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The traditional production planning model based upon the famous linear programming formulation has been well known in the literature. However, the capacity constraints in such a model may not correctly represent the actual situations of the shop floor, as pointed out by Byrne and Bakir (International Journal of Production Economics 59 (1999) 305–311). A hybrid approach was proposed by them, applying simulation and a linear programming model iteratively, to find the capacity-feasible production plan. This paper proposes an extended linear programming model for a similar hybrid approach. At each simulation run, the actual workload of the jobs and the utilization of the resources are identified. The information is then passed to the linear programming model for calculating the optimal production plan with minimum total costs. Through the case study, it is shown that the proposed approach finds the better solution in a less number of iterations compared to the approach by Byrne and Bakir .

The classical linear programming (LP) models for planning production have been around for many years. A typical formulation of the LP planning model has the objective of minimizing the total production-related costs, such as variable production costs, inventory costs, and shortage costs, over the fixed planning horizon (see, for example, [1] and [2].) The usual constraints employed are: (1) inventory balance equations for making the inventory and/or shortages balanced with those from the previous period, production quantity, and the demand quantity, (2) capacity constraints which ensure the total workload for each resource (labor, machine, etc.) not exceed the capacity in each period. A formal description of the classical LP formulation is given in the next section. Even though the classical LP model does consider the limited availability of the resources through the capacity constraints, it is known that such capacity constraints may not correctly represent the complex behavior of the resource consumption in a real production system. Byrne and Bakir [3] recently showed that the solution from the classical LP planning model may be infeasible for the real production system, due to the non-linear behavior of the workloads at the machines. The reason for the infeasibility of the LP solution is that the classical capacity constraints assume the workloads from the decision variables (i.e., the production quantities) to be linear in each period. The total workload in a period is defined as the sum of the processing time multiplied to the production quantity for each product type at the machine. The capacity constraint says the workload should be less than the capacity of the machine in that period. There are two ‘unrealistic’ parts in the classical capacity constraints. First, in the real production system, the workload of the production quantity at a machine as calculated above seldom occurs in the same period as the production quantity is out. For example, let us assume 100 units of a product type are to be produced in period 5. To produce the type, we assume 3 machines are used, and each machine handles one processing step for total of 3 steps. At each machine, 1 hour of unit processing time is required. In the classical capacity constraints, the workload of (1 hour×100 units) will occur at each machine, and it must not exceed the machine work hours in period 5. However, in the real production system, the workload on a machine may not entirely occur in period 5. Instead, the workload may be divided into two periods, say 5 and 6, so that the last machine (machine 3) may finish 100th unit by the end of period 6. Thus, to correctly describe the actual workload at each machine, it is necessary to figure out how much of the total workload will occur in each period. This task is not an easy one due to the fact that the profile of the workload depends upon the production plan, operation rules, and the system configuration. The second unrealistic part in the classical capacity constraint is the right-hand side (i.e., the available amounts of the resources) of the constraint. It is assumed that the full working hours of the machines can be utilized by the production output in the corresponding period. However, even in a deterministic system without failures of the machines or yield loss, it is not possible to achieve it. For example, buffer limits or the operational sequence of the jobs may create some idle times for a machine even if the machine is available. One may adjust the capacity by multiplying the average utilization to the full capacity, but it does not exactly count the ‘feasible’ amount of the resource capacity. The actual amount of the capacity to be allocated to the requirements for each machine in a period is hard to estimate a priori. It also depends upon the product mix, operating rules, and the system configuration. To overcome such shortcomings of the classical capacity constraints, Byrne and Bakir [3] proposed a hybrid approach using a simulation model to support the LP planning model. The proposed approach is based on imposing ‘adjusted capacities’ derived from the simulation model results, which is recursive in structure as follows: Step 1: Solve LP model and find the optimal production plan. Step 2: Using the current plan, run the simulation model. Step 3: If the current plan is found feasible through the simulation, terminate the iteration. The current plan is the final solution. Otherwise, continue. Step 4: Calculate the adjusted capacities for the LP capacity constraints based on the simulation results. Use the new capacities for the capacity constraints of the LP model. Step 5: Go to step 1. In step 4, the proposed adjustment is made through the following formulation: (Adjusted capacity of a machine for period t)=(Full capacity of a machine for period t)×(Fraction of the full capacity of the machine actually consumed in period t during the simulation run). Thus, in each iteration of the