Int. J. Appl. Comput. Math (2021) 7:69 https://doi.org/10.1007/s40819-021-00965-z ORIG INAL PAPER Production System in a Collaborative Supply Chain Considering Deterioration Jaime Acevedo-Chedid1 · Katherinne Salas-Navarro2 · Holman Ospina-Mateus1 · Alina Villalobo1 · Shib Sankar Sana3 Accepted: 24 January 2021 © The Author(s), under exclusive licence to Springer Nature India Private Limited part of Springer Nature 2021 Abstract This research presents a mathematical model for a collaborative planning of the supply chain involving four echelons (supplier, production plants, distribution, retails, or clients). The model seeks to maximize profit (utility) when all members of the chain share information related to demand. It is developed for the aggregate consolidation of different rawmaterials in cement production. The novelty of the model is the consideration of products that deteriorate in the process and thus it has effect on the production times in the plant and lead time. In this supply chain, quality and compliant products and the return of deteriorated products are two flows. The considerations are lead time, inventories with shortages and excesses, production times in normal and extra days, and subcontracting, among others. A mixed integer linear programming with demand scenario analysis is used to optimize and analyze the uncertainty that is consistent with the performance of the construction sector. The model is formed considering two suppliers, two production plants, two distributors, two retailers and two end customers. Four manufacturing inputs (raw materials) are considered for the manufacture of two types of products. A case study of the cement production supply chain of Cartagena (Colombia) is illustrated. The shared benefit is generated around 5 billion pesos (COP) for all members of the chain in a period of 6 months. Keywords Collaborative supply chain · Deteriorating products · Reverse flow B Shib Sankar Sana shib_sankar@yahoo.com Jaime Acevedo-Chedid jacevedo@utb.edu.co Katherinne Salas-Navarro ksalas2@cuc.edu.co Holman Ospina-Mateus hospina@utb.edu.co Alina Villalobo alina.villalobo@ecopetrol.com.co 1 Department of Industrial Engineering, Universidad Tecnológica de Bolívar, Cartagena, Colombia 2 Department of Productivity and Innovation, Universidad de La Costa, Barranquilla, Colombia 3 Department of Mathematics, Kishore Bharati Bhagini Nivedita College, Behala, Kolkata 700060, India 0123456789().: V,-vol 123 69 Page 2 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Introduction The supply chain refers to the set of efficiently integrated companies that seek the best strategies to deal with goods and products in a timely manner at the lowest cost, satisfying the requirements of consumers [14]. The supply chain process integrates organizations and customer–supplier relationships. An integrated, coordinated, and synchronized management provides effective solutions for decision-making in all actors in the chain. The optimality of the supply chain allows profits in inventory, purchasing, transportation, information flow, customer service, and delivery times [2, 56]. The global dynamics of the economy and competitiveness require new challenges to inter- act and satisfy customers. Efficient analysis of the supply chain for the search for competitive advantages must consider all interactions and relationships among suppliers, manufactur- ers, distributors, and customers. A common goal shared by these actors in the chain is to maximize profits and customer satisfaction which translates into greater profitability and competitiveness. Strategies such as cooperation and collaboration of the different actors gen- erate synergies and multiply the efforts of the logistics processes. Collaboration/cooperation in the supply chain guide processes to be more dynamic and competitive based on customer demand, eliminating barriers in the network, and seeking to simplify and make activities more efficiently. This research seeks to develop a collaborative mathematical model in the supply chain for production planning.Problems such as the loss of sales due to low or missing invento- ries, obsolescence and deterioration of products, high transportation, and inventory costs, and uncertainty in the demand informationmotivate the approach of thismathematicalmodel. The proposedmodel considers a case study problem of a company in the mining sector to produce cement in the city of Cartagena in Colombia. The productive chain of the Colombian min- ing sector deals with the exploration, exploitation, and commercialization of non-metallic minerals such as sand, limestones and clays, gypsum that are used to supply materials to cement or concrete in industrial production processes, housing construction and infrastruc- ture [47]. Hence the study of this chain is important in the social and economic development of Colombia [8]. The use of mathematical models becomes an essential tool for the design and implementation of supply chains [57]. Supply chain modeling helps to capture the com- plexity and integrate the resources andmathematical programming represents the best way to approach the methodology for solving the problems of mining supply chains. Next, a literary review is developed in the field of collaboration/cooperation where mathematical modeling is applied in the supply chain. Literature review The concept of collaborative supply chains has taken on importance over the years [6, 19, 23, 31, 38, 40, 62, 64]. The collaborative supply chain is characterized to integrate, relate to the long term, and define common targets and benefits [45, 57]. Joint efforts to achieve the same objective and economic benefits within the chain become a comprehensive solution [60]. Collaboration/cooperation avoid inefficiencies and the bullwhip effect produced by a lack of coordination in the supply chain [26]. Table 1 shows the review of the literature on collaborative supply chain models, based on the structure, approaches to the solution, and their characteristics. In the literary review on models of collaboration/cooperation of logistics chains with different approaches, no models are found that contemplate an integration in the supply 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 3 of 46 69 123 Table 1 Review of the literature on collaborative supply chain models: approaches and considerations Model Class Authors Approaches of solution Levels of SC Considerations GT MS SE SD Others 2 3 M Others Analytical model AHP [51] X X Collaborative information Artificial intelligence-based model [33] X X Uncertainty in demand and supply Conceptual model [46] X X Performance measures [41] X X Development of new products and inter and intra business relationships [52] X X CPFR model based on associations, analysis using structural equation models and confirmatory factor analysis [1] X X SCOR model. Performance measurement [65] X X SCOR model. Benchmarking techniques and performance measurement [67] X X SCOR model. Collaboration of external partners and performance measurement (KPI’s) of the logistics service Inventory management model [16] X X Stochastic demand and life cycle analysis in green supply chains [57] X X Scenarios with uncertainty, quantities of products in inventories [27] X X Coordination of prices and inventory decisions Stochastic programming model [43] X X Cooperative and non-cooperative scenarios for supplier selection [10] X X Production costs for technical advice and discounts 69 Page 4 of 46 Int. J. Appl. Comput. Math (2021) 7:69 123 Table 1 continued Model Class Authors Approaches of solution Levels of SC Considerations GT MS SE SD Others 2 3 M Others [30] X X Cooperative global scenario [61] X X Vertical cooperation [54] X X Shared capacity [35] X X Non-cooperative games in assembly supply chains, decentralized [36] X X Supply shortage [74] X X Contract options [53] X X Sharing information and forecasts [17] X X Decreased Bullwhip Effect [73] X X Simultaneous product distribution with infinite capacity considerations [71] X X Optimal price, and advertising strategies in four-game scenarios [21] X X Market demand, selling price, and marketing expenses [34] X X The problem of allocation of savings in the exchange costs of demand information [25] X X Stochastic demand and delivery times [44] X X Benefits of allocation and stability in cooperative chains [55] X X The price transfer problem [69] X X Two scenarios, one with sufficient supply and the other insufficient from the supplier Int. J. Appl. Comput. Math (2021) 7:69 Page 5 of 46 69 123 Table 1 continued Model Class Authors Approaches of solution Levels of SC Considerations GT MS SE SD Others 2 3 M Others [28] X X Manufacturer dominance over retailers [59] X X Co-op advertising among the supplier, manufacturer, and retailers with a variable demand driven by selling price and advertising costs (fuzzy) Simulation model [37] X X Multi-flow Supply Chain, and inter-company relationships [13] X X Knowledge-based personalization and sharing information for supply chain integration [9] X X Collaborative transport, based on the activation of capacity restrictions [18] X CPFR model Accuracy in estimating forecasts [32] X X Decision-making processes and their impact on CS performance [22] X Collaborative offer Mathematical