Pa ge 1 Pa ge 51 American Journal of Multidisciplinary Research and Innovation (AJMRI) Nonpre-Emptive Integer Nonlinear Goal Programming Model for Multi-Item Inventory Problem: Case Study of a Car Retail Centre in Lagos State. Temitope Ogunleye1*, Mogbojuri A.O1, Adeyeye A. D2 Volume 1 Issue 2, Year 2022 ISSN: 2158-8155 (Online) DOI: https://doi.org/10.54536/ajmri.v1i2.260 https://journals.e-palli.com/home/index.php/ajmri Article Information ABSTRACT Received: May 09, 2022 Accepted: May 20, 2022 Published: May 22, 2022 Most of the real-world optimization problems involve multiple objectives with constraint resources. Retailers oftentimes anticipate demand, they stock, and maintain warehouses with different variety of products at high cost to meet prospecting customers’ demand. In this paper, a Non-preemptive Integer Nonlinear Goal Programming (NINGP) model was devel- oped for obtaining Economic Order Quantities (EOQ) of multi-item inventory problems that satisfy the multiple and conflicting objectives of the Decision Maker (DM). The par- ticular case considered was that of a motor vehicle dealer who sells 10 brands of Tokunbo vehicles and wants to determine the EOQ for each brand such that the deviations from the aspiration level are minimized. Using LINGO 17.0 Software to solve the NINGP model, the EOQ allocated to each brand type from 1 through 10 are 2, 2, 5, 2, 2, 3 3, 3, 3, and 3 cars respectively. The optimal number of cars was 28 with the associated cost of ₦53,825,915. Compared to the estimated budget of ₦60,000,000, the NINGP approach was able to achieve a 10% (₦6,174,085) below budget. With proper modifications considering the as- sociated constraints, related inventory problems can be solved using the NINGP model. Keywords Non-preemptive programming, Nonlinear programming, Goal Programming, Multi-Item Inventory, Economic Order Quantity. 1 Mechanical Engineering Department, Yaba College of Technology, Yaba, Nigeria. 2 Industrial and Production Engineering, University of Ibadan, Ibadan, Nigeria. * Corresponding author’s e-mail: temitope.ogunleye@yabatech.edu.ng INTRODUCTION According to sayed et al. (2014) non-preemptive integer nonlinear goal programming (NINGP) helps to solve problems associated with multi-item inventory decision-making problems. For the most part, real-world optimization problems involve multiple objectives with constraint resources. Retailers oftentimes anticipate demand, therefore stock, and maintain their warehouses with different varieties of products at a high cost to meet prospecting customers’ demand. Multiple items or products are stored in most shops or warehouses to increase profitability, and competition and attract sales from customers with different choices. However, keeping these inventories is quite challenging as their values keep increasing owing to increased cost of importation, fluctuating exchange rates, and changing economic policies (Sayed et al., 2014). Adeyemi (2010) opined that ‘‘to maintain a balance of the conflicting economics, an effective inventory management is required to determine the number inventories in stock at minimum cost. Proper management policies, and control measures must also be in place to monitor and replenish these inventories (Nsikan et al., 2015). Keeping huge and numerous inventories can lead to an expanded running cost, low benefits and tall working capital prerequisites whereas investment in little inventory can lead to break-ups in the organizational handle, loss of customers, and a greatly reduced profit margin (Prabha, and Suchita, 2016). Amini et al. (2017) state that ‘‘the major concern in real- life decision-making situations involve multiple criteria (attributes or objectives) rather than single criteria,’’. These objectives are conflicting and are best approached using a goal programming analytical framework. Goal programming is an operation research technique useful for achieving simultaneously multiple goals with constrained resources (Ajayi-Daniels, 2019). The aim of goal programming (GP) is to find an optimum solution out of a set of feasible solutions that satisfy the real-life constraints and comes in a closed form to the decision- makers stated target value (goals). According to Amini, et al. (2017), Goal Programming is used to analyze all of the objectives with various achievement relations to discover an acceptable solution