Container The size of the containers depends on the airplane model and cargo capacity. The shape of the container is designed to fit the contour of the airplanes shell. Thus, the top of the container is somewhat rounded for the smaller planes (figure 2a), and rounded only around the corners for the larger planes (figure 2b). The bottom is flat and designed to roll on the bed of the plane. One advantage to rounding the top of the container is to limit the choice of packages that can be placed on top to smaller packages. Smaller boxes, usually means lighter, will be placed on top of the larger heavier boxes.
all packages above letter size and below 75 lbs., and 3) the heavy boxes. The packages eligible to pass through the SPSS are bundled in plastic bags and are sorted by region. This system is already installed and packaging has already been optimized by FedEx ground operations. The heavy boxes are transported to their containers on a separate beltline. These boxes are normally loaded separately and will be placed on the airplane around the wing span area to balance the weight. The medium weight boxes, however, are usually manhandled into their containers one at a time. A worker will pickup a box from the loading queue based on personal judgment of weight, size, and possibility of fit in the container since his/her last trip. The decision time is based on visual estimate of space and prior experience with packing a specific size container. Personal expertise in this case depends on the experience of the person working the loading area, and the type of container associated with a specific loading area.
III - THE MODELS A- The Package Arrival Model Since we have decided to decompose the problem into separate problems, it is essential to identify the factors that will contribute to the optimization of each other. We hope that an algorithmic solution can eventually be developed to assist in the efficient packing of the containers. Our most important constraint is time. We cannot afford to have the time allotted for decisions making per package exceed certain boundary conditions. Those conditions have to be set prior to packing the container. The comprehensive model of package arrival will shed some light on these conditions or perhaps tradeoffs for completing the packing process. A1- The First Come First Serve model In this model, workers load the containers on a first come first serve basis. With the help of
Figure 2a Profile of smaller airplanes
Figure 2b Profile of wide body airplanes
The container packaging system consists of three different categories: 1) The Small Package Sorting System (SPSS), 2) the medium weight packages or
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forecasting, we will have prior knowledge of the number of packages destined for a particular flight. However, for the purpose of packing a container, packages with independent weights and sizes show up randomly at the loading zone. Workers load the packages as they come and make decisions on the spot about their placement. Table 1 - Definition of variables k = Number of Boxes. k = Rate of Arrival of boxes. j = The service time per box by a worker. j = Number of workers working the end of queue. TL = Total time allowed per container. Knowing that we have k-boxes destined for this ramp, and that j workers will move these boxes to i containers, a model for this random process can be introduced. Assuming that the containers will be placed at positions that are equidistant from the package pickup point, all workers move at the same speed, and that the belt is operating at constant speed, we can introduce two essential variables. We will call the arrival time of the packages or the rate at which the boxes get-in, k.. We will call the time it takes a worker to pick up the box, move it to the container, place it inside the container, and come back to the pick-up point, j (service time). If jj k , the workers will be standing around waiting for the packages to show up. However, the condition k jj has to be true since we are relying on forecasted data. We know that the containers have to be ready within a certain time window before they are moved and loaded on to the airplanes. We will call the time limit to fill and lock the containers for transportation to the plane, TL. The time it takes to have the k packages available for service is, kk . The time it takes to move and load k packages is, kj . Therefore, we can determine the lower and upper bound times that are needed for loading the containers. If the packages are to be moved one at a time then the upper bound is
TL k/k + kj +
(EQ. 1)
Where is the time slack needed to assure completion of the process prior to the maximum allowed time. However, if we consider that workers are moving packages off the pick-up zone as they, then this scenario will give us the lower bound value for the system TL k/k + j (EQ. 2)
This is the necessary time to have k packages available plus the initial trip to the container and the last return trip to the pick up point. A2- Generalization: Model of the Floating Queue. Having a floating queue will slightly change the problems dimensions. By introducing the idea of a queue, the process takes into consideration the fact that each worker decides which package will have to be picked next. In addition to the previous assumptions, and to avoid working with time averages, we will assume that all workers have the same comprehensive reasoning ability for decision making. This idea introduces a new variable, td, which is the decision time for selecting j boxes by j workers out of the queue. k = Number of Boxes. k = Rate of Arrival of boxes. j = The service time per box by a worker. j = Number of workers working the end of queue. Td = Decision Time per worker. TL = Total time allowed per container. c = Scalar, multiple of number of workers. An = Number of expected boxes in the queue after n-trips. K = Total number of boxes for the process. To introduce such a variable, means that j workers have to choose from multiple j choices of packages. We choose the notation c for a scalar to express the size of the queue, which is cj.