procedure, the simulation results are reflected to the right-hand sides of the capacity constraints by multiplying the utilization fractions. The iteration terminates if the current LP plan is identical to the previous one, i.e., the plan is feasible to the capacities of the resources. The approach by Byrne and Bakir [3] solved the second part of the shortcomings mentioned above. However, the assumption that the workloads by the production quantity in a period occur in the same period still needs to be generalized. To solve this part, Hung and Leachman [4] proposed a similar approach to Byrne and Bakir [3] in which the LP model is revised by the results from the simulation runs iteratively. Hung and Leachman’s [4] approach modified the left-hand sides of the capacity constraints considering the estimated flow times of the production starts from the simulation. The flow times of the production starts are used to calculate the workloads at each machine in each period. Since their calculation procedures are somewhat lengthy, we refer the interested readers to Hung and Leachman [4] for the details. They assumed the full capacity for the right-hand side of the capacity constraint as the classical model. In this paper, we combine and extend the ideas of the above two approaches to find the capacity-feasible production plan using the LP model and the simulation. We propose a methodology to properly modify both the left- and right-hand sides of the capacity constraints in the LP model. The simulation model is adapted as a supporting tool for accurately assess the two measures: (1) how much workload is occurred for each machine in every period by the current production mix and volume, and (2) how much of the full capacity is actually consumed by the current plan for each machine in each period. These two measures are used to modify the capacity constraints of the LP model, and the new LP is solved to generate a new production plan. As in Byrne and Bakir’s [3] procedure, similar iterations are performed until the plan converges. Before we go on to explain the details of the proposed approach, a quick review of the related research follows. Several researchers have proposed methodologies to overcome the drawback of capacity planning in the traditional production control systems, based on the standard manufacturing resource planning (MRP) concept [5]. Some researchers concentrate on capacity adjustment and lead time management, setting due dates in make-to-order company [6] and [7]. Özdamar and Yazgaç [6] develop a production planning system with a multi-stage process flow. In their system, a linear mathematical model is solved for obtaining a capacity-driven master production schedule (MPS) that is feasible in terms of capacity utilization. In MPS module, an iterative algorithm that calculates the adjusted capacity considering setup times is proposed. At each iteration, regular time capacity is further decreased to reserve some space for setup times. The MPS provides order due dates and is used to smooth the total workload by solving a binary LP model. However, this approach adjusts the capacity by only deducting the average of additional overtime from the full capacity over the whole planning horizon. Zijm and Buitenhek [7] propose a framework for capacity planning and lead time management. They develop a rough queueing model to find approximations of the mean and the variance of the lead times, and exploited an aggregate scheduling procedure to calculate workload-dependent planned lead times. Then, they propose a due date setting policy using aggregate scheduling. Tall and Wortmann [8] integrate finite capacity planning and MRP to avoid capacity problem. In their planning algorithm, a production plan is generated against finite capacity with an iterative simulation to estimate the lead times. When solving the capacity problem, the algorithm use alternative routings, resources, safety stock, re-planning of production orders, and demand. Even though the above approaches avoid the disadvantage of the MRP systems, they do not consider the profile of workload explicitly over the planning horizon and not exactly count the ‘feasible’ amount of the resource capacity. In the following section, we describe the details of the proposed approach. A simple numerical case is presented in Section 3 with the comparison against the Byrne and Bakir’s [3] approach. The final section covers the summary and the discussions for further research.

This paper proposed an iterative approach for finding the capacity-feasible production plan, applying the hybrid framework by Byrne and Bakir [3]. An extended formulation of LP model is proposed to consider the workload profile of the production quantity and the actual amount of the capacity to be allocated to the requirements for each machine. The LP model is modified by the information collected from the simulation, and the iteration continues until the final feasible production plan is obtained. Using the same case given in Byrne and Bakir [3], the comparison results show that the proposed approach (with method (A)) generates the plans with less total costs in less numbers of iterations for the cases with and without backloggings. As mentioned in Byrne and Bakir [3], extending the approach to the case of probabilistic system may not be difficult due to the flexibility of the simulation model. However, convergence of the solution may not be guranteed in that case. Further research on finding the solution in a less number of iterations would be possible. Application of the analytical model as a substitute or a supplement for the simulation model may be considered to reduce the total solution time. As mentioned in the previous section, it may be necessary to test and compare the impacts of the different input control policies, different system configurations, and so forth. An integrated and systematic approach utilizing the proposed method to find the best operational controls and configurations for the production system would be possible.