Linear Goal Programming Model Research Articles

A novel intuitionistic fuzzy best-worst method for group decision making with intuitionistic fuzzy preference relations

This paper proposes a new intuitionistic fuzzy best-worst method (IFBWM) for group decision making (GDM) with intuitionistic fuzzy (IF) preference relations (IFPRs). IF values (IFVs) are used to express reference comparisons of criteria. Based on the additive consistency of IFPRs, this paper proposes the definition of additive consistency of IF reference comparisons (IFRCs). Based on the deviation minimization, a linear goal programming model is established to calculate the optimal IF priority weights. By the additive consistency, the consistency index in the closed form is computed by the score function of IFVs. A new approach is devised to enhance the additive consistency of IFRCs. Thus, an IFBWM with the additive consistency of IFRCs is proposed. For GDM with IFPRs, the best criterion and the worst criterion are identified for each decision maker by constructing the score matrix of the IFPR. The individual ranking order of the criteria is obtained by using the proposed IFBWM. Then, the collective ranking order of the criteria is obtained via building a 0–1 integer programming model. Therefore, a GDM method based on the developed IFBWM with IFPRs is proposed. Four examples are analyzed to demonstrate the effectiveness and the advantages of the proposed IFBWM for GDM.

Precision nutrition for type 2 diabetes in Benin: leveraging linear goal programming to optimize diets with emphasis on adequacy, affordability, accessibility, and culture.

Nutrition and diet are critical to managing Type 2 diabetes (T2D). Low-income households often face challenges maintaining a healthy and balanced diet due to food insecurity, availability, and cost. To address this issue, we used a linear goal programming (LGP) model to develop nutritionally adequate, affordable, accessible, and culturally acceptable diets for persons with T2D in Benin, a French-speaking sub-Saharan country. The goal was to help persons with T2D manage their condition more effectively. We compiled a robust list of local commonly consumed foods in Benin, and calculated their nutritional value using West African food composition tables and food costs per serving from a market survey. Using mathematical optimization techniques, we designed dietary plans that meet the daily nutrient intake recommended by the World Health Organization (WHO) to prevent chronic diseases in normal adults. While adhering to dietary constraints of T2D, we developed optimized diet plans with varying energy levels that meet all nutrient requirements while considering availability, acceptability, and budgetary constraints. Fifty-two food items and recipes were evaluated to create six low-cost daily menus. Menu 1 was the most affordable at CFA 1,127 (USD 1.88), providing 1890 kcal of energy, while Menu 6 was the most expensive at CFA 1,227 (USD 2.05), providing 1749 kcal. All the menus met the daily WHO minimum requirements for carbohydrates, fat, cholesterol, and fiber content, while other nutrients such as protein, vitamin C, and iron reached the upper limits of the acceptable value range. Linear goal programming can be an effective tool in helping to obtain optimized adequate, accessible, and culturally acceptable diets at minimal cost by interpreting and translating dietary recommendations into a nutritional model, based on local market prices.

Interval-valued intuitionistic fuzzy best-worst method with additive consistency

Best-worst method (BWM) has received extensive attention because of its fewer pairwise comparisons and maintenance of consistency. Extending it to various fuzzy environments has become a hot research spot. This paper initiates a new interval-valued intuitionistic fuzzy (IVIF) BWM with additive consistency. Firstly, an IVIF fuzzy preference relation (IVIFPR) is viewed as a directed graph with IVIF weights. An approach is offered to distinguish the best and worst criteria by the defined score out-degree and accuracy out-degree of criteria. The concept of IVIF reference comparisons is introduced. The additive consistency of IVIF reference comparisons is acquired by that of IVIFPR using its left and right intuitionistic fuzzy preference relations. We derive the relationship between the IVIF reference comparisons and the IVIF priority weight vector. Based on this relationship, a linear goal programming model is constructed to generate the IVIF priority weights. Two consistency indexes are analytically obtained and the consistency ratio is then defined to check the consistency of IVIF reference comparisons. For the unacceptably additive consistency, a new approach is put forward to enhance the consistency. Thereby, the proposed IVIF BWM is summarized step by step. Three examples of applications are furnished to demonstrate its effectiveness and validation.