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The number of expected boxes in the queue can be then be derived. Originally the queue has in it A0 = cj (EQ. 3)
The result in equation (EQ. 9) can be re-written in the form: (d + j ). k j (EQ.8)
After one trip, the number of expected boxes in the queue is A1 = cj + (k . d ) - j + (j . k ) (EQ. 4)
The total amount of time for this process can now be derived: (cj)/k + ... waiting for cj boxes in the queue (d + j ) + ... 1st pick-up (d + j ) + ... 2nd pick-up (d + j ) ... (K/j) pick-up (EQ. 9) Where K, is the total number of boxes for this process to distinguish it from the previous cases notation. Taking in consideration that: (d + j ). k j (EQ. 10)
Where k . d is the number of boxes coming in while making the decision, and j . k is the number of boxes which came in while the worker was taking the boxes. After the second trip, A2 = A1 + (k . d - j + j . k ) An = An-1 + (k . d - j + j . k ) An = An-2 + 2(k . d - j + j . k ) An = A0 + n(k . d - j + j . k )
the total amount of time allotted for this process can then be re-written in the form: (EQ. 5) TL = (cj)/k + (K/j) (d + j ) (EQ. 11)
Eventually, the second term will be This zero. will yield an important relationship for the number of workers needed for a certain process. k . d - j + j . k 0 or, k . d + j . k j (EQ. 7) (EQ. 6)
Substituting equation (EQ. 7) in equation (EQ. 11) will yield: TL = (cj)/k + K/k (EQ. 12)
To illustrate this tentative solution, we can use a simple example. Assume the arrival rate of packages is k = 1 box/min. and we have 5 workers. The time it takes to load a package and come back is tj = 1 minute. Then the time needed to make the decision on the package is d = 5 boxes/(1 box/min) - (1 box/min * 1 min) / (1 box/min) = 4 minutes.
This analysis will help us determine the tradeoff between the decision time and the size of the queue. The scalar c determines the size of the queue, and the delay time determines the maximum time needed to make a decision per package. A particular solution is when c.j = K, which is the case of an infinite deterministic queue. The formula in this case is transformed into TL = K/k + K/k = 2. K/k or, TL =K/k + (K/j) (d + j ) (EQ. 13)
(EQ. 14)
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This solution gives a direct relationship between the maximum time allotted to fill the container for shipment and the decision time per package. Equation A3.9, however, is the more general form for any size queue. B. Filling the container Model B1- Devising a Plan As mentioned earlier, each dimension of the container will be divided into two-inch intervals, thus forming a grid that spans all three dimensions. Therefore, one can visualize the container as a cavity made up of eight cubic inches volumes. Figure 3 shows a three dimensional illustration of the container.
bottom, one inch to the top, one inch the front, and one inch to the back inside the cube. Thus, any adjacent two cubes centroids in any one of these directions, will be two inches apart. Theoretically, the top boundary of the container will present us with a problem because not all containers will have the same curvature or profile. This will eventually present us with unavoidable margins of error. The error will be in the form of slack and depends on boxes previously placed on the bottom of the profiled area. Figure 4 shows the slack error within the container boundary. Varies with the size and placement of the boxes. This problem can be easily tackled if the profile curvature is in two-inch increments because the top of the container would be expressed as a Riemann sum [5].