model/optimization [66] X X Operations costs, information sharing costs, and information processing costs [70] X X Quantity discounts and incremental quantity discounts [5] X X Collaborative forecasting between manufacturer and supplier [62] X X Joint planning and problem solving [63] X X Shared capacity 69 Page 6 of 46 Int. J. Appl. Comput. Math (2021) 7:69 123 Table 1 continued Model Class Authors Approaches of solution Levels of SC Considerations GT MS SE SD Others 2 3 M Others [12] X X Joint replacement [4] X X Demand forecast and joint replenishment in uncertain scenarios [3] X X Collaborative forecasting between retailer and supplier [50] X X Non-stationary demand [11] X X A multi-product, multi-stage production and distribution planning model, and multi-period which maximizes the benefits, the level of services, and the inventory level [39] X X Mathematical model with uncertainty, whose objective is to minimize the costs of the aggregate plan of the supply chain [49] X X A four-echelon integrated production–distribution in supply chain. The chain is studied in competitive conditions (Cournot, Stackelberg, and Quality and competition-free condition. Two mixed-integer linear programs (MILPs) are developed, which are proved to be unimodular [20] X X Proposing a multi-objective model for a multi-period multi-product multi-site aggregate production planning (APP) problem in a green supply chain Int. J. Appl. Comput. Math (2021) 7:69 Page 7 of 46 69 123 Table 1 continued Model Class Authors Approaches of solution Levels of SC Considerations GT MS SE SD Others 2 3 M Others [15] X X A mixed-integer linear programming model that can handle general supply chains and production processes that require multiple resources [72] X X The model developed an aggregate production-pricing problem with multiple demand classes. Developing a new demand response function with cannibalization [58] X X The objective is to minimize the total cost of the entire supply chain model (SCM) by simultaneously optimizing setup cost, process quality, deliveries, and lot size [24] X X The model determines the production plans of the suppliers to minimize the sum of raw material purchasing costs, production costs, setup costs of the suppliers, transportation costs, and costs for outsourcing semi-finished products. The model is solved with a mixed-integer programming model and a two-step heuristic algorithm Mathematical optimization by This research X X Demand uncertainty, process failure scenarios uncertainty, material arrival uncertainty, reverse flow of defective materials/products GT game theory;MS multi-agent system; SE structural equations; SD system dynamics; M multilevel 69 Page 8 of 46 Int. J. Appl. Comput. Math (2021) 7:69 chain of non-metallic minerals such as cement production. The proposed model helps to plan the production of final products, the distribution of materials, and integrates functions from suppliers to customers, synchronizing, and collaborating among all the actors in the chain. Additionally, the model considers uncertainty in demand, uncertain delays due to process failures, and even uncertainty inmaterial quantities. Themodel contributes to planningwithin the medium and short term at the tactical and operational levels of the supply chain. The present research develops a collaborative supply chain planningmodelwhere themain objective is to maximize the benefits of all the members in the four-echelon supply chain comprising of suppliers, manufacturers, distributors, and retailers. The novelty of the model allows considering a mechanism for detecting products with minor defects or deterioration. These products are purchased at a lower price, avoiding return, which favors less use of transport and environmental impact. Additionally, it is considered a mechanism for detecting products with major defects in buyers (plants and distributors). The model considers returns and changes in delivery times. Likewise, the lead time is considered at the supplier/producer and producer/distributor. The model considers production with work on regular and extra days, and subcontracting also. The content of this research is structured as follows: The introduction in Sect. 1 indicates the context of the problem. The methods in Sect. 2 present the elements for mathematical formulation. Sections 3 and 4 provide the case study and its results. Sections 5 and 6 provide managerial implications and conclusions of the proposed model respectively. Method The proposed model is of a mixed-integer linear programming approach with analysis by scenarios for demand according to the economic performance of the construction sector. The model allows to plan the transport in each echelon, plan the purchases with the suppliers, plan the production of the plant, the sales, the inventory levels, and calculation of the total cost. The chain integrates suppliers, production plants, distribution centers, retailers or cus- tomers.Suppliers deliver raw materials or items to the production plants, which transform them into finished products. The distribution centers receive the products from the produc- tion plants and deliver them to the retailers, who sell the products to the final client. This model maximizes the benefits of all members of the supply chain, considering parameters related to the production process, transport (times and capacities), inventories (capacities and shortage), and costs. Fundamental Assumptions The following assumptions are used to formulate the model: The objective is to guarantee the maximization of the profit margin of all the entities of the supply chain (Revenues-Costs). Multiple suppliers, production plants, and multiple distribution centers are considered. Suppliers and production plants store raw material. There is a fraction of raw materials with deterioration/defect that can be purchased by the plants at a lower price. The lead times of plants and distribution centers are deterministic. There is a fraction of products with deterioration/defect that generate a non-rejection condition, which are bought by the next echelon at a lower price. 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 9 of 46 69 At the beginning of each period new settings are given for production. Defective/deteriorated products purchased at plants from distributors are reprocessed and sold to the retailer as compliant products. Plants, distribution centers, and retailers store finished products. Three scenarios are proposed for the representation of the model under uncertainty, high, medium, and low scenario. 2.2. Notation The following notations are used to develop the model. Declaration of Indices s ∈ S: Suppliers p ∈ P: Plants d ∈ D: Distribution Centers r ∈ R: Retailers c ∈ C : Clients t ∈ T : Periods e ∈ E : Scenarios m ∈ M : Raw Materials j ∈ J : Products q ∈ Q: Production Resources Declaration of Sets Sm : Set of suppliers s that provide raw material m (Sm ⊆ S) S p: Set of suppliers s that provide production plant p (S p ⊆ S) Qp: Set of production resources q of the plants p (Qp ⊆ Q) Dp: Set of distribution centers d that receive finished products j (Dp ⊆ D) Rd : Set of retailers r that receive finished products from distribution centers d (Rd ⊆ R) Cr : Set of clients that receive finished products from retailers r (Cr ⊆ C Jq . Jq : Set of finished products j produced with the production resource q (Jq ⊆ J ) Jm : Set of finished products j produced with the raw material m (Jm ⊆ J ). 2.3. Statement of Parameters Suppliers’ Parameters CCSs,m,t : Cost per unit of raw material, component, or item m at supplier s in period t ($/Ton) HSs,m : Handling cost per unit at supplier s for the raw material m ($/Ton) π+s,m : Inventory excess cost at supplier s for the raw material m ($/Ton) π−s,m : Inventory shortage cost at supplier s for the raw material m ($/Ton) CPSs,m,t : Production cost per unit at supplier s for the raw material m in period t($/Ton) CDSs,m : Disposal cost per unit defective item at supplier s for raw material m ($/Ton) CFT Ss,p,m : Fixed transportation cost at supplier s for the rawmaterialm for the production plant p($) CT Ss,p,m : Transportation cost per unit from the supplier s to production plant p for the raw material m ($/Ton) 123 69 Page 10 of 46 Int. J. Appl. Comput. Math (2021) 7:69 CT 1Sp,s,m : Transportation cost per unit from the production plant p to supplier s for the raw material m ($/Ton) αS( s,m : Expected percentage of defective items at supplier s for the raw material m 0 ≤ )αSs,m ≤ 1 βSs,m : Screening rate (%) of defective items at supplier s for the raw material m CapT Ss,p,m : Transport capacity for the rawmaterialm from the supplier s to the production plant p (Ton) T T Ss,p: Transportation time from supplier s to production plant p (Hours) I oSs,m : Initial inventory level at supplier s for the raw material m (Ton) ImaxSs : Maximum inventory capacity at the supplier s (Ton) FmaxSs,m : Maximum production capacity at the supplier s for the raw material m (Ton) PV Ss,p,m : Selling price of raw material, component, or item m at the production plant p ($/Ton) PV DSs,p,m : Selling price per unit defective item at supplier s for the raw material m for the production plant p ($/Ton) Production Plants’ Parameters CFPp,q, j,t : Fixed cost of changing material