Developing feasible healthy diets for Ethiopian women of reproductive age: a linear goal programming approach.

To develop a healthy diet for Ethiopian women closely resembling their current diet and taking fasting periods into account while tracking the cost difference. Linear goal programming models were built for three scenarios (non-fasting, continuous fasting and intermittent fasting). Each model minimised a function of deviations from nutrient reference values for eleven nutrients (protein, Ca, Fe, Zn, folate, and the vitamins A, B1, B2, B3, B6, and B12). The energy intake in optimised diets could only deviate 5 % from the current diet. Five regions are included in the urban and rural areas of Ethiopia. Two non-consecutive 24-h dietary recalls (24HDR) were collected from 494 Ethiopian women of reproductive age from November to December 2019. Women's mean energy intake was well above 2000 kcal across all socio-demographic subgroups. Compared to the current diet, the estimated intake of several food groups was considerably higher in the optimised modelled diets, that is, milk and dairy foods (396 v. 30 g/d), nuts and seeds (20 v. 1 g/d) and fruits (200 v. 7 g/d). Except for Ca and vitamin B12 intake in the continuous fasting diet, the proposed diets provide an adequate intake of the targeted micronutrients. The proposed diets had a maximum cost of 120 Ethiopian birrs ($3·5) per d, twice the current diet's cost. The modelled diets may be feasible for women of reproductive age as they are close to their current diets and fulfil their energy and nutrient demands. However, the costs may be a barrier to implementation.

Short-term planning of a work shift for open-pit mines: A case study

This work deals with the short-term planning problem of a work shift for open-pit mines. The problem involves ore and waste fronts, shovels, heterogeneous truck fleets, and discharge points. The allocation of trucks is dynamic to allow multiple routes to be assigned to each truck. The problem consists of deciding which fronts must be mined and establishing the number of trucks, their routes, and the amount of material transported by them to each discharge point, satisfying a stripping ratio at the desired level. The objectives are to minimize the deviations from the targets for production, chemical grade, and particle size range of each control parameter at each plant and reduce the number of trucks needed for the process. To solve the problem, we developed a mixed-integer linear goal programming model and tested it using real data from an iron ore mine. The results showed that the proposed approach supports decision-makers in the sizing and allocation of truck fleets and in meeting the production and control parameter targets required by the ore processing plants according to the daily scenario, such as low availability of shovels and trucks, flexibility in ore quality, and need for increased production.

Study of goal programming approach in resource allocation in acute care hospitals

In this study, the goal programming approach is discussed which is used in hospitals to allocate resources. Two linear goal-programming models are included in the process. One model fixes the ratio of different types of cases that doctors see each year, while the other translates case mix decisions into a set of practice changes for doctors. Decision-makers may use the models to find the optimal case mix that maximizes profitability for the institution while minimizing negative impacts on physician compensation and clinical workflow.

A nested multiple-objective optimization algorithm for managing production fleets in surface mines

The fleet management system (FMS) plays a vital role in truck and shovel open-pit mining operations. The list of expectations from the FMS ranges from data recording to making operational decisions. This article proposes a nested fleet management system (N-FMS) for open-pit mining operations. The primary contribution of the proposed FMS is connecting the operation to the strategic plan by incorporating shovel allocation and plant feed optimization. Another contribution of the proposed N-FMS is that it simultaneously optimizes the utilization of the shovel fleet and truck fleet. The proposed FMS makes decisions using two nested multiple-objective mixed-integer linear goal programming models. Results of the implementation of the developed N-FMS in a metal mining case study show that compared to a locked-in operation, the mine experienced a 14.6% improvement in its required truck fleet capacity to meet the production target.

