Contiguous-cells transportation model for multi-objective non-pre-emptive production and maintenance scheduling

The long-standing issue of non-pre-emption in ‘multi-period’ production/maintenance (P/M) scheduling has always been a challenge and this has been the focus of many researchers. Recently, the Contiguous-Cells Transportation Model (CCTM) was proposed to address this problem. However, the CCTM is limited to non-pre-emptive P/M scheduling decision situations where only one objective is of concern. The multiple objective non-pre-emptive P/M scheduling are encountered more frequently in the real world than the single objective case. This study proposed the Multi-Objective Contiguous-Cells Transportation Model (MOCCTM) and the solution procedure for handling multi-objective cases of non-pre-emptive production/maintenance scheduling. The variables and parameters of the CCTM were adopted with modifications to cater for the multi-objective requirements. A composite multiple-objective function was formulated by employing multi-objective optimisation techniques of assigning weights to objectives and normalisation of objectives. An algorithm similar to the conventional least cost method was developed for the solution of the MOCCTM. A biobjective non-pre-emptive maintenance scheduling problem of 5 production machines across an operation and maintenance planning horizon of 10 periods was used to demonstrate the application of the MOCCTM. The MOCCTM is a good approach to solving multi-objective P/M scheduling problems in a non-pre-emptive environment.


Introduction
One of the earliest problems faced by man is the problem of allocation of limited resources. This problem exists in different forms, ranging from production of goods (where there is need to adequately schedule jobs to machines), inventory control, product mix and employment scheduling to rendering of services such as transportation of items from warehouses to customers. Researchers have invested efforts to address this long-standing issue with several models and approaches developed to handle the problem. The model that best describes this multi-dimensional problem is the well-known Transportation model (TM). The Transportation Problem is a special class of linear programs that has to do with shipping of commodities from sources to destination, with the objective of determining the shipping schedule that minimize the total shipping cost while satisfying supply and demand limits [1,2]. The TM is not limited to shipment of commodities due to the fact that the nature of some real-life problems can be described within the context of the TM. As a result, the flexibility of the model has prompted researchers to extend the application of the TM to solve problems of transshipment, inventory control, personnel assignment, production scheduling, etc. Several extensions of the TM are given in [2][3][4][5][6].
One area of application of the TM which is of interest is in Production/Maintenance (P/M) scheduling. The P/M scheduling poses the problem of allocation of periods of time to execute multiple tasks which can either be preemptive or non-pre-emptive. Pre-emption means to temporarily suspend an on-going task for another task of more importance, with the intention of resuming the suspended task later. Non-pre-emptive scheduling on the other hand has to do with completing series of activities without any form of interruption (i.e once the activity begins, it must run to completion without interruption) [7]. Pre-emptive scheduling can occur for reasons such as unavailability of materials and P/M facility breakdown among others. While pre-emption of tasks is the case in many real-life scenarios, there exist situations of P/M scheduling where it is not favourable to pre-empt tasks. A practical case such as petroleum refining is a production situation where pre-emption is not feasible as interrupting any of the production processes might impact greatly on factors such as; set-up cost, set-up time and product quality. Maintenance of a fleet of vehicles (such as; aircrafts, buses, etc.) and maintenance of certain facilities (e.g. steam turbine maintenance) are activities carried out without pre-emption due to the high set-up cost and set-up time associated with them.
While researchers have focused on developing models to handle cases of pre-emption in P/M scheduling, not much is known about cases where pre-emption of tasks is not favourable. This issue of non-pre-emption in P/M scheduling was addressed by the proposition of the Contiguous-Cells Transportation Model (CCTM) [7]. The CCTM was formulated with the aim of determining the set of contiguous periods for preventive maintenance that minimizes total maintenance cost. The CCTM has so far been applied to solve a non-pre-emptive maintenance scheduling problem of a fleet of ships and aircrafts, yielding feasible results. Charles-Owaba et al [7] modelled situations where cost minimisation was the sole objective. Multiple objective decision situations are more common in many real-life scenarios [5,[8][9][10][11]. A practical case is with the maintenance of a fleet of aircrafts. Asides being interested in minimizing the total maintenance cost of the aircrafts, it is also important to ensure that the aircrafts are ready as at when due, in order to reduce the contract penalty associated with tardiness of any aircraft, which imparts negatively on meeting customer demands. A similar situation is the maintenance of a fleet of ambulances in which availability of the ambulances is an important objective to be considered in order to have them available to respond to cases of emergency. These practical situations draw attention to the existence of a multiple-objective case in non-pre-emptive maintenance scheduling which is yet to be addressed by the CCTM. The focus of this study is the development of Multi-Objective Contiguous-Cells Transportation Model (MOCCTM) and solution procedure for non-pre-emptive decision situations in production and maintenance scheduling. The review of literature is presented in the next section followed by model development. Next, numerical example is given followed by the results and discussion and finally, the conclusion is presented.

