A Mixed-Integer Lexicographic Goal Programming Model for Achieving Estimated Targets in Multi-Product Systems

Multi-product systems, mostly business entities, are often faced with the challenge of achieving several target goals within a given period of time which are often conflicting and not measurable in same units. Even as it is most times impossible for such conflicting goals to be entirely optimally achieved, effort is made to minimize deviations from the estimated target of such goals. Priorities are sometimes also given to these goals such that the compromised solution obtained minimizes the deviation from these estimated targets according to a given prioritized order of importance. The Lexicographic Goal Programming technique is an appropriate method for solving such problems. In this paper, we present a Mixed-Integer Lexicographic Goal Programming model for minimizing deviations from estimated target of goals set by multi-product systems. To demonstrate the model, we focused on a multi-product production company (Nigerian Breweries PLC) and developed a Mixed-integer lexicographic model based on the monthly targets established by one of the production factories of the company for the year 2016. The goals considered include the estimated monthly profit target, monthly production target of each of the drinks (Star, Gulder, Maltina, Goldsberg, 33 Export and Fayrouz), estimated machine production time and estimated target distribution cost. These goals were categorized into three priority levels with the second priority normalized. The LINGO Optimization software was used to obtain the satisfactory solution based on the collected data. The result obtained showed that all the target goals were met. This shows that the Mixed-integer Lexicographic goal programming model is an appropriate technique for solving multi-objective problems in multi-product systems.