Evaluation of insurance companies considering uncertainty: A multi-objective network data envelopment analysis model with negative data and undesirable outputs

Uncertainty is an important issue to consider when evaluating entities in both public and private sectors. On the other hand, many operations have more than one stage process when some inputs are fed to the system to produce a number of intermediate measures. The intermediate measures are then transformed into final products in the subsequent stages. The composition method in network data envelopment analysis (NDEA) is a popular method for measuring the efficiency of a two-stage process. The composition method is fractional bi-objective programming that is solved by non-linear programming techniques such as bisection search. In this paper, the two-stage NDEA is extended with negative data and undesirable outputs. First, we propose an alternative linear model based on the goal programming technique to avoid complex non-linear calculations. Then, we use a method to transform negative data into positive and undesirable outputs into desirable ones. Finally, we develop the proposed model using the fuzzy α-cut approach in order to incorporate data uncertainty in the linear goal programming (GP) model. To validate the accuracy of the proposed model, a numerical example is solved. To show the applicability of the proposed model, a real case of 22 insurance companies is examined. We also perform a comparative analysis to specify the benchmark and inefficient companies. Comparative analysis can help managers to recognize where improvement should be investigated with priority.

Introducing a New Approach in Stand Tending Planning and Thinning Block Designation by Using Mixed Integer Goal Programming

Long-term management plans have been developed for nearly all of the forests in Turkey. These plans are applied at a sub-district management unit level and may contain guidance for both intermediate yield and final yield harvests. To implement an intermediate yield plan, which involves the scheduling of forest thinnings (stand tending), consideration in Turkey is given to the advantages of working in the same terrain and the same general area each year. Therefore, compartments are often clumped together to create thinning blocks, taking into consideration the thinning priority of the stands, road conditions, site index, age, and proximity of the compartments. Further, when preparing annual budgets and planning to meet the market’s needs, forest enterprises require an even flow of intermediate wood volume each year. In this paper, we introduce a new approach in stand tending planning designed to schedule an equal amount of intermediate wood volume each year and to create thinning blocks by minimizing the distance to pre-defined ramps (landings). We developed both linear and nonlinear goal programming models to minimize both the deviations from a harvest volume (annual intermediate yield allowable cut) target and the deviations from a target value determined for the distances (total and average) of the centroid of each compartment to the hypothetical forest ramps. By using the extended version of Lingo 16, we solved the problem with different weights for the deviations in volume and distance that ranged from 0.0 to 1.0, in 10% intervals, which created 11 scenarios. We carefully analyzed the results of each scenario by taking into consideration the wood volume and distance of compartments to the ramps. The best scenario using the linear model produced a deviation in volume scheduled for the entire decade of 6 m3, while the deviation in total distance between harvest areas and ramps was 59.7 km. Scenario 5, with weights of 0.6 for volume and 0.4 for distance, produced these results, where compartments were closest to one another. The best scenario using the nonlinear model also produced a deviation in volume of 0 m3 and the total average deviation in distance between harvest areas and ramps was 8.7 km. Scenario 3, with weights of 0.8 for volume and 0.2 for distance, produced these results. The approach and models described through this study may be appropriate for further integration into forest management planning processes developed for the planning of Mediterranean forests.

A multi-objective model for fleet allocation schedule in open-pit mines considering the impact of prioritising objectives on transportation system performance

ABSTRACT This paper deals with optimal decisions regarding truck and shovel scheduling problem in open-pit mines. A mixed-integer linear goal programming model (MILGP) is formulated for solving the problem, aiming at four major goals: (1) maximise production, (b) minimise deviations in head grade, (c) minimise deviations in tonnage to the ore destinations, and (d) minimise fuel consumption of mining trucks. This paper also examines the extent to which the priority of objectives can affect the results and efficiency of the mining operation. Implementation of the model with a copper mine case study demonstrated that the proposed model is effective and efficient.