at plant p with the production resource q for the finished product j in period t ($) CPRPp,q, j,t : Production cost per unit at production plant p for the finished product j working in regular time with the production resource q in period t ($/Ton) CPEPp,q, j,t : Production cost per unit at production plant p for the finished product j working in extra time with the production resource q in period t ($/Ton) CSubPp, j : Purchasing cost per unit subcontracted in the production plant p for the finished product j ($/Ton) ϕp,q, j,t : Fixed handling cost at production plant p for the finished product j with the production resource q in period t($) HPp, j : Handling cost per unit at production plant p for the finished product j ($/Ton) ϕ+p, j : Inventory excess cost at production plant p for the finished product j ($/Ton) ϕ−p, j : Inventory shortage cost at production plant p for the finished product j ($/Sack) CFT Pp,d, j : Fixed transportation cost for finished product j from production plant p to distribution center d ($) CT Pp,d, j : Transportation cost per unit for finished product j from production plant p to distribution center d ($/Ton) MAPm, j : Combination of rawmaterialm necessary to produce the finished product j (Sack) CDPp, j : Disposal cost per unit defective item at production plan p for the finished product j ($/Sack) βPp,m : Screening rate (%) of defective items at production plant p for the raw material m. αPp, j : Expected percentage of defective items at production plant p for the finished product( ) j 0 ≤ αPp, j ≤ 1 μPp, j : Screening rate (%) of defective items at production plant p for the finished product j CapPRPp,q,t : Maximum production capacity at production plant pworking in regular time on the production resource q in period t (Sacks) CapPEPp,q,t : Maximum production capacity at production plant p working in extra time on the production resource q in period t (Ton) 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 11 of 46 69 CapT I Pp : Maximum capacity of transport input in the plant p (Ton) CapT OPp : Maximum capacity of transport output in the plant p. (Sacks) CapT Pp,d, j : Transport capacity for finished product j from production plant p to distribu- tion center d (Ton) T T Pp,d : Transportation time from the production plant p to the distribution center d (Hours) I oPp, j : Initial inventory level at production plant p for the finished product j (Ton) Imax Pp : Maximum inventory capacity at the production plant p (Ton) xpected percentage (%) of finished product j for subcontracting in the production plant p. PV Pp,d, j : Selling price per unit good item at production plant p to distrsibution center d for the finished product j ($/Ton) PV DPp,d, j : Selling price per unit defective item at production plant p to distribution center d for the finished product j ($/Ton) Distribution Centers’ Parameters HDd, j : Handling cost per unit at the distribution center d for the finished product j ($/Sack) γ +d, j : Excess inventory cost at distribution center d for the finished product j ($/Sack) γ−d, j : Shortage inventory cost at distribution center d for the finished product j ($/Sack) CFT Dd,r , j : Fixed transportation cost from distribution center d to the retailer r for the finished product j($) CT Dd,r , j : Fixed transportation cost per unit from distribution center d to the retailer r for the finished product j ($/Sack) CDDd, j : Disposal cost per unit defective item at distribution center d for the finished product j ($/Sack) δDd, j : Screening-rate (%) of defective items at distribution center d for the finished product j . CapT ODd : Maximum capacity of transport output in the distribution center d (Sacks) CapT Dd,r , j : Transportation capacity of finished product j from distribution center d to the retailer r (Sacks) T T Dd,r : Transportation time from the distribution center d to the retailer r (Hours) I oDd, j : Initial inventory level at distribution center d for the finished product j (Sacks) ImaxDd : Maximum inventory capacity at distribution center d (Sacks) PV Dd,r , j : Selling price per unit good item at distribution center d to the retailer r for the finished product j ($/Sack) Retailers’ Parameters. HRr , j : Handling cost per unit at retailer r for the finished product j ($/Sack) ϑ+r , j : Excess inventory cost at retailer r for the finished product j ($/Sack) ϑ−r , j : Shortage inventory cost at retailer r for the finished product j ($/Sack) I oRr , j : Initial inventory level at retailer r for the finished product j (Sacks) Imax Rr : Maximum inventory capacity at retailer r (Sacks) CapT ORr : Maximum capacity of transport input in the retailer r (Sacks) DemCc, j,t,e: Demand rate of client c for the finished product j over scenario e in period t (Sacks) PV Rr ,c, j : Selling price per unit good item at retailer r to the client c for the finished product j ($/Sack) Probe: Probability over scenario e. 123 69 Page 12 of 46 Int. J. Appl. Comput. Math (2021) 7:69 2.4. Statement of Variables Continuous Variables I Ss,m,t,e: Inventory level of raw material m at supplier s in period t (Ton) I S+s,m,t,e: Excess inventory level of raw material m at supplier s in period t (Ton) I S−s,m,t,e: Shortage inventory level of raw material m at supplier s in period t (Ton) I Pp, j,t,e: Inventory level of the finished product j at the plant p in period t (Sacks) I P+p, j,t,e: Excess inventory level of the finished product j at the plant p in period t (Sacks) I P−p, j,t,e: Shortage inventory level of the finished product j at plant p in period t (Sacks) I Dd, j,t,e: Inventory level of the finished product j at the distribution center d in period t (Sacks) I D+d, j,t,e: Excess Inventory level of the finished product j at the distribution center d in period t (Sacks) I D−d, j,t,e: Shortage inventory level of the finished product j at the distribution center d in period t (Sacks) I Rr , j,t,e: Inventory level of the finished product j at the retailer r in period t (Sacks) I R+r , j,t,e: Excess inventory level of the finished product j at retailer r in period t (Sacks) I R−r , j,t,e: Shortage inventory level of the finished product j at the retailer r in period t (Sacks) QSs,m,t,e: Raw material m to purchase from the supplier s in period t (Ton) QPp, j,t,e: Production of the finished product j at the plant p in period t (Sacks) QPPp,q, j,t,e: Product j to be produced in Plant p on Production Resource q in period t. (Sacks) QPRPp,q, j,t,e: Production quantity of the finished product j at the plant p working in regular time on the production resource q in period t (Sacks) QPEPp,q, j,t,e: Production quantity of the finished product j at the plant p working in extra time on the production resource q in period t (Sacks) QSubPp, j,t,e: Subcontracting quantity of the finished product j at the plant p in period t (Sacks) QT Ss,p,m,t,e: Raw material m to transport from the supplier s to the plant p in period t (Ton) QT Pp,d, j,t,e: Product j to transport from the plant p to the distribution center d in period t (Sacks) QT Dd,r , j,t,e: Product j to transport from the distribution center d to the retailer r in period t (Sacks) QT Rr ,c, j,t,e: Product j to transport from the retailer r to the client c in period t (Sacks) QTT Ss,p,t,e: Total quantity to transport from the supplier s to the plant p in period t (Ton) QTT Pp,d,t,e: Total quantity to transport from the plant p to the distribution center d in period t (Sacks) QTT Dd,r ,t,e: Total quantity to transport from the distribution center d to the retailer r in period t (Sacks) QTT Rr ,c,t,e: Total quantity to transport from the retail r to the customer c in period t (Sacks) BSs,e: Expected benefits of the supplier s in period t ($) BPp,e: Expected benefits of the plant p in period t ($) BDd,e: Expected benefits of the distribution center d in period t ($) BRr ,e: Expected benefits of the retailer r in period t ($) 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 13 of 46 69 BTe: Total expected benefit of the supply chain in the scenario e ($) Z: Total expected benefits of the supply chain ($) Binary Variables Xs,p,m,t : 1 if CFT Ss,p,m > 0, otherwise 0 (1 if the transport capacity level from supplier s to plant p is used for raw material, component, or item m in period t) Yp,d, j,t : 1 if CFT Pp,d, j > 0, otherwise 0 (1 if the level of transport capacity from plant p to distribution center d is used for product j in period t) W Dd,r , j,t : 1 if CFTd,r , j > 0, otherwise 0 (1 if the transport capacity level from distribution center d to retailer r is used for product j in period t) V Pp,q, j,t : 1< if CFp,q, j,t > 0, otherwise 0 (1 if the enlistment time in plant p is used on the Production Resource q for product j in period t) U Pp,q, j,t : 1 if CFp,q, j,t > 0, otherwise 0 (1 if a starting change is generated in Plant p on the Production Resource q of product j). OptimizationModel Once the variables have been defined, the formulation is as follows: Objective function and constraints: The expected profit of the whole chain is: ∑∑ ∑∑ ∑∑ ∑∑ Z  Prob SeBs,e + ProbeBP D Rp,e + ProbeBd,e + ProbeBr ,e e∈E sS e∈E pP e∈E dD e∈E r R (1) subject to: ∑∑ ∑ [[ ( ) ] ( ) ] BS  PV S P S P S Ss,e s,p,m 1− βp,m + PV Ds,p,mβp,m 1− βs,m QTs,p,m,t,e t∈T p∈P m∈M ∑∑ ∑ [ ( ) ] − CFT S X S S Ss,p,m