A case study application of linear programming and simulation to mine planning
J.A. de Carvalho JuniorI; J.C. KoppeII; J.F.C.L. CostaII
ICopelmiMineração LTDA, Porto Alegre, Brazil
IIMining Engineering Department, UFRGS, Porto Alegre, Brazil
This paper analyses the impact of the uncertainty associated with the input parameters in a mine planning optimization model. A real example was considered to aid in the building of a mathematical model that represents the coal production process with reference to the mining, processing, and marketing of coal. This model was optimized using the linear programming concept whereby the best solution was perturbed by the stochastic behaviuor of one of the main parameters involved in the production process. The analysis of the results obtained permitted an evaluation of the risk associated with the best solution due to the uncertainty in the input parameters.
Keywords: mine planning, optimization, risk analysis.
Although coal accounts for approximately 50 per cent of the national reserve of fossil fuel, its use in Brazilian energy generation has fallen continuously in the last few years, and coal now contributes less than 1.5 per cent to the total consumption of fossil fuels. This has prompted mining companies to diversify their customer base by supplying thermal coal to other industries such as the petrochemical, ceramic, paper, and cellulose industries, where coal is used for the generation of heat, steam, and/or electrical power for industrial plants; the food industry, where coal is used in the process of drying food grains; and the cement industry, where coal is used in the clinker mix in the kilns.
This diversification has forced the mining companies to develop new products to meet the specific needs of each customer. Companies that produced thermal coal with a calorific value ranging from 3 000 to 4 500 kcal/kg in the past, with practically no restrictions on the sulphur content, have had to increase their range of products. Nowadays, the local market for industrial-purpose coal demands products with higher calorific values (4 700 to 6 800 kcal/kg) and a lower sulphur content (usually below 1.2 per cent). In this context, and with the constant expansion of the global economy, local coal-mining companies are continually looking for alternatives that facilitate the production of better products with competitive prices in comparison with other energy-yielding products such as natural gas, fuel oil, and imported coal. An additional complexity is added to production planning by the fact that major coal producers frequently operate multiple mines with multiple pits feeding multiple processing plants within these mines.
The alternatives commonly chosen to attend to multiple customers in this multiple-option production scheme include the following:
(i) Simultaneously mining multiple mines, multiple pits, and multiple benches (seams) to increase the availability of run-of-mine (ROM) coal with different characteristics
(ii) Washing of coal from each seam separately to enhance the yield at the plant and approximate the characteristics of the washed coal to the desired final product
(iii Minimizing the mining dilution so that the characteristics of the ROM coal are as close as possible to the final saleable product.
The definition of the production strategy to meet a certain market demand is usually achieved by testing a few production scenarios: the finally chosen method should generate the highest economic benefit. However, this methodology does not guarantee that the chosen solution is really the best possible one. With the use of linear programming, it is possible to find the best production strategy to meet the demand of each market.
This problem can be tackled from an operational research (OR) perspective. An adequate thermal coal can be produced with a minimum cost and/or with a higher possible profit. The production depends on a variety of parameters and restrictions such as raw material availability, coal quality (different coal seams), specific product requests, recovery in the processing plant (which depends on the raw material used and on the characteristics of the desired product), and the mining and processing costs. These parameters specify the production of coal within certain limits.
There are various models and mathematical programming tools for solving this kind of problem. Among the several optimization techniques developed, the linear programming (LP) method has been the most studied, and hence the most widely used in several of the applied sciences. LP is not only the simplest technique in mathematical programming, but also the most versatile, offering the possibility of analysing the appropriateness of the chosen model concurrent with a guarantee for obtaining its global best, if it exists.
Due to their complex behaviour, some of the process parameters, can be regarded as stochastic. To quantify the impact of these parameters on the optimized outputs derived from LP, simulation techniques can be used, taking into consideration not only the averages or the deterministic patterns, but also the statistical distributions of the parameters considered.
This paper introduces a methodology for evaluating the impact of the uncertainty associated with input parameters on a given objective function using an optimization program that is based on LP. A case study concerning a major Brazilian coal-producing company illustrates the process by which the economil benefits of using the optimal solution are evaluated in comparison with the solution conventionally used by this particular mining company. Moreover, incorporating the uncertainty associated with the input parameters permits an assessment of the risk in reaching or not reaching the optimized solution.
