Multihead weighers or combinational weighers are used to provide accurate weights at high packing speed and are currently the most used dosing method for many kinds of products, also including those with heterogeneous characteristics [5]. Control of the actual content of the packaging is regulated by the European directive 76/211/EC and must be implemented by factories, packaging plants and importers. The regulations state that consumers be informed about the quantity and be protected against short measures, while allowing businesses the flexibility to control quantity on the production line within specific tolerances [6].

Combinational weighers, designed in the mid 1980s, uses combination weighing techniques to achieve dispensed weights that are closer to the desired target weight than conventional weighing techniques. They have a number of weighing heads that statically weigh the product; these weight data are fed to a computer, which calculates all of the possible combinations of product weights in order to dispense the best combination (closest match to target weight) in a packaging machine. Fig. 1 presents the schematic of the basic combinational weigher components. The weighing system consists of three elements, namely:

A system to automate product feeding to the weighing stations. This is normally in the form of a belt or vibratory feeder (linear vibrators); the belt is controlled to deposit a fraction of the final target weight into a product storage hopper mounted above the weigh hopper. Depending on the layout of the machine, the feed system is configured in either a radial or line form.

Figure 1 Source: The authors.

Figure 2 Source: The authors.

The multihead packing process starts when a quantity of goods is placed into each hopper , [8] and the weight signal is transmitted to the built-in computer. The computer calculates the combinations of weights that come closest to the desired weight T, and the combination of the closest weights is released from the corresponding hoppers. The resulting empty hoppers are supplied with new quantities of goods. The computer continuously repeats this process

If we assume that the weights in the hoppers follow a normal probability distribution, that all the hoppers are independently filled according to the same distribution N (µ, σ) and k hoppers are randomly selected in each packing operation, then the weight of packages is known to follow a normal distribution where the average package mean weight kµ is expected to equal the target T.

The value of (the standard deviation if hoppers were selected at random) is considered to be an index of quality in the packaging process. However, the subset of hoppers to be discharged H' is not actually selected at random but in a driven way, usually such that the total weight is as close to T as possible. Therefore, [12,13].

A summary of the scientific references that can be found in this field will now be presented. Some authors have studied the possibility of improving multiweighing proceedings in packing process performance. For example [12,13] proposed the use of the percentage variability reduction index for the reduction and control of production process variability. [7] Investigated the optimum scheme for determination of operation time of line feeders in automatic combination weighers. [5] Proposed a weighing algorithm for multihead weighers based on bit operation. In [8] a second objective called “priority” is introduced. They formulated the problem as a bi-criteria optimization problem and proposed an algorithm based on dynamic programming. The proposed approach was aimed at heuristically minimizing the maximum duration of items in the system, while attaining a total weight for each package as close to the target weight as possible. Some authors [11,14-16] have studied the possibility of improving the bi-criteria optimization model proposed by [8]. [6] proposed a set of strategies to study the filling setting of hoppers as a way to reduce variability in the packing process. In [9] the process was evaluated taking into count the "priority" of the hoppers (whose content was real weight) and considering that the relative importance of objectives is dynamically managed and adjusted. Other authors investigate different types of packing operations. Several algorithms were developed for double-layered goods packing systems and duplex packing systems, where in an operation two disjoint subsets are simultaneously selected to produce two packages [14,17].

In this work, we propose implementing a modified control chart to monitor and control the multihead weighing process, which is previously optimized and improved through a packaging strategy based on the contributions of [6].

As has been mentioned in section 2, in this work it is assumed that the number of hoppers 𝑘 to be combined in each packing operation is constant and fixed in advance. This means that the supply of product to the weighing hoppers must be set at . However, the packaging strategy proposed considers the general case in which each hopper 𝑖 is expected to be filled with a different average quantity of goods (instead of a common value µ). In this paper we explore the case in which several hoppers weights are set in such a way that they share the same average quantity of goods, as this has been proven to be an efficient strategy to reduce package variability [5,6,13].

In addition, we use the expected coefficient of variation (CV) of the final package (say, if the hoppers were selected at random) for calculation of the standard deviation of weights in every hopper (σ) as an input in the packaging process. e.g., if / gr and k= 2. Theoretically we have gr and therefore σ = 70.71 gr. However, as explained in subsection 2, this does not mean that will be the actual variability obtained in the package produced through our packing strategy.

In our particular case, the packaging strategy proposes dividing the n weighing hoppers into five subgroups (n₁, n₂, n₃, n₄ and n₅ with and provides an unequal amount of product to each subgroup, on average (µ₁, µ₂, µ₃ , µ₄ and µ₅, respectively). During the packing operation the filling setting is establish in the following way: µ₁= µ- 1.5 σ, µ₂= µ - 1σ, µ₃= µ, µ₄= µ+ 1σ and µ₅ = µ+ 1.5σ.

As an example in the calculation of the µ _j values, suppose T= 2000 gr., k= 2, σ= 70.71 gr. In these conditions the µ _j values would be: µ₁= 1000 - 1.5(70.71) = 893.93 gr., µ₂= 1000 - 1(70.71) = 929.29 gr., µ₃= 2000/2 = 1000 gr., µ₄= 1000 + 1(70.71) = 1070.71 gr. and µ₅ = 1000 + 1.5(70.71) = 1106.07 gr. In this way some hoppers would share the same value for µ _j depending on which subgroup (n₁, n₂, n₃ , n₄ and n₅) the hopper belongs to.

The procedure proposed to carry out the packing operation is explained in this section. This enumerative procedure is made for each packed product and can be implemented in software systems installed in control unit of the multihead weigher. The packing algorithm consists in 4 steps, namely:

Step 1. Feed the n weighing hoppers according their respective setting (µ₁, µ₂, µ₃ , µ₄ and µ₅).

Figure 3 Source: The authors.

Step 3 of the algorithm describes a situation in which all the hoppers should be discharged in order to avoid producing packages that would not meet the quality requirements for the final product in terms of weight. When that happens, all of the discharged product could be taken and reused in the process again, for instance. Besides, we will also calculate the percentage of discharge for the confidence level (DCL), as a way to measure the performance of the proposed strategy. Fig. 3 shows the flowchart for the proposed packing procedure.

The packing algorithm was implemented in Pascal and run on a personal computer with Windows 7 Home Premium (64bit), Intel Core i5-3317U CPU (1.7 GHz) and 4 GB memory. Fig. 4 shows the user interface of this prototype for the case in which the number of hoppers 𝑘 to be combined is equal to 5. The current version of the application lets the user enter the probability distribution setting for each subgroup of hoppers, as well as the number of hoppers in each subgroup. Other parameters such as the target weight T, the total number of hoppers n and the total number of packages to produce Q can also be entered by the user. The software outputs are: the proportion of iterations each hopper has been used, the average weight of the number total of packages produced (), the standard deviation of the number total of packages produced () and percentage of discharge for the confidence level (DCL).

The optimization model is formulated to minimize the difference between the target weight (T) and the total weight of the final package (W). In this case, the variable to be minimised is presented by D which is the minimum difference, in absolute value, between T and W.

Figure 4 Source: The authors.

The solution to choose weights in the hoppers can be described in the following way:

Subject to

(5)

(6)

(7)

(8)

(9)

Here, eq. (5) ensures that the value of D is greater than or equal to zero. This represents the minimum absolute value and complies with restrictions Eqs. (6) and (7), i.e., it aims to attain a total weight as close to the target weight T as possible. Moreover, the confidence level constraints are represented by Eq. (8) and Eq. (9). Note that the mathematical model is presented in a generic way and does not depend upon the way in which the 𝑛 hoppers have been subdivided (see subsection 3.1).