s,p,m,t + CTs,p,m + Hs,m QTs,p,m,t,e t∈T p∈P m∈M ∑ ∑ [( ) − CCS S S S + S+ − ] S− s,m,t + CDs,mαs,m Qs,m,t,e + πs,m Is,m,t,e + πs,m Is,m,t,e ∀s t∈T m∈M ∈ S, e ∈ E (2) ∑ ∑ ∑ ∑[[ ( ) ] ( ) ] BP  PV P D P D P Pp,e p,d, j 1− δd, j + PV Dp,d, j δd, j 1− μp, j QTp,d, j,t,e t∈T d∈D q∈Q j∈J ∑∑ ∑ [[ ( ) ] ( ) ] − PV S P S P S Ss,p,m 1− βp,m + PV Ds,p,mβp,m 1− βs,m QTs,p,m,t,e t∈T s∈S m∈M ∑ ∑ ∑[ ] − ϕ P P P P Pp,q, j ,t + CFp,q, j ,t Vp,q, j ,t + CPRp,q, j ,t QPRp,q, j,t,e + CPEp,q, j,t QPEp,q, j,t,e t∈T q∈Q j∈J ∑ ∑ ∑[ ] × CFT P P Pp,d, j Yp,d, j ,t + CTp,d, j QTp,d, j ,t,e t∈T d∈D j∈J ⎡ ⎛ ⎛ ⎞ ⎞⎤ ∑∑∑ ∑ ∑ ∑ − ⎣HP ⎝ ⎝MAP QPP ⎠ + QT P ⎠⎦p, j m, j p,q, j ,t−T T S ,e p,d, j,t,e t∈T s∈S j∈ s,pJ m∈M q∈Qp d∈D ∑ ∑[ ] − CDPp, jαP Pp, j Q p, j ,t,e + CSubP P + P+ − P−p, j QSubp, j ,t,e + ϕp, j I p, j,t,e + ϕp, j I p, j,t e ∀p ∈ P, e ∈ E, t∈T j∈J (3) 123 69 Page 14 of 46 Int. J. Appl. Comput. Math (2021) 7:69 ∑∑∑( ) BD D Dd,e  PVd,r , j QTd,r , j,t,e t∈T r∈R j∈J ∑∑∑[[ ( ) ] − PV P 1− δD + PV DP P Dp,d, j d, j p,d, jβp,mδd, j t∈T p∈P j∈J ( ) ] 1− μPp, j QT P D Dp,d, j,t,e + CDd, j δd, j QT PDdpjt ∑∑∑[ ] − CFT D D Dd,r , j Wd,r , j,t + CTd,r , j QTd,r , j,t,e t∈T d∈D j∈J ⎡ ⎛ ⎞ ⎤ ∑∑ ∑ ∑ − ⎣HD ⎝ p D ⎠ + D+ − D− ⎦d, j QTp,d, j,t−T T P + QTd,r , j,t,e + γd, j Id, j,t,e + γd, j Id, j,t,e ∀d ∈ p,d,et T j∈J p∈P r∈R ∈ D, e ∈ E (4) ∑∑∑( ) ∑∑∑[ ] BR  PV R R D Dr ,e r ,c, j QTr ,c, j,t,e − PVd,r , j QTd,r , j,t,e t∈T c∈C j∈J t∈T d∈D j∈J [ ( )] ∑∑ ∑ ∑ − HR Dr , j QT − D + QT Rd,r , j,t T T r ,c, j,t,e ∈ ∈ ∈ d,r,et T j J d D r∈R ∑∑[ ] − ϑ+ I R+ − R−r , j r , j,t,e + ϑr , j Ir , j,t,e ∀r ∈ R, e ∈ E t∈T j∈J (5) QPPp,q, j,t,e  QPRP Pp,q, j,t,e + QPEp,q, j,t,e ∀p ∈ P, q ∈ Q, j ∈ J , t ∈ T , e ∈ E (6) CapPRPp,q,t kk ∀p ∈ P, q ∈ Q, t ∈ T , e ∈ E (7) ∑ QPEP Pp,q, j,t,e ≤ CapPEp,q,t ∀p ∈ P, q ∈ Q, t ∈ T , e ∈ E (8) j∈J ∑ ∑ Vp,q, j,t  1 ∀p ∈ P, t ∈ T (9) q∈Q j∈Jq Up,q, j,t ≥ Vp,q, j,t − Vp,q, j,t−1 ∀p ∈ P, q ∈ Q, j ∈ J , t ∈ T (10) QSubP P Pp, j,t,e ≤ Subp, j Q p, j,t,e ∀p ∈ P, j ∈ J , t ∈ T , e ∈ E (11) ⎛ ⎞ ∑ ∑ QT S ⎝ P P ⎠s,p,m,t,e  MAm, j QPp,q, j,t−T T S ∀ss,p,e j∈Jm q∈Qp ∈ S, p ∈ P,m ∈ M, t ∈ T , e ∈ E (12) QT S Ss,p,m,t,e ≤ CapTs,p,mXs,p,m,t ∀s ∈ S, p ∈ P,m ∈ M, t ∈ T , e ∈ E (13) QT P Pp,d, j,t,e ≤ CapTp,d, j Yp,d, j,t ∀p ∈ P, d ∈ D, j ∈ J , t ∈ T , e ∈ E (14) QT D Dd,r , j,t,e ≤ CapTd,r , jWd,r , j,t ∀d ∈ D, r ∈ R, j ∈ J , t ∈ T , e ∈ E (15) QT R Cr ,c, j,t,e ≤ Demc, j,t,e ∀r ∈ R, c ∈ C, j ∈ J , t ∈ T , e ∈ E (16) ∑ QTT S Ss,p,t,e  QTs,p,m,t,e ∀s ∈ S, p ∈ P, t ∈ T , e ∈ E (17) m∈M 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 15 of 46 69 ∑ QTT P  QT Pp,d,t,e p,d, j,t,e ∀p ∈ P, d ∈ D, t ∈ T , e ∈ E (18) ∑j∈J QT T D Dd,r ,t,e  QTd,r , j,t,e ∀d ∈ D, r ∈ R, t ∈ T , e ∈ E (19) ∑j∈J QT T Rr ,c,t,e  QT Rr ,c, j,t,e ∀r ∈ R, c ∈ C, t ∈ T , e ∈ E (20) ∑ j∈J QT T S Ps,p,t,e ≤ CapT Ip ∀p ∈ P, t ∈ T , e ∈ E (21) ∑s∈S p QT T P Pp,d,t,e ≤ CapT Op ∀p ∈ P, t ∈ T , e ∈ E (22) d∑∈Dp QT T Dd,r ,t,e ≤ CapT ODd ∀d ∈ D, t ∈ T , e ∈ E (23) r∈∑Rp QT T Rr ,c,t,e ≤ CapT ORr ∀r ∈ R, t ∈ T , e ∈ E (24) c∈C(p ) ∑ I S Ss,m,t,e  Is,m,t−1,e + 1− αS QS Ss,m s,m,t,e − QTs,p,m,t,e ∀s ∈ S,m ∈ M, t ∈ T , e ∈ E (25) ( ) p∑∈P I Pp, j,t,e  I Pp, j,t− + 1− αP P P1,e p, j Q p, j,t,e − QTp,d, j,t,e ∀p ∈ P, j ∈ J , t ∈ T , e ∈ E (26) ∑ d∈D∑ I D D D Dd, j,t,e  Id, j,t−1,e + QD − QTp,d, j,t−T T P ,e d,r , j,t,e ∀d ∈ D, j ∈ J , t ∈ T , e ∈ Ep,d p∈P r∈R ∑ ∑ (27) I Rr , j,t,e  I Rr , j,t−1,e + QRR − QT R ∀r ∈ R, j ∈ J , t ∈ T , e ∈ E (28)d,r , j,t−T T D ,e r ,c, j,t,e d∈∑ d,rD c∈C I S Ss,m,t,e ≤ Imaxs ∀s ∈ S, t ∈ T , e ∈ E (29) ∑m∈M I Pp, j,t,e ≤ Imax Pp ∀p ∈ P, t ∈ T , e ∈ E (30) ∑j∈J I Dd, j,t,e ≤ ImaxDd ∀d ∈ D, t ∈ T , e ∈ E (31) j∑∈J I R Rr , j,t,e ≤ Imaxr ∀r ∈ R, t ∈ T , e ∈ E (32) j∈J I Ss,m,t,e  I S+ S−s,m,t,e − Is,m,t,e ∀s ∈ S,m ∈ M, t ∈ T , e ∈ E (33) I Pp, j,t,e  I P+ P−p, j,t,e − Ip, j,t,e ∀p ∈ P, j ∈ J , t ∈ T , e ∈ E (34) I D D+ D−d, j,t,e  Id, j,t,e − Id, j,t,e ∀d ∈ D, j ∈ J , t ∈ T , e ∈ E (35) I R  I R+ − I R−r , j,t,e r , j,t,e r , j,t,e ∀r ∈ R, j ∈ J , t ∈ T , e ∈ E (36) The variables (QSs,m,t,e, Q P p, j,t,e, QP P P P P p,q, j,t,e, QPRp,q, j,t,e, QPEp,q, j,t,e, QSubp, j,t,e, QT S P D R S P Ds,p,m,t,e, QTp,d, j,t,e, QTd,r , j,t,e, QTr ,c, j,t,e, QTTs,p,t,e, QTTp,d,t,e, QTTd,r ,t,e, QTT R S+ S− P+ P− D+ D− R+ R−r ,c,t,e„ Is,m,t,e, Is,m,t,e, Ip, j,t,e, Ip, j,t,e, Id, j,t,e, Id, j,t,e, Ir , j,t,e, Ir , j,t,e) are nonneg- ative here. Equations 1–5 contain the general objective function of the model. Equation 2 represents the benefits obtained by raw material suppliers, because of the sales of products (materials) and the recoverable fraction of defective products sold at a lower price to the plants. This equation also deducts handling costs (reception and dispatch), purchase costs of raw mate- rials, cost of excess and shortage inventory, costs of inventory of the recoverable defective products, and costs of transporting materials. Equation 3 represents the benefits obtained by the plants because of the income from the sale of conforming products and with minor defects that are accepted by the distribution centers at a lower price. This equation also 123 69 Page 16 of 46 Int. J. Appl. Comput. Math (2021) 7:69 deducts fixed manufacturing costs, material handling costs, total manufacturing costs (nor- mal and overtime), subcontracting costs, material reception and dispatch costs, raw material purchase costs, cost of excess and shortage inventory, including the costs of defective product inventory and the cost of transportation. Equation 4 represents the benefits obtained by the distribution centers because of the income from the sale of products to retailers, deducting the costs of receipt and dispatch, the total cost of purchases from the plants, cost of excess and shortage inventory, defective product inventory costs, and transportation costs. Equation 5 represents the benefits obtained by retailers because of the income from the sale of products to customers, deducting the costs of receipt and dispatch, the total cost of purchases from the plants, cost of excess and shortage inventory, including the inventory of defective products detected and transportation costs. Equation 6 represents the amount of total production considering regular hours and over- time products. Equations 7 and 8 denote the maximum production capacity available during regular working hours and overtime. Equations 9 and 10 specify that the manufacturing plant prepares for production or a possible change in a period. Equation 11 specifies the maximum amount of outsourcing. Equations 12–16 specify the quantity to be produced in a period. Equations 17–20 specify the total quantities transported for each product in a period. Equa- tion 21 specifies the maximum inbound transportation capacity to the plant from suppliers. Equations 22–24 specify the maximum outbound transportation capacity from the plant to the distribution center and from the distribution center to the retailer, and from the retailer to the customer, respectively. Equations 25–28 correspond to the inventory balance equations at each stage of the supply chain. Equations 29–32 represent the maximum inventory capacity in each of the stages of the chain. Equations 33–36 regulate the level of total inventory as the occurrence of excess inventory or shortage inventory at each stage of the supply chain. Case Study This section presents a numerical example to illustrate the proposed model, considering a cement company that produces and distributes cement and concrete. Figure 1 shows the schematic diagram of the cement manufacturing process includingthe grinding lime- stone, homogenization, preheating, clinkerization, cooling, clinker storage, cement grinding, cement storage, packing, and delivery. Cement is produced from raw materials likelimestone or calcium carbonate, chert, iron ore, gypsum, coal, and slag. The limestone is extracted in quarries and transported in trucks to the grinding limestone section,where the size of the rock is reduced and send to raw material storage with a capacity of 50.000 tons. The raw materials are transported and dosed to the crude mill with the iron ore to adjust the levels of silica and iron oxide in the mix, which is discharged into the ponds with a capacity of 8000 tons (Fig. 1). The mix produced in the mills is homogenized in each pond and send to the preheater tower. In the rotary kiln, physical and chemical reactions allow the formation of clinker. The heat exchange occurs through heat transfers between the homogenized crude and the hot gases that are obtained from the preheater tower at high temperatures. The clinker obtained is subjected to a rapid cooling process, then it is taken to the clinker silos. The clinker is conducted to the cementmillswhere it is ground togetherwith the aggregates or additives, gypsum, and slag. At this point, the cement is ready to be transported and deposited in storage silos. The cement delivery process includes the extraction of cement 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 17 of 46 69 Fig. 1 Benefits of the supply chain per month. (echelons vs. Profit ($COP Million)) from the silos, transport, and loading.Modeling is assumed and validated in the supply chain of cement production. In the supplier stage, there are two (S1 and S2). It has two plants (P1 and P2). In distribution, two independent national