A New Hybrid LGPMBWM-PIV Method for Automotive Material Selection

Efforts are continuously being made by researchers to improve fuel efficiency and to reduce CO 2 emissions from the passenger cars. To achieve these goal, recent trend is to make the cars components light in weight for which manufacturing car roofs using natural fiber reinforced composites (NFCs) is one of the method. Several natural fibers (NFs)are available as alternative reinforcements for the fabrication of NFCs. Different NFs possess different properties and therefore, it is necessary to select the most appropriate natural fiber for fabrication of the composites which in turn will lead to the desired performance of the vehicle. Selection of the optimal natural fiber, amongst the several alternatives, is basically a multi criteria decision making (MCDM) problem as selection is based on the evaluation of several conflicting criteria. In this study, twelve alternative natural fibers (Flax, Hemp, Jute, Kenaf, Ramie, Okra, PALF, Coir, Isora, Cotton, Banana and Sisal) and six evaluation criteria (Tensile strength, Stiffness, Failure strain, Density, Degradation temperature and Moisture gain) are considered and selection of the optimal NF is made using a newly developed hybrid MCDM method i.e. Linear goal programming model for Best-Worst method (LGPMBWM) and Proximity index value method (PIV). Results of the study reveal that among all considered natural fibers, Ramie fiber is the most suitable alternative for the fabrication of composites and coir fiber is the worst candidate for the same.

Using Mixed Integer Goal Programming in Final Yield Harvest Planning: A Case Study from the Mediterranean Region of Turkey

A mixed integer goal programming model is developed to address the regeneration planning problems of even-aged forests in the Mediterranean region of Turkey. The unique aspect of the goal programming formulation is to minimize deviations in scheduled wood product volumes and the size of harvest areas within each time period, as these are important goals for the management area. About 98% of the forests in Turkey are considered even-aged, and 2% are uneven-aged. Therefore, an age class method is used for the planning of even-aged forests. For the areas where this method is applied, reaching the optimal age class structure is the first priority. This involves implementing final harvests (clearcuts) to regenerate an amount of forest area into each age class. To meet the local market’s needs, forest enterprises also require the final yield to be fairly equal each year. Further, it is desired that the harvest area (regeneration area) is relatively equal each year, to address operational considerations. A linear goal programming model is developed to address the problem. The minimization of deviations from both the harvest area and harvest volume targets are incorporated as goals in the objective function of the model. Several scenarios are solved using the extended version of Lingo 16. A scenario with weights of 0.8 for area and 0.2 for volume produces the best results. Here, the total deviation for 20 years is 3.8 ha in area and 2889 m3 in volume. In the actual regeneration plan, the area deviation for 10 years is 54.72 ha (6.2% of total regeneration area), and the volume deviation is 20,472 m3 (9.8% of harvest volume). The model described through this study can be developed further and integrated into forest management planning software and processes used for the planning of even-aged forests in the Mediterranean region.

Gender Differentials and Optimal Combination of Crop Enterprises under Limited Resource Conditions in South Western Nigeria: A Case Study of Cocoa-based Farming Systems

This study investigated gender differentials and an optimal combination of crop enterprises in southwestern. Specifically, it described the socio-economic characteristics of households in cocoa-based farming by gender; determine enterprise combinations of households in the cocoa-based farming systems and profile gender participation in optimal farm plan activities in cocoa-based farming systems. A multistage sampling procedure was used to select 200 respondents, comprising 147 males and 53 females. Data were collected on socioeconomic characteristics such as age, household size, years of schooling and year of farming experience among others. Data were analyzed using descriptive statistics and linear goal programming. Descriptive statistics show that majority of the respondents aged above 60 years comprising of 46.5% male and 2% females, married comprising of 69.5% male and 20% female, and had a household size of between 7 and 12 persons, with males accounting for 25.5% and females 12.5%. Most of the respondents had primary school education in the study area, with 73.8% male and 26.2% female. Goal ranking showed that food security; increased income; and reduced production cost are the most important goals among farmers in the cocoa-based farming system. The linear goal programming model shows that Cocoa and Banana; Cocoa, Oil Palm and Plantain and Cocoa, Maize, Cassava and Orange are the optimal enterprise combination. It further showed that farmers will incur a cost of N38, 833.56/ha on the three basic activities to achieve the stated goals. The shadow prices for the fully utilized resources were N9348, N1352 and N6.52 for land, hired labour and capital respectively. The result further showed that 67 percent of the goal components was achieved. Percentages of female-headed households in cocoa/banana, cocoa/oil palm/plantain and cocoa/maize/cassava/orange enterprises were 45%, 31% and 0% respectively. For the goals of food security, increased income and reduced farm production costs to be accomplished in the study area, farmers should produce 0.2 ha of cocoa/banana, 1.2 ha of cocoa/oil palm/plantain and 1.25 ha of cocoa/maize/cassava/orange in the study area. Also, conscious efforts should be made by policymakers to ensure gender equality in the access and use of productive farm resources.

