JOURNAL OF OPERATIONS MANAGEMENT

Vol. 5. No. 4. August 1985

A Resource Allocation Problem with the Aid of an Integer Zero-One Programming Model in the Plastics Industry

ABRAHAMMEHREZ*

EXECUTIVE SUMMARY

This article represents an integer zero-one linear programming model which gives an optimal solution to a cost problem that determines the division of products in the plastics industry using the injection method. The structure of the model and its limitations will be studied, and the efficiency of the model by analyzing a real case illustrated.

The case concerns the plastics factories of Kibbutz Ruchama in Israel. The problem is to select the mold injection machine and the size of the mold that wiIl be used for the production of a given product and its parts, when the demand for production or the size of the production series is a parameter that is determined exogenously vis-a-vis the production layout, according to market demands. The cyclical production process uses various raw materials, injection molding machines, auxiliary equipment and manpower to produce the final product.

The example that was studied was characterized by “economies of scale.” Indeed, the reduction in unit cost turned out to be almost linearly related to the size of the production series.

BACKGROUND

A routine problem in plastics factories in general and specifically in the plastics factory of Kibbutz Ruhama in Israel is that of product design. These factories produce various plastic products using the “injection molding” method. The production process uses various raw materials such as polystyrene and polyethylene, injection molding machines, auxiliary equipment, and manpower. Molds give form to the raw materials and convert them to the final product. The production process in this factory is cyclical; the injection molding machine heats up the raw material that arrives in the form of plastic pellets and injects them into a mold that is capable of producing parts of a product (one or more items). After the material cools inside the mold, it is ejected and the mold recloses. A common short-term problem of the plastics factory is the selection of the mold injection

machine and the size of the mold that will be used for the production of a given product and its parts, when the production lot size is predetermined.’ The basic assumption, beyond

* Ben-Gurion University of the Negev, Beer-Sheva, Israel. ’ The problem presented here is a special case of a set of process selection problems. This set is described by

Johnson and Montgomery, Operations Research in Production Planning, Scheduling, and Inventory Control, 1974.

Journal of Operations Management 455

the model suggested here to solve the problem, is that a given product and its parts compete only among themselves for capacity and molds.

In the next section, the structure of the model that solves the problem will be presented. The economic parameters will then be explained. Next, the solution method will be described. Finally, the model is applied to a specific problem.

THE MODEL

Min 2 = 2 C CjtiYij + 2 2 PiYij/S i j i j

Set. 2 Yij = 1 V C nt

S C tiY, G Tj V j ni

where e=l,. . . ,M j=l i= 1:: 1: 1

K, is an index for labelling machine 2M - 1 is an index for labelling molds

Y, is a zero one variable that gets the value if mold i is assigned to machine j and 0 otherwise.

The parameters of the model are:

nj - the group of molds suitable for machine j (can withstand overweight) nc - the group of molds and machines that are suitable for producing item C. S - size of series (the required amount of products)

Tj - operation time of machine j ti - time cycle of mold i (this is the duration of time that passes between two identical

points in the work cycle of a given production process) Cj - cost of one hour operating time of machine j Pi - cost of mold i.

Within the framework that is defined above, the production cost of one unit of a product is brought to a minimum where production lot size is designated S. Alternately, it is possible to minimize the total cost of producing S units. The cost per unit is twofold: the machine’s operating costs, and the molds’ investment cost. These costs do not take into account the long run costs of investment in machines and auxiliary equipment.

The constraints of the model are: 2 Yij = 1 nt

is a constraint that ensures that every item will be produced in exactly one mold.

S z tiYij G Tj V j nj

is a constraint that ensures an operation of a given machine within its allocated time. It is clear that the number of constraints is M + IS.

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TABLE 1 Machine Data

j Machine Injection Weight pi ci

I 80 gr. 2,000,OOO Shekel 300 shekel/hour 2 200 gr. 4,500,OOO Shekel 700 shekel/hour 3 900 gr. 12,500,OOO Shekel 2 100 shekel/hour

DESCRIPTION OF THE ECONOMIC PARAMETERS OF THE MODEL

To produce a given product in a production lot-size S, several predetermined decisions must be made with regard to the amount of raw material, auxiliary equipment, and man- power that should be allocated. In addition, decisions are made in advance regarding the types of machines used, their allocated time, and the size of the molds that are being used in the process.

The efficient use of mold injection machines and molds depends on the following param- eters: Cj is the cost of the jth machine-hour that depends on the injection capability of the machine. The better a machine is able to inject a greater weight of plastic material, the more expensive is the cost per hour. This is the result of the size of the machine, its con- sumption of electricity and oil, and the use of spare parts.

In practice, it is possible to produce a certain weight on any machine. This required weight can meet the constraint of the maximal injection weight of the machine. It should be pointed out that in practice, the production capability of the machine depends on an additional factor that is not considered in this work, the locking force of the machine (surface area of the item multiplied by injection force) where the surface area depends on the form of the item and the injection force depends on the form of the item, and the various types of raw materials used. In practice, the locking force increases with the size of the machine. It should be noted that the model defined here is useful for an external problem that considers locking force as a constraint.