distributors (D1 and D2) are considered. In the stages of retailers, they include the national companies that market the largest volumes of products to small builders and hardware stores (R1 and R2). The physical architecture of the stages of the supply chain to be applied is shown in Fig. 2. Two production resources (Q1, Q2) and two final products (A, B) are considered. The demand is assumed for six months. The data used to validate the modelis provided by the cement company. The data contains the parameters of suppliers, production plants, distribution centers, and retailers considering selling prices, handling costs, inventory excess costs, inventory shortage costs, production costs, disposal costs, fixed transportation costs, purchasing costs and transportation costs. Also, the percentage of defective items, screening rates, transport capacities, transportation times, inventory levels, the combination of materials, production capacities and demand rates 123 69 Page 18 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Fig. 2 Physical architecture of the stages of the supply chain Table 2 Sets of the model Sets s Suppliers 1,2 m Raw materials Gypsum, slag, clinker, sacks j Products A, B p Plants 1,2 q Production resources Q1, Q2 d Distribution centers 1,2 r Retailers 1,2 c Clients 1,2 e Scenarios High, medium, low t Periods 1,2,3,4,5,6 are included. The sets of the model are presented in Table 2. The demand rate of client is provided in Table 3. The proposed model considers three scenarios for the demand rate of the client: high, medium, and low (SeeTable 4) and the probability of each scenario is 33.33%.The parameters related to incomes and costs of suppliers, production plants, distribution centers and retailers are presented in Tables 4–8. The parameters of production and transportation capacities, inventory levels, and defective items of distribution centers are presented in Appendices A–D. 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 19 of 46 69 Table 3 Demand rate of clients Demand rate of client (tons) c j 1 2 3 4 5 6 1 A 17,000 1,000 15,000 10,000 8,000 15,000 1 B 8,000 14,000 8,000 8,000 7,000 9,000 2 A 10,000 14,400 8,000 9,000 10,000 8,000 2 B 8,000 9,000 9,000 3,000 4,000 1,000 Table 4 Demand rate of clients Demand rate of client (tons) for each scenario c j e 1 2 3 4 5 6 1 A Low 15,000 800 13,000 8,000 6,000 13,000 1 B Low 6,000 10,000 5,800 9,000 10,000 9,000 2 A Low 9,000 12,400 6,000 7,000 6,000 5,000 2 B Low 1,500 2,000 3,200 2,000 10,000 800 1 A Medium 17,000 1,000 15,000 10,000 8,000 15,000 1 B Medium 8,000 14,000 8,000 8,000 2,000 9,000 2 A Medium 10,000 14,400 8,000 9,000 8,000 8,000 2 B Medium 8,000 9,000 9,000 3,000 7,000 1,000 1 A High 19,000 1,200 17,000 12,000 10,000 17,000 1 B High 10,000 20,000 8,500 8,500 8,000 11,000 2 A High 17,000 18,400 10,000 11,000 11,000 10,000 2 B High 9,000 10,000 10,800 4,800 5,000 1,200 Results The proposedmodel for themaster planning of operations in the supply chainwith uncertainty seeks to maximize the profit margin of all the actors in the supply chain and it is validated through the data of a cement company. The model is solved with GAMS (General Algebraic Modeling System) under the XPRESS solver and likewise, the data is processed in Neos Solvers web support. In Table 9, the solution of the model under uncertainty is shown in the three established scenarios. Profit in high scenario is $ 5,146,132,791 COP. Profit in the middle scenario is $ 5,048,779,281 COP. Profit in low scenario is $ 4,532,596,643 COP. Additionally, the expected total benefit value is $ 4,909,164,663 COP. Likewise, the benefits derived for each actor in the supply chain are shown in Table 10. Plant P1 is the largest producer, mainly producers of type A products. This product has the highest demand, and its manufacturing costs are lower. The plants distribute their production between resource Q1 and Q2, manufacturing larger quantities in Q2 for periods 3 and 4. Plant 1 is transporting the largest number of products, directing it to distribution center D2 in a higher proportion than D1. Products are transported from distributors to retails, with product A being the most marketed compared to product B. Retailer R2 receives a greater quantity of products than the retailer R1. Product A (61%) is supplied to the customer from the retailers in greater quantity. Retailer R1 is the one that supplies the largest quantities of product B. Retailer R1 is the one that supplies the largest quantities of product B. Retailer R2 sup- plies a greater proportion of product A. The most significant costs at the suppliers are the 123 69 Page 20 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Table 5 Parameters of suppliers PV Ss,p,m Selling price of raw material ($- COP) s p Gypsum Slag Clinker Sacks 1 1 20,825 20,298 59,358 2,958,501 1 2 20,825 20,298 59,358 2,958,501 2 1 20,630 20,546 59,849 2,988,086 2 2 20,630 20,546 59,894 2,988,086 PV DSs,p,m Selling price per unit defective item ($-COP) s p Gypsum Slag Clinker 1 1 13,101 11,979 44,712 2 1 12,958 12,158 44,809 CPSs,m,t Production cost per unit of raw material ($-COP) s m 1 2 3 4 5 6 1 Gypsum 1,101 1,101 1,101 1,101 1,101 1,101 1 Slag 1,979 1,979 1,979 1,979 1,979 1,979 1 Clinker 4,712 4,712 4,712 4,712 4,712 4,712 1 Sacks 218,876 218,876 218,876 218,876 218,876 218,876 2 Gypsum 1,101 1,101 1,101 1,101 1,101 1,101 2 Slag 1,979 1,979 1,979 1,979 1,979 1,979 2 Clinker 4,712 4,712 4,712 4,712 4,712 4,712 2 Sacks 218,876 218,876 218,876 218,876 218,876 218,876 CTSs,p,m Transportation cost per unit from the supplier to production plant ($- COP) s p Gypsum Slag Clinker Sacks 1 1 3,374 3,099 1,200 69,344 1 2 3,585 3,148 1,150 70,037 2 1 3,189 3,281 1,230 69,691 2 2 3,368 3,292 1,180 68,994 HSs,m Handling cost per unit at supplier for the raw material ($-COP) s Gypsum Slag Clinker Sacks 1 641 500 483 6,693 2 565 630 579 6,760 π+s,m Inventory excess cost at supplier for the raw material ($-COP) s Gypsum Slag Clinker Sacks 1 367 276 430 8,284 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 21 of 46 69 Table 5 continued π+s,m Inventory excess cost at supplier for the raw material ($-COP) s Gypsum Slag Clinker Sacks 2 251 269 383 8,367 π−s,m Inventory shortage cost at supplier for the raw material ($-COP) s Gypsum Slag Clinker Sacks 1 5,348 4,889 16,607 887,550 2 5,289 4,964 16,408 887,550 CDSs,m Disposal cost per unit defective item at supplier for raw material ($-COP) s Gypsum Slag Clinker Sacks 1 367 276 430 284 2 251 269 383 367 manufacturing costs of the raw material. The costs associated with detecting defective raw material at the supplier are higher in S2 since a greater quantity is handled. The costs of purchasing raw materials represent an important part of the costs. Manufacturing costs are also important within the total costs since the cement manufacturing process involves high energy consumption. Figure 2 shows the benefits of the supply chain per month. The results show that suppliers participate in 47% of the total profits, plants receive 30%,distribution centers and retailers receive 23%. Distributors and retailers have a lower contribution. In addition to inventory costs(these assume the costs for defective or deterioration of the products), these results show the positive effects of the chain in a collaborative system and, if not, the total percentages of profits between retailers and distributors would be less than 10%. The results of the model include the optimal (values )of the variables’ pro- (duction qua)ntity of raw material to purchase Q S s,m,t,e and to transport QT Ss,p,m,t,e from the suppliers to the plants and the total quantity to trans-( ) port QTT Ss,p,t,e . The production plantspresent thenumber of finished items to( ) produce QP , QPP P P P Sp, j,t,e p,q, j,t,e, QPRp,q, j,t,e, QPEp,q, j,t,e, QTp,d, j,t,e, QTT( s,p,t,e) andtransport from the plants to the distribution centers QT P Pp,d, j,t,e, QTTp,d,t,e . (Also, the quantitiesto) transport from the distribu(tion centers to the) retailers QT Dd,r , j,t,e, QTT D d,r ,t,e , from retailers to clients QT R R ( ) r ,c, j,t,e , QTTr ,c,t,e , and the inventory levels I P− D+ R+p, j,t,e, Id, j,t,e, Ir , j,t,e, I R− r , j,t,e are presented in Appendix E. The optimal production quantity of finished products in each production plant is presented in Table 11 and the optimal values of total quantities to transport from suppliers, production plants, distribution centers, and retailers are presented in Tables 12–15. 