Model of linear programming
According to Prado1, LP is a technique that allows the merging of several variables on the basis of a linear function of effectiveness (objective function) while simultaneously satisfying a group of linear restrictions for these variables. The construction of a LP model follows three basic steps (Ravindran et al.2):
(i) Identifying the unknown variables to be determined and representing them using algebraic symbols
(ii) Listing all the restrictions of the problem and expressing them as linear equations in terms of the decision variables defined in the previous step
(iii) Identifying the objective or criterion for optimizing the problem and representing it as a linear function of the decision variables. The objective can be either a maximizing or a minimizing function.
Given an LP problem consisting of many variables in the maximization or minimization mode from a specific linear function, which is also known as the objective function (OF), the variables are submitted to a group of restrictions that are also linear. Generally, a problem in LP is formulated as follows. The restriction of 'no negativity in the decision variables' constitutes a necessary condition for the application of the solution algorithm. Although this restriction usually holds good due to the nature of the variables used in the model, under certain circumstances it can lead to situations where the variables become unrestricted. In this situation, an artificial variable should be used to substitute each unrestricted variable by the difference between the other two variables for which the restriction of 'no negativity' is applied (Loesch and Hein3).
Briefly, the algorithm can be described as follows: (i) Maximize or minimize the OF of the type or
(ii) Submit the variables to the following restrictions:
Where xn are the decision variables, cn are parameters of the OF, amn are parameters of the restriction equations, bnn is an independent term of the restriction equations, n is the number of decision variables, and m is the number of functional restrictions.
The values of all the coefficients, or control parameters, are known during the modelling of the problem. These coefficients can have either deterministic or probabilistic characteristics depending on the nature of the problem modelled (Montevevechi4).
Algorithms for the solution
The simplex method is the algorithm typically employed in the solution of linear programming problems (LPPs). This technique uses the available tool in linear algebra to determine the best solution for the LPP by an algebraic iterative method.
Generally, the algorithm is created by a viable solution of the system of equations that constitute the restrictions of LPPs, and starting from that initial solution, it identifies either new viable solutions for the same or a better value for the problem than the current one. The algorithm, therefore, possesses (i) a choice criterion that provides the opportunity for always finding new and better vertexes of the convex boundary of the problem, and (ii) a stop criterion that can verify whether the chosen vertex is the one giving the optimum solution (Goldbarg5).
The basis for the simplex algorithm is in the formatting of the limited area for the restrictions, common in all OR problems (Dantzig6). Such an area is called simplex. Any two points selected in the outline of a feasible region, when united by a line, result in a line positioned entirely inside the simplex. Starting from the verification, the search for the solution in OR problems becomes limited to the extreme points of the simplex area. This observation has facilitated the development of an algorithm of low computational complexity for solving this problem by Dantzig6.
The algorithm of LP can be explained using a simplified graphical sketch; however, the illustration is limited to two decision variables. The example used to illustrate the method uses the following OF:
The OF is subjected to the following restrictions:
After plotting the lines corresponding to the above restrictions, an area within which all restrictions are simultaneously satisfied (hatched region in Figure 1) is defined. The polygon defined by the corners A, B, C, D, and E is known as the 'admissible region'.
To define the optimum solution, the line corresponding to the OF, for a specific value of Z, is initially plotted. In the next step, various values are defined for Z, and the OFs for each defined Z are plotted. A group of parallel lines, known as the lines of the OF levels, is obtained (traced red lines in Figure 2). The OF grows towards the direction of its gradient (arrow in Figure 2), and the optimum solution is obtained at the maximum OF that intersects the feasible region.
In a two-dimensional solution, the feasible region is shown in plan. The equation that represents the OF can be depicted by a vector, so that it is possible to reach the highest point by following the direction of improvement of the OF as determined by the vector inside the space of feasible solutions. The search guarantees that (i) one of the extreme points respectively maximizes or minimizes the OF, and its value is the maximum or the minimum, respectively; (ii) once the search for the highest point is restricted to the space of viable solutions of the problem, the optimum vertex satisfies the group of restrictions composing the problem of LP (Fogliatto7).
This reasoning can be extended to problems of larger dimensions. Bazaraa8 establishes that while searching for the highest point during the tracking of the space of viable solutions for an LP problem, the solution should correspond to one of the extreme points of the simplex. In practice, this can correspond to a point, a straight line, a plan, or any other pattern with a larger dimension.