by reducing deviations from expected values. It also analyzes how much a proposed optimal solution deviates from each target though there are deviation variables defined, for each pair of stated goals (Moumita & De, 2016). This concept has enjoyed significant applications by many researchers over time, in solving different inventory problems with multiple constraints resources. According to Aouni & Kettani (2001), the increase in the use of goal programming was simply because it is exceptionally simple to comprehend and applied in many areas, such as the manufacturing, production, retail shops, transportation, medicine, agriculture, academic institutions, and construction companies alike. In the study of Ajayi-Daniels (2019), a goal-programming model was developed to optimize resources in a fashion firm where goals, were prioritized according to importance. The result-achieved bases on the priority level show a reduction in the overtime hours from 10hours to 8 hours, which is the optimum time, and a target profit margin was achieved. Also with efficient use of resources the set goal of 3 garments per day was achieved. Kliestik et al. (2015), developed a unique GP model that management companies with numerous plans can use to implement a strategic goal. Yahia-Berrouiguet & Tissourassi (2015), proposed a goal-programming model for the allocation of https://doi.org/10.54536/ajmri.v1i2.260 https://journals.e-palli.com/home/index.php/ajmri mailto:temitope.ogunleye%40yabatech.edu.ng?subject= Pa ge 52 https://journals.e-palli.com/home/index.php/ajmri Am. J. Multidis. Res. Innov. 1(2) 51-55, 2022 time and cost to three different projects with preemptive goals. They discussed the illogical allocation of zero (0) time to project planning and concluded that a project is bound to fail if the planning phase is not given proper consideration with an allocation of time. Jana & Das (2016) previewed a multi-item two-warehouse inventory model with a nested discount on unit cost and inventory costs over a fixed period. The proposed a multi-objective genetic algorithm with a varying populations (MOGAVP) for multi-item two-warehouse inventory problems with nested price breaks. The optimal shipments, lot size of the two warehouses, shipment size, and maximum profit are determined by maximizing the profit function. Huseyinov & Bayrakdar (2019) developed Non- Dominated Sorting Genetic Algorithm-III (NSGA-III) and Strength Pareto Evolutionary Algorithm (SPEA2) for solving a simulated multi-objective single-period multi-item inventory problem. The study proved that the SPEA-II outperformed NSGA-III with spacing, generational distance, and hypervolume as performance evaluation metrics. Waliv et al. (2020) proposed a multi-objective, multi- item fuzzy stochastic inventory model for deteriorating items under limited storage space as well as capital investment. Their proposed model considered some parameters to be vague and random, by representing the vagueness parameters membership function and randomness parameters as a probability distribution. In their study, fuzzy nonlinear programming (FNLP) and intuitionistic fuzzy optimization (IFO) methods to solve the multi-objective fuzzy stochastic inventory model were developed. Their study showed that FNLP performed better than IFO in the area of minimizing shortage costs. Das (2020) developed the multi-item inventory model in the fuzzy environment. The formulated Multi-objective inventory problem was used to solve by different approaches such as the Geometric Programming (GP) approach, the Fuzzy Programming Technique with Hyperbolic Membership Function (FPTHMF), Fuzzy Nonlinear Programming (FNLP) technique, and Fuzzy Additive Goal Programming (FAGP) technique. The study of Teymouri et al. (2020) presented a multi- item supplier selection, order allocation, and pricing method with stochastic demand and purchase from the suppliers who provide goods with all-unit discount. They developed a mixed nonlinear integer single-objective model and applied a genetic algorithm to solve the model. The result showed that increasing the variance of the random variable of demand results in decreasing the profit level. The study also shows that the genetic algorithm is considered an appropriate alternative to solve a mixed nonlinear integer model of supplier selection, order allocation, and pricing. Cárdenas-Barrón et