123 69 Page 22 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Table 6 Parameters of production plants PV Pp d j Selling price per unit good item at production plant ($-COP), , p d A B 1 1 12,907 13,022 1 2 12,907 13,022 2 1 12,505 13,210 2 2 12,505 13,210 PV DPp d Selling price per unit defective item at production plant ($-COP), , j p d A B 1 1 9,680 9,766 1 2 9,680 9,766 2 1 9,449 9,684 2 2 9,449 9,684 CPRPp q j t Production cost per unit at the production plant for the finished product working, , , in regular time ($-COP) p q j 1 2 3 4 5 6 1 Q1 A 4,374 4,577 4,741 4,316 4,697 4,588 1 Q1 B 4,340 4,250 4,843 4,078 4,514 4,472 1 Q2 A 4,505 4,848 4,104 4,282 4,591 4,087 1 Q2 B 4,751 4,508 4,446 4,972 4,824 4,269 2 Q1 A 4,737 4,201 4,693 4,351 4,031 4,525 2 Q1 B 4,710 4,224 4,940 4,739 4,383 4,660 2 Q2 A 4,241 4,320 4,068 4,738 4,033 4,724 2 Q2 B 4,972 4,449 4,779 4,706 4,402 4,409 CPEPp q j t Production cost per unit at the production plant for the finished product working, , , in extra time ($-COP) p q j 1 2 3 4 5 6 1 Q1 A 14,374 14,577 4,741 4,316 4,697 4,588 1 Q1 B 14,340 14,250 4,843 4,078 4,514 4,472 1 Q2 A 14,505 14,848 4,104 4,282 4,591 4,087 1 Q2 B 14,751 14,508 4,446 4,972 4,824 4,269 2 Q1 A 14,737 14,201 4,693 4,351 4,031 4,525 2 Q1 B 14,710 4,224 4,940 4,739 4,383 4,660 2 Q2 A 14,241 4,320 4,068 4,738 4,033 4,724 2 Q2 B 14,972 4,449 4,779 4,706 4,402 4,409 CSubPp j Purchasing cost per unit subcontracted by the production plant ($-COP), p A B 1 15,000 15,000 2 15,000 15,000 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 23 of 46 69 Table 6 continued HPp Handling cost per unit at production plant ($-COP), j p A B 1 397 397 2 338 338 CT Pp d j Transportation cost per unit for the finished product from production plant to, , distribution center ($-COP) p d A B 1 1 517 517 1 2 562 562 2 1 595 595 2 2 598 598 ϕ+p j Inventory excess cost at production plant for the finished product ($-COP), p A B 1 444 444 2 482 482 CDPp j Disposal cost per unit defective item at production plan for the finished product, ($-COP) p A B 1 244 244 2 215 215 ϕ−p j Inventory shortage cost at production plant for the finished product ($-COP), p A B 1 3,872 3,907 2 3,890 3,937 Managerial Implication • In the operational context, thismodel presents a significant contribution to themanagement of the production system, as it supports decision-making related to the integration of the chain actors, the programming and control of production and distribution, efficient management of resources, and optimization of the level of customer service. Likewise, it guarantees the best use of capacities and optimizes inventory management in each of the chain’s echelons, reducing the impact of the "bullwhip effect" on the chain. • In the economic context, the model allows the maximization of collective benefits, consid- ering the collaboration between the actors in the chain. There is a possibility to purchase of products withminor defects at lower price which not only allows the generation of adjacent income but also reduces the costs associated with transportation because of returns, and environmental consideration. 123 69 Page 24 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Table 7 Parameters of distribution centers PV Dd r j Selling price per unit good item at distribution center to the retailer for the finished, , product ($-COP) d r A B 1 1 17,216 17,365 1 2 17,216 17,365 2 1 17,296 17,500 2 2 17,296 17,500 CT Dd,r Fixed transportation cost per unit from distribution center d to the retailer r for the finished product, j ($-COP) d r A B 1 1 314 314 1 2 338 338 2 1 315 315 2 2 342 342 HDd j Handling cost per unit at the distribution center for the finished product ($-COP), d A B 1 22 22 2 23 23 γ +d j Excess inventory cost at distribution center for the finished product ($-COP), d A B 1 69 69 2 69 69 γ−d j Shortage inventory cost at distribution center for the finished product ($-COP), d A B 1 5,165 5,209 2 5,189 5,250 CDDd j Disposal cost per unit defective item at distribution center for the finished product ($-COP), d A B 1 840 840 2 820 820 • Another fundamental aspect provided by the model is related to quality management by considering factors for detecting defects in the materials and products received, which guarantee customer satisfaction throughout the downstream chain. • Finally, the model presents a good approximation of the behavior of demand in the real world. 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 25 of 46 69 Table 8 Parameters of retailers PV Rr,c, j Selling price per unit good item at retailer to the client for the finished product ($-COP) r c A B 1 1 21,813 22,007 1 2 21,813 22,007 1 1 21,913 22,178 1 2 21,913 22,178 HRr j Handling cost per unit at the retailer for the finished product ($-COP) r A B 1 13 13 2 13 13 γ +r, j Excess inventory cost at retailer for the finished product ($-COP) r A B 1 87 88 2 88 89 ϑ−r, j Shortage inventory cost at retailer for the finished product ($-COP) r j A B 1 A 6,544 6,602 2 B 6,574 6,653 Table 9 Total benefits in the BT(r) ($ COP) BT exp(r) ($ COP) supply chain by scenarios Low 5,146,132,792 1,715,375,882 Medium 5,048,779,282 1,682,924,744 High 4,532,596,643 1,510,864,037 4,909,164,663 Table 10 Expected benefits in the supplier, plant, distribution center, and retailer Scenario S BS(s) Million ($ P BP(p) Million D BD(d) Million R BR(r) Million COP) ($ COP) ($ COP) ($ COP) Low S1 1,063 P1 716.8 D1 222.4 R1 206.1 Medium S1 1,151 P1 716.4 D1 228.6 R1 341.3 High S1 1,151 P1 716.4 D1 228.6 R1 438.9 Low S2 1,068 P2 704.9 D2 225.7 R2 324.7 Medium S2 1,157 P2 744.4 D2 256.1 R2 453.2 High S2 1,157 P2 744.4 D2 256.1 R2 452.9 123 69 Page 26 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Table 11 Optimal values of QP Production quantity of the finished product j at the plant p in production quantities of finished p, j ,t,eperiod t (sacks) products p j e 1 2 3 4 5 6 1 A High 21,120 7,751 11,080 15,962 9,187 1 A Medium 21,120 7,751 11,080 15,962 9,187 1 A Low 21,120 9,571 11,080 11,711 9,187 1 B High 2,811 6,768 1 B Medium 2,811 6,768 1 B Low 991 11,019 2 A High 5,755 216 10,589 9,620 2 A Medium 5,755 216 10,589 9,620 2 A Low 3,919 216 11,194 9,620 2 B High 10,522 22,018 605 5,960 12,714 2 B Medium 10,522 22,018 605 5,960 12,714 2 B Low 4,881 22,018 1,718 12,714 Table 12 Optimal values of total QTT Ss,p,t,e Total quantity to transport from the supplier s tothequantities to transport from plant p in period t (ton) suppliers s p e 1 2 3 4 5 6 1 1 High 1,098 549 576 1,182 478 1 1 Medium 1,098 549 576 1,182 478 1 1 Low 1,098 549 576 1,182 478 1 2 High 846 1,156 582 310 1,161 1 2 Medium 846 1,156 582 310 1,161 1 2 Low 458 1,156 582 89 1,161 2 1 High 1,098 549 576 1,182 478 2 1 Medium 1,098 549 576 1,182 478 2 1 Low 1,098 549 576 1,182 478 2 2 High 846 1,156 582 310 1,161 2 2 Medium 846 1,156 582 310 1,161 2 2 Low 458 1,156 582 89 1,161 Discussion and Conclusions The mathematical model has been developed under the approach of collaboration and uncer- tainty in the demand in a supply chain of the cement sector which is proved to be a valuable tool for making decisions towards the maximization of profits in all actors of the supply chain. The proposed model considers critical decision variables, and it isused in investi- gations such as defective products, excesses, and shortages. Validating the model in a real scenario, common benefit exceeds 5 billion for all members in a period of six months. The present model identifies the benefits of collaborative planning [68] and it allows the reduction in cycle times, greater flexibility in the processes associate with orders and deliveries, and the decrease in inventory levels for maximizing the profit of channel members. 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 27 of 46 69 Table 13 Optimal values of total quantities to transport from plants QTT Pp d t e Total quantity to transport from the plant p to the distribution center d in period t (sacks), , , p d e 1 2 3 4 5 6 1 A High 10,453 5,562 10,026 14,174 9,000 10,090 1 A Medium 10,453 5,562 10,026 14,174 9,000 10,090 1 A Low 10,453 7,270 10,026 14,174 9,000 10,090 1 B High 11,116 4,876 921 8,290 76.62 11,290 1 B Medium 11,116 4,876 921 8,290 76.62 11,290 1 B Low 11,116 3,166 921 8,466 76.62 11,290 2 A High 7,325 10,910 400 6,200 10,910 2 A Medium 7,325 10,910 400 6,200 10,910 2 A Low 4,177 10,910 172 6,200 10,910 2 B High 9,264 11,124 10,683 5,906 15,923 10,910 2 B Medium 9,264 11,124 10,683 5,906 15,923 10,910 2 B Low 5,004 11,124 10,910 1,703 15,923 10,910 Table 14 Optimal values of total quantities to transport from distribution centers QTT Dd r t e Total quantity to transport from the distribution center d to the retailer r in period t (sacks), , , d r e 1 2 3 4 5 6 1 1 High 6,800 6,800 6,800 6,800 6,800 6,800 1 1 Medium 6,800 6,800 6,800 6,800 6,800 6,800 1 1 Low 5,470 6,690 6,400 6,800 6,800 6,800 1 2 High 8,400 8,400 8,400 8,400 8,400 8,400 1 2 Medium 8,400 8,400 8,400 8,400 8,400 8,400 1 2 Low 8,400 8,400 8,400 8,400 8,400 8,400 2 1 High 8,200 8,200 8,200 8,200 8,200 8,200 2 1 Medium 8,200 8,200 8,200 8,200 8,200 8,200 2 1 Low 4,400 7,690 6,800 4,600 8,200 8,200 2 2 High 7,800 7,800 7,800 7,800 7,800 7,800 2 2 Medium 7,800 7,800 7,800 7,800 7,800 7,800 2 2 Low 7,340 7,800 7,800 7,800 7,800 7,800 Additionally, themodel allows theflexibility of the processes as indicatedbyBinder andClegg [7]. If resources are required, and when these are not available, proper planning will support the exchange of resources between agents in the supply chain. The approach of themodel that allows validating the defective products within the chain, helps to improve the quality within the processes as mentioned by Sarkar et al. [58]. In this model, the collaboration between suppliers and customers will help to reduce poor quality costs and thus efficiently impact customer service. Finally, product inventories are kept in their economic balance as stated by Huiskonen [29]. As a result, collaborative planning improves the availability of products for customers. 