e
D
Figure 4 Errors from container profile
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C e n tro id
2
This error will present us with a constraint if we were to optimize this model. If the profile increment is wider than two inches, the two-inch volumes can easily be implemented. If is smaller, than a lower cube resolution should be considered. Currently, a worker would pick up a box, move inside the container, look around for a spot, place the box based on personal judgement and/or box size estimate, then leave the container to go pick another box with the last container layout in mind. To model the placement of packages in a container based on the above description, we have to consider two methods. The two methods depend on two
2 Figure 3 Illustration of Container Grid
We will assume that the centroid of a cube represents a cube. In our model, the centroid of each of these cubes will be considered a vertex. Hence, each side of the vertex in a cube spans one inch to the left, one inch to the right, one inch to the
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distinct human behaviors in placing objects in enclosed areas. The first method depends on visual estimation of how much space a box should take. In other words, it is based on the number of empty two inch increments a box will fill in three dimensional space. An analogy is the shape fitting game given to most preschoolers. The second method depends on a worker using relative distances with respect to other boxes, walls, and ceiling. An analogy to this method is the tile counting routine used to fit a piece of furniture in a tiled room. C- First method The memoryless system This method depends on limiting your decision of box fitting to the immediate area the worker can get to. Thus, the area in question is localized and its parameters do not affect the rest of the container. To model this approach, we will have to take each two-inch volume independently. To clearly explain this method, we have to define some notation. V w d h = A certain volume of a centroid in the container Height Depth Width Vbwdh = The volume of a certain box b. Wb = Width of box b. Db = Depth of box b. Hb = Heigth of box b. Xwdh= Decision Variable. Each two-inch volume in the container will be assigned a volume variable V. The position of V in the container will be defined as Vwdh . This means, starting from right to left, every centroid will be adjacent to six other centroids through their vertices (Fig. 5). Vwd(h+1) Vw(d-1)h
Figure 5 A volume vertex and adjacent vertices Xwdh will be a decision variable. It is equal to 0 if the vertex is empty and is 1 if the vertex is full. 0 X wdh = 1 (EQ. 15)
C1- Constraints For a box to be placed at a vertex Vwdh, conditions on the decision variables are: Xwdh = 0 and Xwd(h-1) = 1. (EQ. 16)
The box placement has to start at a corner vertex. Without loss of generality we can start from any of the four bottom corners. In our analysis, we will start from the right deepest corner of the container. Any of the boxes will have a volume of:
b Vwdh = Wb Db H b
(EQ. 17)
Originally, the container will be empty. We will call the time at which the process begins t = 0. Thus, @t = 0 X wdh = 0 w, d , h (EQ. 18)
V(w-1)dh Vwd MAESC'99 Vw(d+1)h
V(w+1)dh
C2- Algorithm 1 To develop an algorithm for this process, we have to decompose the process in two pieces. First, we have to decide on a position to place the box; check if there is anything under it to support it; try to place it; and if it does not work, rotate in any one of six mutations, then try the process again. Second, after the box is placed readjust the Xwdh values to acknowledge that the vertices taken up by the box
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are not available any more. The process can be explained in the form of a flowchart (Fig. 6).
Container empty
Wb Db H b 2 2 2
(EQ. 19)
Pick-up box to place
Does it have Support?
No
where the first term is the number of vertices in the width, the second term is the number of vertices in the depth, and the third term is the number of vertices in the height. Now we can try to write a pseudocode to describe each of the functional blocks of the flowchart in figure 6. The first block will check whether or not a candidate position has place in the container that will support it. In other words, we want to check if the volume of a box:
b Vwdh = Wb Db H b
Yes
(EQ. 20)
Is position Available?
No
has a place to lay on. A typical pseudocode will have the following form: PSEUDOCODE 1 Check Bottom of box for support if( h == 1) Done // Were on bottom of container true = 1; i = 0; j = 0; k = 0; while( j <= Db/2) while(i <= Wb/2) if(X(w+I)(d+j)(h-1)= 0) true = 0; Call (check for availability function) The next pseudocode will check for the availability of a position in the container for a box WbDbHb. The availability of a position will be tested at a boxs reference vertex Vbwdh. PSEUDOCODE 2 Checking for availability of a position. true =1; i = 0; j = 0; k = 0; while(k <= Hb/2) while(j <= Db/2) while(i <= Wb/2) if(X(w+i)(d+j)(h+k) = 1 )
Yes
Place Box
Does it fit?
No
Rotate Box
Re-Adjust Xs
The routines that describe this process are easily programmed. The complexity of the iterative process will vary with the number of two-inch volumes that are present inside the container. In our case since we chose a resolution of two inches in any direction, we can write that each box will fill vertices in three dimensions in increments of two inches. In other words, the numbers of vertices affected by the placement of a box are
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X wdh [a r l Where,
f
ba o u ]
(EQ. 21)
After we find a position to place the box, we need to update the weights of the vertices covered by the box to 1. To adjust the weights, we also need to know the volume of the box. The number of vertices affected by the process was mentioned in equation (EQ. 19). The next pseudocode can perform this simple task once we know the boxs volume represented at its vertex Vbwdh. PSEUDOCODE 3 Updating weights of container vertices for ( k = 0 ; k <= Hb/2 ; k++ ) for( j = 0 ; j <= Db/2 ; j++ ) for( i =...