Consistency Analysis and Priorities Deriving for Pythagorean Fuzzy Preference Relation in the “Computing in Memory”

Pythagorean fuzzy set, characterized by membership function and non-membership function, has received increasing attention in recent years. In this paper, a new approach to decision making is proposed based on Pythagorean fuzzy preference relation and its additive consistency. Firstly, the concepts of Pythagorean fuzzy preference relations and its additive consistency are introduced, and followed by a discussion of their desirable properties. Then, a linear goal programming model is proposed to determine the consistency of PFPRs. For the PFPRs that does not satisfy the consistency, the consistency index is defined to measure the degree of consistency, and a consistency adjustment algorithm is proposed. Finally, based on the additive consistency, a new algorithm for decision making is presented. An example of CIM(Computing In Memory) is provided, and in comparison with other methods, the validity and rationality of the proposed method are verified.

A two-step approach to incorporate cut-off grade and stockpiling in oil sands mine planning optimization framework

In achieving maximum benefit in oil sands mining, the time and sequence of removing ore, dyke material and waste from the final pit limit is essential to the long-term production schedule. In-pit waste management strategy requires the simultaneous construction of dykes with the advancement of mining operations. This paper seeks to determine: (1) the time and sequence for removal of ore, dyke material and waste to maximize Net Present Value (NPV); (2) the quantity of dyke material required for dyke construction to minimize construction costs; and (3) the impacts of stockpiling and stockpile reclamation with limited time duration. An Integrated Cut-Off Grade Optimization (ICOGO) model was used to generate an optimum cut-off grade policy and a schedule for mining ore and waste, as well as overburden, interburden and tailings coarse sand dyke materials in long-term production planning. Subsequently, a Mixed Integer Linear Goal Programming (MILGP) model was developed to generate a detailed production schedule for removal of ore, waste and dyke materials from the final pit limit. The cut-off grade profile and schedule generated by the ICOGO model are used as guides to define the grade constraints and production goals required by the MILGP model. The developed models feature stockpiling with limited duration for long-term production scheduling. The models were applied to an oil sands case study to maximize the NPV of the operation. In comparison, whereas the ICOGO model solved the optimization problem faster, the MILGP model results provided detailed mining-cut extraction sequencing for practical mining.

A consensus-based method for group decision making with incomplete uncertain linguistic preference relations

In practical group decision-making (GDM) problems, it is nature for experts to express their preference relations in uncertain linguistic pattern, due to the ambiguity of the external environments and the internal nature of human judgments. In this paper, a consensus process is introduced to deal with GDM problems with incomplete uncertain linguistic preference relations (ULPRs). Firstly, a consistency level for ULPRs is defined. Secondly, a linear goal programming model is furnished to estimate the missing uncertain linguistic preference values with respect to consistency property. Thirdly, an additively consistent ULPR is constructed. What’s more, one can obtain the collective ULPR by aggregating individual additively consistent ULPRs, where the order-inducing variable is consistency level (CL), based on the thought of more importance will be given to the larger CL. Furthermore, a consensus reaching process is presented to make experts’ opinions achieve to a predefined level. In the end, a new algorithm is proposed to solve the GDM problem with incomplete ULPR, which is further applied to the selection of best cooperator in a water conservancy project.

