1. Introduction
Optimal production, safety inventory, and manpower are concurrently determined given a set of manufacturing resources and limitations, Baykasoglu [
1] mentioned; the APP spans the production planning time to a range of 2 to 18 months, during which the planners may take into consideration each product category the overall production levels to meet predictable fluctuating demands, which is justified from an aggregate point of view [
2]. At the same time, such APP can provide production information to effectively utilize an organization’s resources to satisfy various demands. Aggregate production planning determines not only which output levels have to be met and what appropriate input mix of resources should be used, but also how to meet prescribed requirements at minimum cost without violating capacity limits [
3].
It is essential to aggregate the information for it is almost impossible to consider every detail related to the production planning without sacrificing its effectiveness in the long run. This planning approach can calculate an aggregate number of production problems, such as average item, weight, volume, production time, or dollar value. Productionquantities for each item or item group can be specifically determined upon the creation of the aggregate production plan that includes anticipated constraints. One of the benefits of an APP is that it provides multiple selections of mixes that contains either one or a combination of strategies to respond to demand fluctuation, such as overtime/idle time production rate adjustment, workforce size regulation, level production plus inventory policy, or sales shortfalls [
4,
5,
6]. In addition, managers may have the option of using subcontracting as a suitable alternative for part of the production schedule to reduce internal capacity shortages.
This work proposes a multi-objective planning (MOP) method to determine the most effective means of satisfying forecast demand by adjusting production rates, hiring and layoffs, inventory levels, overtime work, back orders, and other controllable variables. The objective functions of this APP decision problem are to minimize total production costs, carrying and backordering costs, also changing workforce levels under production constraints.
While pervasive approaches–such as global criteria methods, goal programming, fuzzy programming and interactive approaches–consider only the single criterion of shortest distance from goal(s) or the positive ideal solution (PIS), the technique for order preference via the similarity ideal solution (TOPSIS) provides a broader principle of compromise for solving multiple criteria, decision-making problems. The compromise control (ideal solution) minimizes the distance measure, provided that the nearest solution has the shortest distance to the positive ideal solution (PIS) and the longest distance to the negative ideal solution (NIS). The proposed method transforms multiple objectives into two objectives. The bi-objective problem can then be solved by balancing satisfaction through a max–min operator to resolve the conflict between the new criteria based on PIS and NIS. The development of the proposed approach is motivated by the following facts: (1) it combines MOP and TOPSIS to provide an easy way to solve a complex APP problem; (2) it can be efficiently coded when the problem is large in scale; and (3) this combined decomposition-based method gives better results than traditional methods for solving MODM problems [
7]. The efficiency of the decomposition-based method increases sharply with the size of the problem.
This work is organized as follows. First,
Section 2 presents some approaches to solving APP problems.
Section 3 presents the formulation of the model.
Section 4 provides a guide for developing the optimal overall production plan using ideal-solution principles. A case study in
Section 5 demonstrates the applicability of the proposed model to practical APP decision problems for developing the optimal overall production plan. Finally, conclusions are drawn in
Section 6.
3. Problem Formulation
The APP problem can be described as follows. It is assumed that a firm produces a single product to satisfy market demand over the planning horizon. The problem is to determine the most effective manufacturing mix that satisfies the projected demand by adjusting production rates, workforce sizes and loads, inventory levels, order status, supplier collaboration, and other controllable variables. The objective functions of this APP decision problem are to maximize sales revenue, to minimize total production costs, and to minimize repair costs as well. Thus, the APP problem can be derived into a mathematical model under the following assumptions:
Notations
The notations used in the model are given as follows.