The detailed mathematical presentation of the simplex algorithm can be found along with practical examples in most of the articles referring to LP; see, for example, Ravindran et al.2, Loesch and Hein3, Goldbarg5, Dantzig6, Bazaraa8, and Bronson9.
While creating a representative model of any industrial process in which the behaviours of all the parameters are not perfectly known, it is not possible to be certain about the results generated by the model. However, it is always possible to use methodologies of risk analysis to mitigate the effects of the uncertainty and to visualize the risk problem (Motta10).
For each group of restrictions that encircle the problem, a group of answers can be obtained from the model using simulation techniques to imitate the stochastic behavior of the process parameters that are not deterministic. This group of answers can be associated with the uncertainty space and can be used to quantify the inherent risk of the given process.
According to the functional visualization proposed by Menner11, in simulation studies, the models possess (i) a number of entrances or inputs x1, x2,..., xr; (ii) a number of parameters related to the system p1, p2,..., pt; and (iii) a number of exits or results y1, y2,..., ys, which are the results defined by a function of the type f (x, y).
Simulation in the sense used in this study constitutes some mathematical techniques used to mimic any type of real operation or process (Freitas12). There are a few other definitions for the term 'simulation'. According to Schriber13, simulation in modelling implies a process or system that imitates the responses of the real system during a succession of events occurring over an extended period.
Pegden et al.14 present a more comprehensive definition of the simulation process. According to these authors, a simulation should fulfill the following objectives:
(i) Describe the behaviour of a certain system
(ii) Help in building a theoretical framework and explain an arbitrary hypothesis, given the observations provided by the simulation model
(iii) The model finally adopted should be able to predict new responses due to either a modified system or modifications in the production processes.
Contrary to the optimization methods, simulation does not yield an optimal result. Simulations provide a set of responses for a certain group of variables that condition the system (Figure 3).
The main advantages of a simulation process include the following (Pegden et al.14; Banks and Carson15):
(i) Simulation is simpler to use than an analytical solution
(ii) Although analytical optimization models normally require many simplifications to obtain a treatable solution, simulation models do not require these simplifications
(iii) Any hypothesis about how certain variables influence the system can be evaluated using the simulation model
(iv) An appraisal of the variables that are more relevant to the performance of the system and details of the interaction between the input variables leading to a certain response becomes possible
(v) New variations in the production process can be screened before practical implementation and their performances can be assessed.
Some systems do not show any probabilistic variables and hence are known as deterministic. In these circumstances, the responses are conditioned only by the input variables and their interrelations. In various situations, the input Variables system is controlled by random input variables and, consequently, a stochastic simulation is used to obtain the response of the system to a set of inputs. In this latter situation, the responses are also variable and are modelled using a probability function.
In the following section, a simulation model is presented that mimics the production process of thermal coal. The model is constituted by the following factors: (i) the input variables, x: the amount of ROM coal originating from different seams or mines; (ii) the parameters of the system, p: the operational cost associated with the mining and processing of the ROM, the indexes of recovery in the improvement process, and the selling prices of the different products; and (iii) the exit variables, y: the cash flow and the values of the liquid assets available. In these terms, the two groups that contain the input variables and the parameters of the system comprise the elements associated with uncertainty, and these can therefore be treated as random variables.
Settings, products, and production routes
Similar to other mineral commodities, coal production can be divided into two main stages: exploitation and processing. Coal mining occurs by open-pit mining, whereby the topsoil and other sedimentary formations that cover the coal seams, constituting the overburden, are stripped first, uncovering the coal seams underneath; these are later mined in an individual and selective manner (Hartman16). Generically, the mining operation involves the production of large amounts of waste material for each ton of coal extracted, constituting one of the more costly operational stages of the productive process.
The case under consideration is at Copelmi Mineração Ltda. Copelmi is the largest private coal-mining company in Brazil, supplying 80 per cent of the industrial market and 18 per cent of the total Brazilian coal market. Its facilities are located at Rio Grande do Sul State, in the southernmost part of the country.
The company currently operates four coal mines, commercializing approximately 1.5 Mt of coal per year. Its flexibility in producing at multiple mines enables a diverse product range with calorific value products ranging from 3100 kcal/kg up to 6000 kcal/kg.