al. (2021) considered the multi-item inventory lot-sizing problem with supplier selection and proposed a facility location extended formulation with the preprocessed based on the cost structure and describe new valid inequalities in the original space of variables. An effective MIP (mixed integer programming) heuristic using the extended formulation was the model to implement the proposed formulation. In the study, the results showed that the preprocessing approach can significantly reduce the size of the formulation and also allow both an increase in the number of instances solved to optimality within the time limit and a reduction in the average time to solve them. The performance of the approaches can vary considerably under different cost structures. Meanwhile, little/ less attention is on the inventory of foreign (Tokubo) car retail business. Therefore, this paper developed the NINGP model to determine the EOQ required in a Car Retail Centre in other to achieve the Decision Maker’s competitive priorities with stated constraints. MATERIALS AND METHOD The approach of the Non-preemptive goal programming involves establishing a specific numerical value (target) for the objectives of the Decision Maker (DM). Deviations from these targets are not desirable, therefore the DM seeks a compromise solution. Brief Description of the Problem A Second-hand (Tokunbo) vehicle dealer in the metropolitan city of Lagos-state wishes to determine the economic order quantity (EOQ), for the vehicles, he sells. He sells ten (10) different and affordable brand of vehicles with a yearly budget of sixty million, three hundred and fifteen thousand naira’s (N60, 315,000.00). The available retail space has the capacity for 25 or more Second-hand cars. Due to the retail space and other possible constraints, the dealer wishes to determine the optimal mix of these vehicles that will fill up the retail space and still be able to meet prospecting customers’ demand. Assumptions of Model The model assumptions are: i. The inventory system involves multiple items ii. For every replenishment, order for a single delivery is made iii. There is a constant Lead time iv. Shortages are not allowed v. Discounts are not allowed for any item ordered vi. Equal space is available for all items ordered vii. Inventory costs (Carrying cost and ordering cost) are known viii. Demand is known and constant. ix. The inventory parameters are preset and constant x. The goals are of equal weight as well as the deviations from the target xi. The goals are of equal importance Mathematical Formulations The general model is of the form Subject to: https://journals.e-palli.com/home/index.php/ajmri Pa ge 53 https://journals.e-palli.com/home/index.php/ajmri Am. J. Multidis. Res. Innov. 1(2) 51-55, 2022 Table 1.1: Notations with Description Indices Description i Integer index for items (i= 1, 2, 3 …n) j constraints Index (j = 1, 2 …m) aij The average investment per unit of item i oqi The decision variable (ordered quantity of item i) bj The right-hand side value associated with constraint j di ̅ ,dj ̅ The negative deviation variables of the NINLGP from the item (i) and constraint (j) (underachievement) di +,dj + The positive deviational variable of the INLGP from the ith goal and constraint j (over-achievement) Soci Sum of ordering cost and holding cost for ith item Tic Total inventory cost Sci Setup cost for item i Gi The target value for the item i Sc Storage capacity Goal Formulation The NINGP model objective and constraints are taken as the model goals with the introduction of deviational variables. Total Inventory Cost Goal Deviational variables added to the inventory cost to give: Investment Goal With the deviational variables the targeted investment goal gives: Inventory Space Goal The targeted inventory space becomes; Weighted Average Structure Accordingly, the weighted average structure of the Non-preemptive Integer Nonlinear Goal Programming (NINGP) is stated as; Find Oq ̅ (Oq1,Oq2,…Oqi) to Minimize: Subjected to Where, since all goals are presumed to be of equal importance to the DM. Model Application The cost detail for each brand of “Tokunbo” car is tabulated below. Table 1.2: Cost details for each brand of cars Car Type (Ci)(N) ‘000 Setup cost (Si)(N) ‘000 Target value (Gi)(N) ‘000 Average Investment (aij)(N) ‘000 Oq1 Toyota Camry (2004) 3963 2600 10000 7613 Oq2 Toyota