Parameters
= sales revenue for product i ($/unit)
= production cost of regular time for product i ($/unit)
= production cost of overtime for product i ($/unit)
= inventory carrying costs for product i in each period ($/unit period)
= stock-out cost for product i in period t ($/unit period)
= regular payroll cost per worker in period t ($/man hour)
= cost of hiring one worker in period t ($/man-day)
= cost of firing one worker in period t ($/man-day)
= repair cost for product i ($/unit)
= defect rate for product i
= labor time for product i (man-hour/unit)
= regular working hours per worker per day
= fraction of regular workforce available for overtime use in period t
= initial workforce level (man-day)
= maximum workforce level available in period t
= planning horizon or number of periods
= initial inventory level (units)
= initial backorder level (units)
= maximum demand for product i in period t (units)
= minimum demand for product i in period t (units)
= regular time machine capacity in period t (machine-hour)
= fraction of regular machine capacity available for overtime use in period t
= machine time for product i (machine-hour/unit)
= the storage space limitation in period t
= fraction of subcontracting output available for product i
= maximum capacity limitation of subcontracting in period t
= minimum order quantity for subcontracting in period t
Decision Variables
= inventory level for product i in period t (units)
= backorder level for product i in period t (units)
= the unit of regular time production for product i in period t
= the unit of overtime production for product i in period t
= the unit of subcontracting production for product i in period t
= the number of workers in period t
= the number of workers hired in period t
= the number of workers laid off in period t
Objective Functions
This proposed APP problem considers three objectives: maximizing sales revenue, minimizing total production cost, and minimizing repair cost. The sales revenue indicates the market share. When market share is a main concern for a company, maximizing sales revenue becomes a primary objective in the APP problem. Repair costs can represent inner failure costs in the production process. In many cases, inner failure costs may be a chief goal because they impact the utilization of production resources. When inner failure costs are high, a company may be forced to use overtime production or more resources. In these cases, repair costs should be considered a separate objective, instead of a component of production costs. In this case, minimizing total production costs, maximizing sales revenue, and minimizing repair costs are all important considerations for the case company. Therefore, it is more appropriate to model them as three separate objectives so that the APP model can find a Pareto optimum that balances these three goals. Thus, we formulate a three-objective, multiple-period APP model for the case study as follows:
Maximize Sales Revenue
Minimize Production Costs
Minimize Repair Costs
The first objective function in (1) is to achieve the highest possible return from the quantities generated by regular production, overtime production, and contract production, including inventories and back orders. The second objective function in (2) is to minimize production costs, which contain three components. The first component involves production costs, subcontracting costs, inventory, and backorder level costs. The second component entails extra production loading or how many workers would have to be laid off to reduce overhead. The last component is the labor costs associated with regular-time workers. The third objective function in (3) considers the quality issue. Defect rates differ slightly across products. Management has set an acceptable (if not desirable) amount that the company is willing to pay in each period for repair costs. The first objective function is to maximize sales revenue and increase production quantities to achieve the highest possible goal. However, attaining the highest possible revenue may result in increased production costs in the second objective function. Similarly, increased repair costs may raise total expenses, which impacts total sales revenue. Most conflicts are generated by improving each objective function. Attempting to enhance each objective function may result in further conflicts. Hence, this paper proposes a compromised solution to solve this problem.
Constraints
After the three objective functions formulated in the previous section, nine constraints related to the APP model were set up as follows.
Constraint (4) ensures that the limit of the maximum available labor is respected in each period. Constraint (5) ensures that the available workforce in each period is equal to the workforce in the previous period, plus or minus any change in the workforce in the current period. The change in headcount may be due to the hiring of additional workers or the laying off of redundant workers. Constraint (6) states that because either net hiring or net laying off of workers occurs in a period but not both simultaneously. Constraints (7) and (8) ensure that the hours worked to produce products during regular and overtime hours are limited to the available regular and overtime hours. Constraints (9) and (10) specify that the quantity of products sold is between the minimum known demand and the maximum forecast demand. Constraint (11) ensures that either inventory or backorders are included in the solution, but not both. Constraints (12) and (13) ensure that the machine times for manufacturing the products are limited to the regular and overtime machine capacity in the manufacturing facility. Constraint (14) ensures that the inventory does not exceed the maximum storage space limit. Constraint (15) ensures that procurement quantities can meet the minimum order quantity, but do not exceed the maximum supplier capacity. Constraint (16) ensures that all decision variables are non-negative.