Introduction
Submitted to Data Research, page 34-42 www.itspoa.com/journal/data Goal programming (GP) is a mathematical technique for solving multiple objective decision making problems in which the objectives may even be conflicting. The Goal Programming technique has been applied in a wide range of planning, resource allocation, policy analysis and functional management problems [1]. Hence the design and application of GP models in solving multi objective models, especially in industry, is a popular problem area. Lexicographic Goal programming (LGP) is a special case of the goal Programming technique. The distinguishing feature of the LGP variant from other Goal Programming variants is the existence of a number of priority levels with each priority level containing a number of unwanted deviations to be minimized. In the lexicographic GP approach in a multi-product system, management sets some estimated goals for a certain period of time and assigns priorities to them. In applying this technique management only has to say which goal is more important than the other, it need not say how much more. With this information the LGP model tries to minimize the deviations from the estimated targets that were set by beginning with the most important goal and continue in such a way that the less important goal is considered only after the most important ones are satisfied or have reached the point beyond which no further improvement is desired [2]. In the final solution even if all the goals are not fulfilled to the fullest extent (of estimated targets) to give an optimal solution, the deviations will be the minimum possible giving what is called a satisficing solution. The LGP approach as well as other GP variants has been studied by many researchers and successfully applied to many diverse real life problems especially in industry.
Gasin in his study proposed a Lexicographic goal Programming approach with different scenarios to solve the aggregate production planning model with conflicting multi-objective functions in order to maximize the total net profit with limited investments, such as, budget, limited storage space, production capacity and resources of the company [3]. The LINGO software was used to obtain the optimal solution of the problem for which some goals were achieved while some others were not for the different scenarios. Also, Kellegoz presented a Lexicographic GP model for multiassembly multi-manned assembly line balancing problem with the primary objective of minimizing the total number of multi-manned stations (line length), minimizing the total number of workers as the secondary objective and smoothing the number of workers at the stations as the tertiary objective [4]. An experimental study was performed based on benchmark instances ranging in size from small to large for different target values of problem parameters. From the result obtained it was seen that only small size instances could be optimally solved while if the instances had target values attainable, then its optimal solution may be found more easily. Meanwhile a Lexicographic GP approach was presented for the optimal deployment of traffic police with different roads segments and shifts considered in [5]. The model was demonstrated with focus on the metropolitan city of Delhi (central) as a case study. The problem was executed using the GP package developed in visual C++. The satisfactory solution obtained showed the possible allocation of patrol cars to different road segments. Sensitivity analysis was also performed on the budget and number of patrol cars to identify the best possible solution. An efficient solution for solving Lexicographic linear GP problems has been presented in [6]. The procedure considered goal constraints as both objective function and constraints and then the problem is solved in an iterative tabular procedure in which the objective function becomes the prioritized deviational variables and is solved sequentially from the highest priority level to the lowest. The procedure was illustrated with an example and solution obtained which showed the proposed method as efficient and its formulation representing a better model than existing ones. An aggregate production planning model that best serves those companies whose aim is to have the best utilization of their resources in an uncertain environment while trying to keep an acceptable degree of quality and customer service level simultaneously while also taking into account the performance and availability of production lines has been developed in [7]. The proposed model which was a fuzzy model was first converted to an equivalent crisp multi-objective model and then Goal Programming was applied to the converted model. Using data obtained from an automobile parts manufacturing company, the IBM ILOG CPLEX optimization studio software was used to obtain the solution which showed significant importance of performance and availability of production lines in developing a real and practical aggregate production plan. A procedure based on analytic hierarchy process combined with a mean variance and Goal Programming model that made integrated asset management possible was suggested in [8]. The objective was to suggest a flexible approach that is relatively simple to use and that makes it possible to incorporate all factors, both objective and subjective, that are likely to influence the asset allocation decision. The procedure was illustrated with data from Canadian Mutual Fund over a period of five years and three months. The result obtained showed that the portfolios constructed in this manner performed better than the SDP/TSX 60 index, which is reference portfolio for the Canadian market. Also, a multi-objective integrated production planning model which was constructed doubly by resource and materials has been constructed in [9]. The model took delivery-on-time, reduced inventory, reduced overtime work, maintained safety inventory and its optimization objectives and achieved the integrated optimization of production planning, material requirement planning, resource requirement planning, inventory planning and overtime work planning. The LINGO software was used to derive the solution and simulation analysis which showed that the optimization method provided by the model has strong feasibility and effectiveness. A satisficing method of the Fuzzy goal programming model which holds that a more important objective has to be achieved as much as possible was presented in [10]. Compared to the weighted and lexicographic model, the relaxed preemptive priority requirement in the proposed model provides a more efficient, flexible and practical decision analysis support. The performance of the model was evaluated by comparing it with six existing models in literature to prove its efficiency over them. A proposal of a multiperiod and multi-stage model with multi-choice goals under inventory management constraints formulated by 0-1 mixed integer linear programming was made in [11]. The design task of the problem involved the choice of the pop up stores to be opened and the distribution network design to satisfy the demand with three multi-choice goals. The Revised multi-choice goal programming approach is applied to solve the mixed integer linear programming model and optimal solution was obtained that satisfied the demands with the three multi-choice goals. A goal programming model for the capital investment problem where the financial decision-makers preferences will be explicitly incorporated through the concept of satisfaction functions was formulated in [12]. The proposed models was illustrated using data from the Italian venture capital fund and in result obtained from the analysis not all the investment goal were satisfied.
In this paper, we present a mixed-integer Lexicographic Goal programming (MILGP) model for attainment of management objectives in multi-product systems. The achievement function of the MILGP model presented is such that some priority levels might be combination of goals while some others contain just single goals.
Some variables of the model are both non-negatively and integer constrained while others are just non-negatively constrained.

The Lexicographic Linear Goal Programming Model
Production planning is one of the most important activities in a production factory and entails how best to use the different resources available to achieve an optimal production plan that minimizes cost and maximizes profit [13,14]. Here, the general lexicographic goal programming model with m goals and priority levels for minimization of deviations from the estimated targets as presented in [15] will be used and is given as: Goal constraints represents the vector of n decision variables.

Proposed Mixed-integer Lexicographic Goal Programming Model for Multi-Product Systems
The mixed-integer lexicographic goal Programming model is hereby presented as follows.
For a multi-product system with products given as ̅ ( ) with q goals and l priority levels with ∑ for a given priority level as the case may be.
Preferential weights associated with the minimization of in the prioriy level The proposed model will have the achievement function given as Subject to ∑ Goal constraints System constraints ,

Assumptions of the Model
The following assumptions are stated for the Lexicographic GP model presented above Additivity: the level of penalization for an unwanted deviation from a target level is independent of the levels of unwanted deviations from other goals.
Proportionality: The penalization for an unwanted deviation from a target level is directly proportional to the distance away from the target level.
Mixed-integer value constraints: Some decision variables are integer constrained while some others just non-negatively constrained.
Certainty: The data coefficients of the model are known with certainty while the goal targets are regarded as initial estimates.
Priority Levels: Some priority levels have just one goal while some are the combination of goals which are taken to be of equal importance.