123 69 Page 28 of 46 Int. J. Appl. Comput. Math (2021) 7:69 Table 15 Optimal values of total QTT Rr ,c,t,e Total quantity to transport from the retailer r to the client c inquantities to transport from period t (Sacks) retailers r c e 1 2 3 4 5 6 1 1 High 8,000 200 3,100 2,600 6,800 1 1 Medium 10,740 2,740 2,600 1,600 6,800 1 1 Low 13,800 6,600 2,600 3,600 7,400 9,200 1 2 High 7,450 14,800 15,000 12,700 13,200 9,000 1 2 Medium 5,060 13,060 15,800 11,200 11,200 9,000 1 2 Low 1,500 2,400 9,200 6,200 5,200 5,800 2 1 High 14,650 2,600 10,400 13,100 13,400 14,000 2 1 Medium 8,500 5,860 15,000 15,400 13,400 16,200 2 1 Low 7,200 4,200 16,200 13,400 8,600 12,800 2 2 High 1,550 13,600 5,800 3,100 2,800 2,200 2 2 Medium 7,700 10,340 1,200 800 2,800 2 2 Low 9,000 12,000 2,800 4,800 In addition to the previously mentioned benefits, the proposed collaborative model pro- vides a competitive advantage focused on generating transparency in production processes, reducing response times, and minimizing potential conflicts between chain actors as estab- lished by Sarkar et al. [59]. Optimization shares the benefits, giving importance to all the actors in the chain, and generating confidence so that everyone cooperates as indicated by Zhang and Huang [73]. Finally, among other novelty, the reverse flow of defective products is studied in the proposed model. Another novelty of this proposed model is related to the objectives of a sustainable chain that helps to reduce the carbon footprint [42] and additionally integrates concepts of reverse and green logistics for the use of raw materials. The proposed model can be expanded in future research, including the production safety inventory. The model can incorporate the features of uncertainty so that the design is com- pletely stochastic based on different policies of collaboration and integration in the chain. It is possible to combine the collaborative approach, grouping the actors, and mixing the type of collaboration (information, capacity, inventories, among others). Amodel can be proposed that includes Shapley Value, where agents are part of the chain who collaborate and form coalitions in such a way that costs are minimized, and better profits are obtained. Author contributions JAC, KSN, HOM, AV: literature review, model formulation, mathematical analysis and numerical solution, data analysis; SSS: overall supervision, model formulation. Funding None. Compliance with ethical standards Conflict of interest The authors do hereby declare that there is no conflict of interests of other works regarding the publication of this paper. This article does not contain any studies with human participants or animals performed by any of the authors and materials of third party are not used in this article. 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 29 of 46 69 Appendix A: Parameters values of production and transportation capacities, inventory levels, and defective items of suppliers I oSs,m Initial inventory level at supplier for the raw material (ton) s Gypsum Slag Clinker Sacks 1 20 135 561 20 2 30 100 528 12 CapT Ss,p,m Transport capacity for the raw material from the supplier to the production plant (ton) s p Gypsum Slag Clinker Sacks 1 1 140,000 135,000 173,500 112,880 1 2 140,000 135,000 173,500 112,880 2 1 138,000 134,000 172,000 111,019 2 2 138,000 134,000 172,000 111,019 ImaxSs Maximum inventory capacity at the supplier (ton) s 1 380,186 2 370,759 FmaxSs,m Maximum production capacity at the supplier for the raw material (ton) s Gypsum Slag Clinker Sacks 1 50,000 55,000 53,500 22,880 2 48,000 44,000 82,000 21,019 αSs,m Expected percentage of defective items at supplier for the raw material s Gypsum Slag Clinker Sacks 1 0.016 0.013 0.011 0.018 2 0.015 0.017 0.015 0.017 123 69 Page 30 of 46 Int. J. Appl. Comput. Math (2021) 7:69 βSs,m Screening rate of defective items at supplierfor the raw material s Gypsum Slag Clinker Sacks 1 0.98 0.99 0.99 0.97 2 0.98 0.99 0.99 0.96 T T Ss,p Transportation time from supplier to production plant (hours) s Gypsum Slag Clinker Sacks 1 0.98 0.99 0.99 0.97 2 0.98 0.99 0.99 0.96 Appendix B. Parameters values of production and transportation capacities, inventory levels, and defective items of production plants I oPp j Initial inventory level at production plant for the finished product (sacks), p A B 1 275 427 2 239 226 Imax Pp Maximum inventory capacity at the production plant (sacks) p 1 89,580 2 89,200 CapT Pp d j Transport capacity for the finished product from production plant to distribution center (ton), , P d A B 1 1 10,026 10,926 1 2 19,026 10,926 2 1 10,910 10,910 2 2 10,910 10,910 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 31 of 46 69 CapPRPp,q,t Maximum production capacity at production plant working in regular time on the production resource in period time (Sacks) p q 1 2 3 4 5 6 1 Q1 10,547 10,562 10,512 10,684 10,563 50,522 1 Q2 10,573 10,559 10,540 10,681 10,547 10,554 2 Q1 10,543 10,563 10,565 10,688 10,591 10,592 2 Q2 10,537 10,554 10,597 10,669 10,576 10,524 CapPEPp,q,t Maximum production capacity at production plant working in extra time on the production resource in period time (Sacks) p q 1 2 3 4 5 6 1 Q1 547 562 512 684 563 522 1 Q2 573 559 540 681 547 554 2 Q1 543 563 565 688 591 592 2 Q2 537 554 597 669 576 524 CapT I Pp Maximum capacity of transport input in the plant (ton) p 1 49,580 2 49,200 CapT OPp Maximum capacity of transport output in the plant (sacks) p 1 48,643 2 42,249 μPp j Screening rate of defective items at production plant for the finished product, p A B 1 0.99 0.98 2 0.97 0.99 αPp j Expected percentage of defective items at production plant for the finished product, p A B 1 0.012 0.011 2 0.010 0.009 123 69 Page 32 of 46 Int. J. Appl. Comput. Math (2021) 7:69 βPp.m Screening rate of defective items at production plant for the raw material p Gypsum Slag Clinker Sacks 1 0.99 0.97 0.99 0.98 2 0.98 0.98 0.97 0.97 MAPm j Combination of raw material necessary to produce the finished product (sack), p Gypsum Slag Clinker Sacks 1 0.0025 0.0095 0.038 0.002 2 0.0025 0.007 0.0405 0.002 SubPp j Expected percentage of quantities to manufacture the finished product for subcontracting at, production plant p A B 1 0.1 0.1 2 0.1 0.1 T T Pp d Transportation time from the production plant to the distribution center (Hours), p 1 2 1 0 0 2 0 0 Appendix C. Parameters values of production and transportation capacities, inventory levels, and defective items of distribution centers I oDd Initial inventory level at distribution center for the finished product (sacks), j d 1 2 1 1,250 700 2 1,000 820 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 33 of 46 69 ImaxDd Maximum inventory capacity at distribution center (sacks) d 1 5,800 2 6,200 CapT ODd Maximum capacity of transport output in the distribution center (sacks) d 1 15,800 2 16,200 δDd j Screening-rate of defective items at distribution center for finished product, d A B 1 0.99 0.97 2 0.98 0.99 T T Dd r Transportation time from the distribution center to the retailer (sacks), d 1 2 1 0 0 2 0 0 Appendix D. Parameters values of production and transportation capacities, inventory levels, and defective items of retailers I oRr j Initial inventory level at retailer for the finished product (tons), d A B 1 200 250 2 220 240 Imax Rr Maximum inventory capacity at retailer (sacks) r 1 15,800 2 16,200 123 69 Page 34 of 46 Int. J. Appl. Comput. Math (2021) 7:69 CapT ORr Maximum capacity of transport input in the retailer (sacks) r 1 15,800 2 16,200 Appendix E. Optimal values of decision variables QSs,m,t,e Quantity of raw material m to purchase from the supplier s in period t (tons) s m e 1 2 3 4 5 1 Gypsum High 75 83 57 73 80 1 Gypsum Medium 75 83 57 73 80 1 Gypsum Low 56 83 57 62 80 1 Slag High 197 253 213 244 271 1 Slag Medium 197 253 213 244 271 1 Slag Low 139 257 214 203 271 1 Clinker High 896 1,323 857 1,135 1,243 1 Clinker Medium 896 1,323 857 1,135 1,243 1 Clinker Low 595 1,318 856 972 1,243 1 Sacks High 56 67 45 58 64 1 Sacks Medium 56 67 45 58 64 1 Sacks Low 41 67 45 50 64 2 Gypsum High 64 83 57 73 80 2 Gypsum Medium 64 83 57 73 80 2 Gypsum Low 45 83 57 62 80 2 Slag High 233 254 214 215 272 2 Slag Medium 233 254 214 215 272 2 Slag Low 175 258 215 204 272 2 Clinker High 933 1,328 861 1,139 1,248 2 Clinker Medium 933 1,328 861 1,139 1,248 2 Clinker Low 631 1,324 859 976 1,248 2 Sacks High 64 67 45 58 64 2 Sacks Medium 64 67 45 58 64 2 Sacks Low 49 67 45 50 64 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 35 of 46 69 QT Ss.p.m.t .e Quantity of raw material m to transport from the supplier s to the plant p in period t (Tons) s p m e 1 2 3 4 5 1 1 Gypsum High 53 26 28 57 23 1 1 Gypsum Medium 53 26 28 57 23 1 1 Gypsum Low 53 26 28 57 23 1 1 Slag High 201 93 105 199 87 1 1 Slag Medium 201 93 105 199 87 1 1 Slag Low 201 98 105 188 87 1 1 Clinker High 803 408 421 881 349 1 1 Clinker Medium 803 408 421 881 349 1 1 Clinker Low 803 404 421 891 349 1 1 Sacks High 42 21 22 45 18 1 1 Sacks Medium 42 21 22 45 18 1 1 Sacks Low 42 21 22 45 18 1 2 Gypsum High 40.69 55.59 27.99 14.90 55.84 1 2 Gypsum Medium 40.69 55.59 27.99 14.90 55.84 1 2 Gypsum Low 22.00 55.59 27.99 4.30 55.84 1 2 Slag High 128.33 156.18 104.83 41.72 180.39 1 2 Slag Medium 128.33 156.18 104.83 41.72 180.39 1 2 Slag Low 71.39 156.18 106.34 12.03 180.39 1 2 Clinker High 644.83 899.94 426.89 