A Consistency-Based Method to Decision Making with Triangular Fuzzy Multiplicative Preference Relations

Triangular fuzzy multiplicative preference relations (TFMPRs) enable decision makers to apply triangular fuzzy numbers to denote their preferences, which can express their vagueness and fuzziness. Consistency analysis is an important research topic that can guarantee logical ranking order. Considering previous research regarding the consistency of TFMPRs, several issues and limitations exist with these relations. The researches cannot cope well with certain situations, such as unacceptably consistent TFMPRs and incomplete TFMPRs. In this paper, a new consistency definition for TFMPRs is defined that can overcome the issues in the previous concepts. Next, several desirable properties of the definition are discussed, and the relationship between the new definition and two existing ones is presented. A linear goal programming model is also built to judge the consistency of TFMPRs, and a method is introduced to obtain an acceptably consistent TFMPR from an inconsistent one. Furthermore, several goal programming models are constructed to estimate missing values in an incomplete TFMPR. Finally, a decision-making method with TFMPRs is developed. Illustrative examples are offered to demonstrate the concrete application of the developed procedure, and a comparison analysis is also made.

Optimizing Student Team and Job Assignments for the Holy Family Academy

Holy Family Academy (HFA) is a private Catholic high school in which funding for students is provided largely by organizations, many of whom sponsor work-study positions for students. Each student must work one day per week. HFA seeks to group its students into teams based on the day of work-study and then assign students to specific jobs. We develop weighted mixed-integer linear goal-programming models as a means of decision support for HFA to assign students to teams for each school year, and to assign each student on each team to a job. The models increase the transparency of student assignments for HFA administrators, while effectively determining well-balanced, diversified student team and job assignments. The decision support tool provides tremendous value to HFA in terms of time and reproducibility, requiring only minutes to solve, rather than weeks. This enables HFA to conduct quick what-if analyses, such as varying the number of students assigned to each team, a task that was impossible to perform manually.

Optimal formulations of local foods to achieve nutritional adequacy for 6–23-month-old rural Tanzanian children

ABSTRACTBackground: Achieving nutritional goals of infants and young children while maintaining the intake of local and culture-specific foods can be a daunting task. Diet optimisation using linear goal programming (LP) can effectively generate optimal formulations incorporating local and culturally acceptable foods.Objective: The primary objective of this study was to determine whether a realistic and affordable diet that achieves dietary recommended intakes (DRIs) for 22 selected nutrients can be formulated for rural 6–23-month-old children in Tanzania.Design: Dietary intakes of 400 children aged 6–23 months were assessed using a weighed dietary record (WDR), 24-hour dietary recalls and a 7-days food record. A market survey was also carried out to estimate the cost per 100 g of edible portion of foods that are commonly consumed in the study area. Dietary and market survey data were then used to define LP model parameters for diet optimisation. All LP analyses were done using linear program solver (LiPS) version 1.9.4 to generate optimal food formulations.Results: Optimal formulations that achieved DRIs for 20 nutrients for children aged 6–11 months and all selected nutrients for children aged 12–23 months were successfully developed at a twofold cost of the observed food purchase across age groups. Optimal formulations contained a mixture of ingredients such as wholegrain cereals, Irish potatoes, pulses and seeds, fish and poultry meat as well as fruits and vegetables that can be sourced locally.Conclusions: Our findings revealed that given the available food choices, it is possible to develop optimal formulations that can improve dietary adequacy for rural 6–23-month-old children if food budget for the child’s diets is doubled. These findings suggest the need for setting alternative interventions which can help households increase access to nutrient-dense foods that can fill the identified nutrient gaps.

A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations

To express uncertain information in decision making, triangular fuzzy reciprocal preference relations (TFRPRs) might be adopted by decision makers. Considering consistency of this type of preference relations, this paper defines a new additive consistency concept, which can be seen as an extension of that for reciprocal preference relations. Then, a simple method to calculate the triangular fuzzy priority weight vector is introduced. When TFRPRs are inconsistent, a linear goal programming framework to derive the completely additive consistent TFRPRs is provided. Meanwhile, an improved linear goal programming model is constructed to estimate the missing values in an incomplete TFRPR that can address the situation where ignored objects exist. After that, an algorithm for decision making with TFRPRs is presented. Finally, numerical examples and comparison analysis are offered.