Data Analysis on the Mixed-integer Lexicographic GP Model
The mixed-integer Lexicographic GP model formulated above will be illustrated using data collected from a multi-product manufacturing company. Data was collected from a production factory of Nigerian Breweries Plc on production of the drinks-Star, Gulder, Maltina, Fayrouz, Goldsberg, and 33 exports. The data which are estimated values is shown in appendix A and includes the average monthly production quantity (in crate), average number of brews produced in a month, average number of cartons produced per brew for a month, quantity of raw material used per brew, average machine bottling time per truckload in a month, estimated monthly profit per truckload, and estimated distribution cost (fueling and drivers allowance) per truckload for each drink for the year 2016. The objective is the minimization of the following-underachievement of the estimated profit target, underachievement of the estimated production target levels for each of the drinks Star, Gulder, Maltina, Fayrouz, Goldsberg, and 33 exports, overachievement of the available machine bottling time and overachievement of the estimated distribution cost. This gives us 9 goals which are grouped into three priority levels as follows; Priority 1: Achieve the Profit goal.
Priority 2: Achieve the production targets goals for each drink and production time goal.
Priority 3: Achieve the distribution cost goal.

With
The achievement function of the model is given as a vector to be lexicographically minimized;

Subject to
Goal constraints (Production target for Star goal) (10)

Results and Interpretation
The mixed-integer lexicographic GP model on Nigerian Breweries Plc was solved using the LINGO optimization software. The formulated LINGO programme for the problem and the results are shown in the appendix B. The result is summarized in table 1 below. From the summary of the results shown in table 1 above we can see that each the goals in the different priority levels were satisfied. The monthly production targets of 506, 250, 72, 220, 397, 264 truckloads for each of the drinks Star, Gulder, Maltina, Fayrouz, Goldsberg, and 33 export respectively were met, the monthly distribution cost was satisfied as targeted while there was a slight overachievement by 2452.26 Naira in the profit target as well as the machine bottling time for the production of all the drinks been achieved in 628.2 hrs out of the targeted 720 in the month. Also the quantity of the raw materials-malted sorghum, malted barley, white sorghum and sugar used in the production was 74166kg, 227156kg, 312312kg and 36492kg respectively while the quantity of brews used was 86309.0 hectoliters.

Checking Solution for Lexicographic Redundancy
The solution obtained was checked for lexicographic redundancy by checking the solutions in the priority levels 1, 2 and 3. From the solution obtained it was seen that the satisfaction of the goals keeps improving as each priority level is considered from priority 1 to 3 and the goals are completely satisfied when all the priority levels are considered in the final analysis. Hence there is no problem of lexicographic redundancy.

Conclusions
This paper presents mixed-integer lexicographic goal programming (MILGP) model that can be applied by management for multi-product systems in achieving set targets for a given period of time. Although in most goal programming problems it is difficult to guarantee an optimal solution or a solution in which all the goal targets are achieved, but from the data illustration of the MILGP model in this work we have achieved a satisficing solution. The MILGP model presented in this paper has an achievement function in which a priority level can be either a single goal or a combination of goals and the variables are mixed-integer constrained. The functional constraint of the model is made up of both the goals constraints and the system constraints. The MILGP model which can be applied in any system with multiple goals and given priority levels was illustrated with data from Nigerian Breweries PLC with the objective of minimizing the underachievement of the estimated profit target, underachievement of the estimated production target levels for each of the drinks Star, Gulder, Maltina, Fayrouz, Goldsberg, and 33 export, overachievement of the available machine bottling time and overachievement of the estimated distribution cost. The satisficing solution obtained using the LINGO software showed that all the goals according to the priority levels were achieved. Hence the model can be considered as efficient and can be applied in other multi-product multi-goals systems. The model may still be improved upon by considering the case of an increase or decrease in per unit penalty function instead of the per unit penalty of the weights for every unit of deviational variable from the target level as considered in this paper.