Literature review
The TM, being a special class of linear program associated with resource allocation has been the interest of many researchers because the model has been found useful in solving real-life problems. Although the TM was initially formulated and applied to finding an optimal schedule for transportation of items with the aim of minimizing cost, researchers have explored the extent to which the model can be applied including practical cases that involve multiple and conflicting objectives as reported in [12,13]. The existence of such problems led to the development of the Multi-Objective Transportation Problem (MOTM). The MOTM is given by [14][15][16] and described as follows: (1) Where; Agl = supply amount at source g bgl = demand at destination Z1, Z2, Z3…. Zk = Objectives of interest to Decision Maker. Scheduling has a wide range of definitions depending on the context in which it is used. In [7], the scheduling problem was defined from a production perspective to be the case of finding the feasible and optimal sequence in which jobs pass through machines.
Maintenance scheduling was explained in [16] as the process of developing the sequence in which items are to be maintained as well as defining the methods and procedures to accomplish maintenance tasks. One underlying fact behind scheduling is that it has to do with the allocation of one form of resource or the other. Scheduling is associated with allocation of maintenance resources, plan production processes, etc. which in many practical cases, are just a few of many issues that have to be addressed. With the TM associated with the systematic allocation of resources, the scheduling problem situation can be modelled within the framework of the TM making the TM a useful decision-making tool for P/M scheduling.
In many practical situations of P/M scheduling, two or more objectives are relevant as a broad class of P/M scheduling objectives were identified in [7,8]. In these cases of multi-objective P/M scheduling, the problem is that of minimizing, maximizing, or a combination of minimizing and maximizing all the objectives simultaneously. The solution to such problems relies on multi-objective optimisation techniques which can only provide "compromise solutions" as against "optimal solutions" due to the conflict associated with the objectives. In most cases, the objectives in question are of different dimensions which cannot be evaluated over different scales. This inconsistency in dimensions of objectives also brings about the issue of skewness in computations which calls for scaling of the objectives by the process of normalisation. Various normalisation techniques and the effect of these various techniques in a multi-criteria decision making (MCDM) environment were studied by [2,[17][18][19]. Mathematical models for expressing the multi-objective scheduling problem are presented in [4,20]. The approach presented in [20] follows multi-objective optimisation principles which has to do with assignment of relative weights to each objective and aggregating all the objectives into a single function called the Linear Composite Objective Function (LCOF). The LCOF is as in equation (1) Over the years, it has been the concern of researchers to apply different optimisation approaches to solve P/M scheduling problems. Some of the approaches adopted to solve the problems are presented in [21,22,23].
The first attempt at formulating the scheduling problem as a TM was proposed in [24] by identifying the problems of balancing production overtime and inventory storage costs associated with production scheduling, so as to yield the least cost of production. An extension of the TM to maintenance scheduling is presented in [25]. By defining a gantt charting problem of scheduling machines on periods, Charles-Owaba [25] developed an extension of the TM for preventive maintenance scheduling of mass transit vehicles, with the objective of minimizing total preventive maintenance cost. Advanced works on the approach in [25] focused on optimisation of preventive maintenance cost and operational activities using fuzzy linguistic approach [26] and evaluation methods for preventive maintenance cost [27]. Although the situations in [25][26][27] expressed and solved the maintenance scheduling problem as a TM, the models developed were based on the assumption of pre-emption. The existing extensions of the TM also depict structures that can only handle pre-emption which renders them unfit to solve the long-standing issue of non-preemption of tasks in P/M scheduling. To address this issue, the contiguous-cells transportation model (CCTM) was proposed in [7]. However, the CCTM considers a single objective, as against multiple and conflicting objectives which is obtainable in many practical cases. This limitation of the CCTM is addressed by the development of the MOCCTM

Problem definition
The Multi-Objective Contiguous-Cells Transportation Problem is stated mathematically as follows: Given Q objectives to be considered, with a relative weight of importance assigned to a specific objective; M distinct items for processing; T periods available for alternate operations and processing, (in which processing is carried out without pre-emption); determine the set of contiguous periods yij (i.e. contiguous-cells) required for processing each item, which provides a compromise solution for the Q objectives.