Pi, the cost of the mold i, is the main cost among the components of production costs. The following two factors affect the cost of the mold i which fits a certain machine: the nature of the product and the number of items in the mold. A technologically advanced product that requires greater precision or denser tolerances increases the mold cost. In addition, the mold cost increases as the mold includes more items. The mold cost per unit of output is computed with respect to S where the cost decreases as S increases.

In practice, S is estimated with some degree of uncertainty; the estimated value depends

TABLE 2 Item Data

Item number 1 2 3 4 5 6 7 8 9 Weight in grams 30 28 30 90 186 40 285 210 375 Time in minutes 1 1 2 2 2 1 3 3 3

Journal of Operations Management 457

on external requirements and its marketability. These factors are not known for certain to the production manager. Therefore, it is important to solve the problem or to measure the unit cost as a function of S. In some cases, where the sensitivity of S is demonstrated, the result may be to further improve the estimation process of S. This process often results from costly marketing research or other studies.

METHOD OF SOLUTION

The flow chart of Figure 1 describes the steps required to generate the database for the planning model. First, 2M - 1 molds are considered to produce a product of M items. A matrix A of size (2M - 1) X (M) that describes the structure of all the possible combinations of molds is derived. The elements of the matrix are either zero or one. It requires the value one in the (i, C) assignment if and only if the ith mold includes the Cti item. The multiplication of the ith row of this matrix in the vector of item weights

determines the total injection weight that is required to produce the ith mold.

FIGURE 1 The Model Data

PROVIDED DATABASE \7il, ?, S, M, K

I BUILD MATRIX A I

1

GENERATE NJ, Nj

REMOVE Tj DUE TO

TIME CONSTRAINTS

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The calculation of AG allows for the immediate determination of the groups:

“j = set of molds that can withstand the weight load of the jth machine ne = the set of molds and machines that can produce the C* item.

It is possible to produce this by examining all the rows of the matrix A that includes in the lth column the value one.

The multiplication of the ith row of A in the two vectors

LfN-l of the item times determines the total time ti required to produce the i* mold. By multiplying ti by S and comparing it to Tj, all the zero-one variables whose time deviates from the operating time of the given machine are removed.

EXAMPLE

In the following, a production problem that was faced by the plastics division in Kibbutz Ruhama will be presented and the solution will be compared to the previous solution that was implemented by the division.

In principle these data are fed into the zero-one linear programming model that was solved by the MPOSP package. If the number of variables is too large to use the package’s algorithm efficiency, then the number of variables can be reduced by using different heuristic rules. From experience, it is suggested that one consider only variables that use at least a certain percentage of the injection capability of the machine.

From Table 3 we observe that on the small machine j = 1; one mold is assigned. On the medium sized machine j = 2; two molds are assigned. On the large machine, two molds are assigned.

SENSITIVITY ANALYSIS

In addition, the solution of the problem depends on S. In general, a dual approach could be used to run a sensitivity analysis for the solution with respect to S. Instead, we preferred to run the program for different sizes of S and thus analyze the relationship among S and

TABLE 3 Solution

Machine in Use Weight in grams Data of Molds and Items

123456789

j=i 80 00 1000000 j=2 158 010101000 j=2 900 100000111 j=3 186 0000 10000

Total 1324 9

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TABLE 4 Sensitivity Analysis

S Z

The Objective

10,000 2506 20,000 1331 30,000 938 50,000 626

the unit cost and the assignment of molds to machines. In Table 4, how sensitive the solution is to S is illustrated.

Note that for all the values of S reported in Table 4, the assignment of molds to machine was the same. In fact, the change in the solution reflects only the value of the objective function. This change demonstrates the economics of scale. Indeed, the reduction in unit cost is almost linearly related to S within the studied range.

In principle, the following factors affect the sensitivity of the solution to S: Data of items. This determines the capability to produce them with different machines.

The more machines that can produce these items, the larger are the number of feasible solutions and the more sensitive the solution is with regard to S. In an extreme case, where the items can be produced on only one machine, the solution is completely insensitive to S. This is clearly an extreme case that is not encountered often.

Objective function coefficients. The objective function is based on two components. One is independent of S (Cj X Ti), and the other is (Pi), which is divided by S. Clearly, the solution is sensitive to changes in these components.

The values of S. In the range of values where S is close to zero, the sensitivity of the solution to S is much more significant than in cases where the values of S are suffi- ciently large.

COMPARISON TO PREVIOUS SOLUTION

In the previous solution the product was produced by two molds that were assigned to the large machine j = 3 and therefore Z was equal to 27 10. The percentage of saving in the unit cost that was obtained by the optimal solution was about 7.5%.

Finally, it is noted that the structure of the problem suggested here could be extended to consider factors that vary in the long run. This means that the predetermined decisions regarding the determination of auxiliary equipment and the machine (types and sizes) could be decided within a framework of a programming model. This extension would lead to a profit maximization problem instead of the cost minimalization problem. In the presence of uncertainty regarding the size of S, and within the organizational framework of the given factory, the profit problem is irrelevant. Thus, the suggested model is practical and can be adjusted to changes either in S or in the types of products produced.

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