Corolla (2006) 4325 2463 10000 8013 Oq3 Toyota Corolla (2004) 1550 2325 8000 4225 Oq4 Peugeot 307 (2002) 2688 2380 7000 5768 Oq5 Peugeot 307 (2004) 3525 2500 9000 7163 Oq6 Honda Accord (2004) 2550 2500 8200 5750 Oq7 Nissan Quest 2162 2225 8000 5000 Oq8 Honda Accord (2003) 2505 2363 9000 5629 Oq9 Honda Civic (2003) 2437 2363 9000 5509 Oq10 Toyota sienna (2003) 2637 2325 9000 5645 Total 28,342 22,044 87,200 60,315 Inventory Cost goals The inventory cost goals for brands 1 through 10 respectively are presented in equations (11–20) below using the data in Table 2: https://journals.e-palli.com/home/index.php/ajmri Pa ge 54 https://journals.e-palli.com/home/index.php/ajmri Am. J. Multidis. Res. Innov. 1(2) 51-55, 2022 Investment constraint Since the cost of each vehicle ordered is fixed the equation becomes: Inventory constraint The total carrying capacity of the warehouse is 30 cars. Thus, the goal equation appears a RESULTS AND DISCUSSION The NINGP Model was tested and the result is provided in table 2. Table 2: The optimal mix of various types of cars and their respective deviations Car type Quantity (EOQ) Positive (+)/Negative (-) Deviation (N) from the budget Oq1 2 -1204275 Oq2 2 -1348312 Oq3 5 529619 Oq4 2 -760805 Oq5 2 -1054150 Oq6 3 830042 Oq7 3 -696021 Oq8 3 -815128 Oq9 3 -790410 Oq10 3 -864645 Total 28 -6174085 Table 2 shows that the dealer can stock up the car Centre with 10 different types of cars occupying 93% of the inventory space. For instance, at the retail Centre two types of Toyota Camry (2004), Toyota Corolla (2006), Peugeot 307 (2002) and Peugeot 307 (2004) must always be on display. There must always be five types of Toyota Corolla (2004) on display and 3 types each for the following car types; Honda Accord (2004); Nissan Quest; Honda Accord (2003); Honda Civic (2003); Toyota sienna (2003). Also the table shows an under-achieved budgeted goals for each car type. The solution saved the decision-maker the sum of ₦6,174,085, which represent 10.3% of the decision maker’s annual budget estimate, which can also purchase two or three Tokunbo cars. The optimal cost of the cars is ₦53,825,915 as against the initial cost of ₦60,315,000 in table 2 from the saved cost of ₦6,174,085). The highlighted table 2 is supposed to be written is table 1 CONCLUSION This paper presents the application of the Nonpre- emptive Integer Nonlinear Goal Programming (NINGP) model and the procedure for solving a multi- item inventory problem in other to obtain the EOQ required to achieve the aspirations of a local car dealer. All conflicting objectives and constraints were taken into consideration and with the use of an optimization tool in LINGO 17.0 software, an optimal solution was obtained. Most Tokunbo car dealers used to display several types of cars in any available area, without properly analyzing the best mix of different car types to display and their associated costs. This makes it harder for car dealerships to achieve their objectives. They will benefit from the proposed NINGP model since it is sufficiently adaptable to meet related situation-specific objectives and constraints when the goal is to minimize costs and deviations from defined goals. Hence the NINGP model can be used analytically to solve related multi-item inventory management problems. There is to test the model for a large number of foreign “Tokunbo” and types of cars. Further study may suggest using other mathematical programming models like, stochastic and dynamic programming models, etc., to solve multi-item inventory foreign (Tokunbo) cars problem to obtain the EOQ. Also, a combined optimization algorithm and mathematical programming may be considered or involved to solve the same problem. REFERENCES Ajayi-Daniels, E.E. (2019). Resource Optimization in a Fashion Firm: A Goal Programming Approach. International Journal of Management and Fuzzy Systems. 5 (1), 14-20. Amini, A., Alinezhad, A., & Heydarlou, B.N., (2017). A model for solving the fuzzy multiple-objective problem using a satisfying optimization method. 6th, International Conference on new challenges in management and business. Dubai University. Aouni.B & Kettani.O. (2001). Goal programming: A glorious history and a promising future. European Journal of Operational Research.133, 225-231. Cárdenas-Barrón, L. E., Melo, R. A., & Santos, M. C. (2021). 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