Notations
Notations of parameters and variables of the MOCCTM are similar to those from the contiguous-cells transportation model, as presented by Charles-Owaba et al [7]. However, considerations are given to cater for the multi-objective situation of the problem. Notations used in the model formulation are as follows; i : Index indicating the item to be processed Observe that si = ki if activities or processing commence in item i as soon as it is ready or arrives and si ≥ ki if activities commence on i at a later period.
yij : A contiguous variable defined as;

Model Assumptions
Assumptions for the MOCCTM general formulation for preventive maintenance scheduling are adopted from the contiguous-cells transportation model, as presented in [7], with inclusions to cater for multi-objective situation of the problem. The assumptions of the model are as stated below: The total numbers of periods (T) are fixed and contiguous.
An item is either on queue or in for processing at any moment.
The time (period) in which an item is ready for processing is known.
An item visits the processing facility only once within a planning horizon.
An item that is ready for processing has its activities commencing only when resources are available. Otherwise, it waits.
The problem parameters are known.
Many objectives are considered and span over the contiguous period.
The length of a period may be in any unit of time (i.e. seconds, minutes, hours, days, weeks, month etc.).
When an item is ready, activities commence at the beginning of the period while completion is at the end of the period.
Pre-emption is not allowed (i.e. once processing begins on an item, it remains until completion without interruption).
A member in the set of objectives considered has to be independent.

General MOCCTM formulation
Let Q be the set of objectives considered; α be a decision objective and Uijα be the expression for objective α, with known dimensions. {where α = 1, 2, 3, 4…. Q} For any specific situation, the expression for Uijα must be known. Let Vijα be the dimensionless equivalent of Uijα. Then: Where rα connotes normalisation factor of α th objective. Adopting the Linear Min-Max normalisation method in [20], Vijα in equation (2) Also, let wα be the relative weight of importance of objective α, as decided by the model user and Vij• be the total value of all the K objectives considered. Then: Where; Since the contiguous-cells concept is that of completing the processing of items without pre-emption, then the objectives are evaluated as 'cumulative' over the set of contiguous periods. For contiguity, if ki is the period item i is ready for processing and H is any period in which processing is completed on item i (where ki and H are members of T); and Ṽij • is the cumulative value of all Q objectives, from the beginning of period ki to the end of period H. Then: '-' in the equation implies that the period is not contiguous with other periods.

Development of objective function and constraints
For the objective function, the actual period for processing item i is considered since processing of the item may not start at the beginning of the arrival period. Now let Ɣij be the value of the combined objectives for item i from the beginning of the start period (i.e ki or si) to the end of finish period H. Then Ɣij is defined as follows: Note that Ѵij represent the value of the combined Q objectives for processing item i in any period j within the set of contiguous periods for item i. Since Ɣij is for all Bi contiguous-cells, then Ѵij is given by the expression: For item i across T periods, the 'actual' cumulative value of the objectives becomes: For all M items, the cumulative value of the objectives across T periods (i.e. the objective function) is given by: The demand and supply constraints are given in equations (12) and (13) respectively as expressed by [10]: The supply constraint which caters for pre-emption is given by: Re-writing equation (11) from equations (8) and (9); and combining equations (4), (5), (12) and (13), the general MOCCTM is given by:

The Multi-Objective Contiguous-Cells Transportation Algorithm (MOCCTA)
The algorithm to be employed for solving the MOCCTM is similar to the conventional 'least cost' method of solving transportation problems. This algorithm will make use of allocation by maximum/minimum combined objectives (Ṽij•) and is stated as follows: Step 1 For each item, select the Ṽij• value in the last (Tth) period on the contiguous-cells transportation tableau, and divide the Ṽij• value by the total number of contiguous cells to obtain the average Ṽij• value.
Step 2 Start with the item having the lowest/highest average Ṽij• for maximize/minimize cases respectively.
Step 3 Allocate all the supply in that row (to satisfy the contiguity condition) whilst simultaneously meeting the demand constraint.
Step 4 Repeat the procedure from step 1 if the duration in the row and capacity in the column are satisfied for the scheduled item. But in case there is no capacity to accommodate the duration for the item, move to the next available period that can accommodate he capacity constraint for the duration on that row and schedule the item. (Note that moving to the next period for any item connotes re-scheduling of the item, thereby changing the start period from ki to si).

Solution procedure for the MOCCTM
Step 1 Obtain the values of the parameters of the problem (Q, T, M, , Bi, Aj, ki,).
Step 2 Convert the values of all Q objectives into dimensionless (i.e. normalised) values, using the linear min-max normalisation method in equation (3).
Step 3 Assign relative weights of importance to all Q objectives (as perceived by the model user), such that the sum of weights for all objectives must be equal to 1.
Step 4 Compute Vij• (i.e the total value of all the Q objectives) to be the value in each cell using equation (4), and by computing Ṽij• using equation (6), develop the MOCCTM transportation tableau as depicted in Table (1).
Step 5 Apply the MOCCTA to the tableau to determine the optimal yij for the contiguous perio, d. The optimal yij gives the optimal schedule considering all the Q objectives.
Step 6 Using the optimal yij obtained from the MOCCTM transportation tableau, determine the compromise solution to the Q objectives independently.