241.38 880.48 1 2 Clinker Medium 644.83 899.94 426.89 241.38 880.48 1 2 Clinker Low 346.59 899.94 425.37 69.58 880.48 1 2 Sacks High 32.55 44.47 22.39 11.92 44.67 1 2 Sacks Medium 32.55 44.47 22.39 11.92 44.67 1 2 Sacks Low 17.60 44.47 22.39 3.44 44.67 2 1 Gypsum High 52.80 26.41 27.70 56.83 22.97 2 1 Gypsum Medium 52.80 26.41 27.70 56.83 22.97 2 1 Gypsum Low 52.80 26.41 27.70 56.83 22.97 2 1 Slag High 200.64 93.31 105.26 199.02 87.28 2 1 Slag Medium 200.64 93.31 105.26 199.02 87.28 2 1 Slag Low 200.64 97.86 105.26 188.39 87.28 2 1 Clinker High 802.56 408.38 421.04 880.66 349.10 2 1 Clinker Medium 802.56 408.38 421.04 880.66 349.10 2 1 Clinker Low 802.56 403.83 421.04 891.29 349.10 2 1 Sacks High 42.24 21.12 22.16 45.46 18.37 2 1 Sacks Medium 42.24 21.12 22.16 45.46 18.37 2 1 Sacks Low 42.24 21.12 22.16 45.46 18.37 2 2 Gypsum High 41 56 28 15 56 2 2 Gypsum Medium 41 56 28 15 56 2 2 Gypsum Low 22 56 28 4 56 123 69 Page 36 of 46 Int. J. Appl. Comput. Math (2021) 7:69 QT Ss.p.m.t .e Quantity of raw material m to transport from the supplier s to the plant p in period t (Tons) s p m e 1 2 3 4 5 2 2 Slag High 128 156 105 42 180 2 2 Slag Medium 128 156 105 42 180 2 2 Slag Low 71 156 106 12 180 2 2 Clinker High 645 900 427 241 880 2 2 Clinker Medium 645 900 427 241 880 2 2 Clinker Low 347 900 425 70 880 2 2 Sacks High 33 44 22 12 45 2 2 Sacks Medium 33 44 22 12 45 2 2 Sacks Low 18 44 22 3 45 QPPp q j t e Quantity of product j to be produced in plant p on production resource q in period t (sacks). . . . p q j e 1 2 3 4 5 1 Q1 A High 10,547 7,751 4,600 1 Q1 A Medium 10,547 7,751 4,600 1 Q1 A Low 10,547 9,571 349,437 1 Q1 B High 2,811 6768 1 Q1 B Medium 2,811 6768 1 Q1 B Low 991 11,019 1 Q2 A High 10,573 11,080 11,362 9,187 1 Q2 A Medium 10,573 11,080 11,362 9,187 1 Q2 A Low 10,573 11,080 11,362 9,187 1 Q2 B High 1 Q2 B Medium 1 Q2 B Low 2 Q1 A High 2 Q1 A Medium 2 Q1 A Low 2 Q1 B High 10,522 11,126 11,182 2 Q1 B Medium 10,522 11,126 11,182 2 Q1 B Low 4,881 11,126 11,182 2 Q2 A High 5,755 216 10,589 9,620 2 Q2 A Medium 5,755 216 10,589 9,620 2 Q2 A Low 3,919 216 11,194 9,620 2 Q2 B High 10,892 605 5,960 1,532 2 Q2 B Medium 10,892 605 5,960 1,532 2 Q2 B Low 10,892 1718 1,532 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 37 of 46 69 QPRPp q j t e Production quantity of the finished product j at the plant p working in regular time on the. . . . production resource q in period t (sacks) p q j e 1 2 3 4 5 1 Q1 A High 10,547 7.751 3,916 1 Q1 A Medium 10,547 7.751 4,600 1 Q1 A Low 10,547 9,571 1 Q1 B High 2,811 6768 1 Q1 B Medium 2,811 6084 1 Q1 B Low 991 10,684 1 Q2 A High 10,573 10,540 10,681 8,640 1 Q2 A Medium 10,573 10,540 10,681 8,640 1 Q2 A Low 10,573 10,540 10,681 8,640 1 Q2 B High 1 Q2 B Medium 1 Q2 B Low 2 Q1 A High 2 Q1 A Medium 2 Q1 A Low 2 Q1 B High 10,522 10,563 10,591 2 Q1 B Medium 10,522 10,563 10,591 2 Q1 B Low 4,881 10,563 10,591 2 Q2 A High 5,755 10,589 9,620 2 Q2 A Medium 5,755 10,589 9,620 2 Q2 A Low 3,919 216 10,597 9620 2 Q2 B High 10,554 8,449 5,291 957 2 Q2 B Medium 10,554 8,449 5,291 957 2 Q2 B Low 10,338 1,718 957 123 69 Page 38 of 46 Int. J. Appl. Comput. Math (2021) 7:69 QPEPp q j t e Production quantity of the finished product j at the plant p working in extra time on the, , , , production resource q in period t (sacks) p q j e 1 2 3 4 5 1 Q1 A High 684 1 Q1 A Medium 1 Q1 A Low 350 1 Q1 B High 1 Q1 B Medium 684 1 Q1 B Low 335 1 Q2 A High 540 681 547 1 Q2 A Medium 540 681 547 1 Q2 A Low 540 681 547 1 Q2 B High 1 Q2 B Medium 1 Q2 B Low 2 Q1 A High 2 Q1 A Medium 2 Q1 A Low 2 Q1 B High 563 591 2 Q1 B Medium 563 591 2 Q1 B Low 563 591 2 Q2 A High 215.83 2 Q2 A Medium 215.83 2 Q2 A Low 597 2 Q2 B High 338 597 669 576 2 Q2 B Medium 338 597 576 2 Q2 B Low 554 576 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 39 of 46 69 QT Pp d j t e Quantity of product j to transport from the plant p to the distribution center d in period t (Sacks), , , , p d j e 1 2 3 4 5 6 1 1 A High 10,026 4,472 10,026 7,974 9,000 9,000 1 1 A Medium 10,026 4,472 10,026 7,974 9,000 9,000 1 1 A Low 10,026 6,290 10,026 7,802 9,000 9,000 1 1 B High 427 1,090 6,200 1,090 1 1 B Medium 427 1,090 6,200 1,090 1 1 B Low 427 980 6,200 1,090 1 2 A High 11,116 3,186 921 7,796 77 9,600 1 2 A Medium 11,116 3,186 921 7,796 77 9,600 1 2 A Low 11,116 3,186 921 3,769 77 9,600 1 2 B High 1,690 494 1,690 1 2 B Medium 1,690 494 1,690 1 2 B Low 4,698 1,690 2 1 A High 2,253 2 1 A Medium 2,253 2 1 A Low 434 172 2 1 B High 5,073 10,910 400 6,200 10,910 2 1 B Medium 5,073 10,910 400 6,200 10,910 2 1 B Low 3,743 10,910 6,200 10,910 2 2 A High 3,684 214 10,483 9,523 2 2 A Medium 3,684 214 10,483 9,523 2 2 A Low 3,684 214 10,910 9,523 2 2 B High 5,580 10,910 200 5,906 6,400 10,910 2 2 B Medium 5,580 10,910 200 5,906 6,400 10,910 2 2 B Low 1,320 10,910 1,703 6,400 10,910 123 69 Page 40 of 46 Int. J. Appl. Comput. Math (2021) 7:69 QT Dd r j t e Quantity of product j to transport from the distribution center d to the retailer r in period t, , , , (Sacks) d r j e 2 3 4 5 6 1 1 A High 3,400 3,400 3,400 3,400 3,400 3,400 1 1 A Medium 3,400 3,400 3,400 3,400 3,400 3,400 1 1 A Low 3,400 3,400 3,400 3,400 3,400 3,400 1 1 B High 3,400 3,400 3,400 3,400 3,400 3,400 1 1 B Medium 3,400 3,400 3,400 3,400 3,400 3,400 1 1 B Low 2,070 3,290 3,000 3,400 3,400 3,400 1 2 A High 5,600 5,600 5,600 5,600 5,600 5,600 1 2 A Medium 5,600 5,600 5,600 5,600 5,600 5,600 1 2 A Low 5,600 5,600 5,600 5,600 5,600 5,600 1 2 B High 2,800 2,800 2,800 2,800 2,800 2,800 1 2 B Medium 2,800 2,800 2,800 2,800 2,800 2,800 1 2 B Low 2,800 2,800 2,800 2,800 2,800 2,800 2 1 A High 4,400 4,400 4,400 4,400 4,400 4,400 2 1 A Medium 4,400 4,400 4,400 4,400 4,400 4,400 2 1 A Low 4,400 4,380 4,400 800 4,400 4,400 2 1 B High 3,800 3,800 3,800 3,800 3,800 3,800 2 1 B Medium 3,800 3,800 3,800 3,800 3,800 3,800 2 1 B Low 3,310 2,400 3,800 3,800 3,800 2 2 A High 5,200 5,200 5,200 5,200 5,200 5,200 2 2 A Medium 5,200 5,200 5,200 5,200 5,200 5,200 2 2 A Low 5,200 5,200 5,200 5,200 5,200 5,200 2 2 B High 2,600 2,600 2,600 2,600 2,600 2,600 2 2 B Medium 2,600 2,600 2,600 2,600 2,600 2,600 2 2 B Low 2,140 2,600 2,600 2,600 2,600 2,600 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 41 of 46 69 QT Rr c j t e Quantity of product j to transport from the retailer r to the client c in period t (sacks), , , , r c j e 1 2 3 4 5 6 1 1 A High 8,000 200 1 1 A Medium 10,740 1,000 2,740 1 1 A Low 12,980 2,200 7,400 1 1 B High 3,100 2,600 6,800 1 1 B Medium 1,740 2,600 1,600 4,060 1 1 B Low 820 6,600 400 3,600 7,400 1,800 1 2 A High 7,600 7,800 7,900 8,200 9,000 1 2 A Medium 4,060 7,400 8,200 7,200 8,000 1 2 A Low 2,400 6,000 4,200 3,200 5000 1 2 B High 7,450 7,200 7,200 4,800 5,000 1 2 B Medium 5,060 9,000 8,400 3,000 4,000 1,000 1 2 B Low 1,500 3,200 2,000 2,000 800 2 1 A High 10,560 8,600 7,700 8,000 9,800 2 1 A Medium 6,260 10,200 10,000 8,000 11,260 2 1 A Low 2,020 800 10,800 8,000 6,000 5,600 2 1 B High 4,090 2,600 1,800 5,400 5,400 4,200 2 1 B Medium 2,240 5,860 4,800 5,400 5,400 4,940 2 1 B Low 5,180 3,400 5,400 5,400 2,600 7,200 2 2 A High 10,800 2,200 3,100 2,800 1,000 2 2 A Medium 4,760 10,340 600 800 2,800 2 2 A Low 9,000 10,000 2,800 4,800 2 2 B High 1,550 2,800 3,600 1,200 2 2 B Medium 2,940 600 2 2 B Low 2,000 I P−p j t e Shortage inventory level of the finished product j at plant p in period t (Sacks), , , p j e 1 2 3 4 5 6 1 A High 18,600 1 A Medium 18,600 1 A Low 18,600 1 B High 2,780 1 B Medium 2,780 1 B Low 2,780 2 A High 2 A Medium 2 A Low 2 B High 21,820 2 B Medium 21,820 2 B Low 21,820 123 69 Page 42 of 46 Int. J. Appl. Comput. Math (2021) 7:69 I P−p j Shortage inventory level of the finished product j at plant p in period t (Sacks), ,t,e p j e 1 2 3 4 5 6 1 A High 18,600 1 A Medium 18,600 1 A Low 18,600 1 B High 2,780 1 B Medium 2,780 1 B Low 2,780 2 A High 2 A Medium 2 A Low 2 B High 21,820 2 B Medium 21,820 2 B Low 21,820 I D+d j t e Excess Inventory level of the finished product j at the distribution center d in period t (sacks), , , d j e 1 2 3 4 5 6 1 A High 4,528 1,026 1 A Medium 4,528 1,026 1 A Low 2,710 1,198 1 B High 5,800 5,800 1 B Medium 5,800 5,800 1 B Low 5,800 5,800 2 A High 6,200 1,804 2 A Medium 6,200 1,804 2 A Low 6,200 2,231 2 B High 6,200 6,200 2 B Medium 6,200 6,200 2 B Low 5,000 6,200 123 Int. J. Appl. Comput. Math (2021) 7:69 Page 43 of 46 69 I R+r , j t Excess inventory level of the finished product j at retailer r in period t (Sacks), ,e r j e 1 2 3 4 5 6 1 A High 1 A Medium 400 600 1 A Low 400 4,600 1 B High 1 B Medium 2390 850 2,990 1 B Low 2,390 3,400 1,200 5,800 2 A High 460 460 460 460 460 460 2 A Medium 460 460 460 460 2 A Low 5,200 2 B High 2 B Medium 460 460 2 B Low 2,800 1,000 I R−r j t e Shortage inventory level of the finished product j at the retailer r in period t (Sacks), , , r j e 1 2 3 4 5 6 1 A High 100 500 1,700 1 A Medium 2,740 2,340 1 A Low 4,980 1 B High 700 1,100 700 1 B Medium 1,150 2350 750 1 B Low 2 A High 2 A Medium 2 A Low 2 B High 2 B Medium 2 B Low References 1. Angerhofer, B.J., Angelides, M.C.: A model and a performance measurement system for collaborative supply chains. Decis. Support Syst. 42(1), 283–301 (2006) 2. Arns, M., Fischer, M., Kemper, P., Tepper, C.: Supply chain modelling and its analytical evaluation. J. Oper. Res. 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