Numerical example
The proposed MOCCTM approach is illustrated with a hypothetical bi-objective non-pre-emptive maintenance scheduling problem of 5 production machines, across an operation and maintenance planning horizon of 10 periods with the objective of minimizing the total preventive maintenance cost and total production losses. The capacity of the maintenance facility is 2 machines per period.

Step1 Identify all Problem Parameters
The problem parameters are given in Tables 2, 3 and 4 below.
Number of objectives (Q) = 2; Note that the 'demand per period' is taken as 'available maintenance capacity per period', and the 'supply per machine' is taken as 'maintenance duration per machine.

Step 2 Normalization of the objectives
The normalized maintenance cost for machine 1 in period 9 (U191) is given by equation (3). From Table 3 Uij1(max) = 25, Uij1(min) = 9 and Uijα = 10. Using equation (3), the normalized = 0.94. Similarly, the normalized value for the second objective which production quantity for machine 3 for period 7 is 0.19. The normalized value of the maintenance cost and production for all the machines and periods are shown in Tables 5 and 6 as follows:  Table 6 Normalized production loss for all machines and period

Step 3 Assign Weights Objectives For the purpose of illustration
A weight of 0.6 is assigned to maintenance cost and 0.4 is assigned to production loss and each entry in Tables 5 and  6 were multiplied by 0.6 and 0.4 respectively, as shown in Tables 7 and 8. Observe that weights may be elicited from the Decision Maker or use any of methods mentioned in [20].

Step 4 (Computation of Vij•, Ṽij• and the MOCCT Tableau)
Vij• is computed using equation (4), by adding the corresponding entries in Tables 7 and 8. The Vij• values are computed and shown in Table 9.

Step 5 Application of MOCCTA
On applying the MOCCTA to Table 10, the order of schedule from the first machine to the last machine is obtained as follows: Since the case under study is a minimisation problem, we start allocation of cells with the machine with the highest average Ṽij• value and continue in descending order (i.e. machine, 5 -3 -1 -2 -4) as shown in Table 11: Table 11 Final Schedule

Step 6 Determination of compromise values of maintenance cost and production loss
Given the final schedule in Table 11, the respective values of maintenance cost and production loss are obtained from Tables 3 and 4 as presented in Table 12. The relevant parameters are defined as follows; MCi: Maintenance cost of i th machine over the Bi periods PLi: Production loss of i th over the Bi periods when it was under maintenance PLIi: Production loss of i th machine due to idleness. While waiting for maintenance work to commence on it.
OPi: Number of periods i th machine spent in operation producing units of products. Note that OPi = T -Bi.
IPi: Number of periods, the i th machine stayed in the maintenance facility while waiting for maintenance to commence. Observe that IPi = si -ki TPLi: Total production loss due to waiting time on queue and maintenance (i.e. total time in the maintenance system). That is, TPLi= PLi + PLIi

Results and discussion
The maintenance costs, production losses due to waiting time before maintenance, production losses due to time taken to actually carry out the maintenance and their respective durations are presented in Table 12.  Table 12 shows that the algorithm gives total operations period {3, 6, 7, 7, 7} with idle periods {4, 0, 0, 1, 0} for machines {1, 2, 3, 4, 5} respectively. The production loss for machines 2, 3 and 5 are losses accrued across the contiguous periods of maintenance. However, there is an additional production loss of 437 units for machine 1; and 75 units for machine 4 due to the idleness across periods 3 to 6; and 7 respectively on Table 11. The idleness is as a result of unavailable maintenance capacity on arrival of machine 1 and 4, leading to a re-scheduling. This takes the total production loss of machine 1 and 4 to 830 and 450 units respectively. The total preventive maintenance cost and production loss over the planning horizon are 214 million and 2,556 units respectively.
The merit of the proposed approach is the ability to give the best compromise preventive maintenance schedule in a multiple objective decision situation under the existing conditions. The computed values provide insight for management on areas requiring improvement for the system to be more efficient. For instance, in real world situations, attention of management would be how to reduce production loss due to time spent on queue at the maintenance facility.

Conclusion
This study proposed the Multi-Objective Contiguous-Cells Transportation Model as an extension of the Contiguous-Cells Transportation Model, to cater for cases where more than one objective is of interest in non-pre-emptive scheduling. The model was developed by adequately identifying and defining the variables and parameters of the problem. An efficient solution procedure of was also proposed, with a numerical example to illustrate the application of the model. The results obtained from the numerical example show that the model is a good approach that researchers can employ to solve multi-objective preventive maintenance scheduling problems in a non-pre-emptive environment. A proposition for further research is to apply the model to a real-life problem and to study the behaviour of the model in response to varying the parameters of the model.