Copelmi's main products are steam coal for a power plant, a petrochemical plant, and a large paper mill, all located within a maximum distance of 150 km from the mines.
During the period analysed, four mines were under simultaneous operation: Recreio Mine, ButiáLeste Mine, Faxinal Mine, and the Cerro Mine. Each mine exploited a number of coal seams with different qualities and reserves.
After the mining stage, the coal from the mines is transported to a washing plant, where it is processed to obtain a saleable product. The coal from each mine is processed at the closest washing plant. During this processing stage, impurities such as pyrite and clay present in the ROM coal are removed to reduce the emission of ash and gas during coal combustion. When the mine production exceeds the feed capacity of the plant due to any operational contingency, the excess ROM is diverted to the stockpile for future use. The washing process yields several types of products. The different production strategies are explained in the specifications of the OF section of the paper.
One of the most relevant operational parameters in the washing process is the yield or recovery. This parameter corresponds to the ratio between the amount of product generated and the amount of ROM coal fed into the plant, which may be stated as follows:
Where R is the yield (%), Product is amount of product generated (t), and ROM is amount of run-of-mine coal fed into the washing plant (t).
In addition to the product, the washing plant generates tailings, which are disposed at the tailings dam. These tailings are predominantly composed of clay and pyrite; however, due to the inefficiency of the processing plant, organic matter can be added to the tailings. In the latter instance, depending on the ash content of the tailings, the tailings can be redirected to be blended with other products.
Mathematical modelling of the production process
The central subject of the mathematical modelling is the description of the production process and its decision variables. The quantity (in tons) to be extracted and processed from each seam can be determined, thus fulfilling the demand for each product commercialized by the company. Optimization is achieved when the production process generates the largest possible operational contribution after abiding by all the restrictions imposed on the production process, in addition to considering the demands of the consumer market.
The decision variables are defined as follows:
Qij: amount of ROM coal of j seam extracted from mine i
Eji amount of ROM coal of j seam used (or added) from the stock of mine i
Ajilk: amount of ROM coal of j seam extracted from mine i and processed by the washing plant to obtain a product k.
The process parameters are, among others: coal yield (plant recovery), production costs, selling prices of the products, and the tonnage and quality of the product specified by the consumers.
Once the decision variables are defined, the next step is the definition of the OF. This function is represented by Equation , and its goal is to maximize the operational contribution satisfying a certain market demand.
The operational contribution is the financial result obtained by the commercialization of the generated products (gross earnings), except for the costs exclusively associated with the production process (production costs) (Equation).
This function can be better understood if it is subdivided into four corresponding portions: mining costs (MC), costs of stocks (SC), washing costs (WC), and gross earnings (GE). For this variant, a series of parameters representing the economical, operational, commercial, and geological factors are used, which can present either deterministic or probabilistic behaviours depending on the methodology used. The four portions that constitute the OF can be expressed in the form of Equations  to .
Where: i = index of the mines; j = index of the seams; l = index of the improvement plants; k = index of the generated products; REMi = stripping ratio of mine i (m3/t ROM); CEi = cost of waste removal from mine i (R$/m3); CDi= blasting cost of mine i (R$/t ROM); CCi = cost of extraction and transport of coal from mine i (R$/t ROM); Qji = amount of coal from seam j extracted from mine i (R$/t ROM); CUEi = cost of handling unit stock of ROM coal in mine i (R$/t ROM), Eij = amount of ROM coal of seam j moved on the basis of the stock of the mine i (t ROM); CBlk = processing cost of ROM coal in the washing plant for the production of substance k (R$/t ROM); Ajilk = amount of seam j of mine i fed and processed in the washing plant to obtain product k (%); Rjilk = yield from seam j of mine i processed in washing plant to obtain product k(%); and PVk = selling price of the product k (R$/t).
The restrictions defined in the outline conditions form the system and define the domain of the possible values for the decision variables. In the current case study, the group of restrictions comprises the following:
Geological proportion of the seams (percentage of seam j in the total quantity mined) in each mine
Maximum capacity of ROM coal production from the mines
Maximum coal tonnage able to be fed into each washing plant
Recovery of each product at each washing plant
Table I shows typical numeric values for the restrictions in one of the scenarios examined in the case study.
This group of restrictions defines a mathematical formulation formed by 64 linear equations represented in terms of decision variables. The theoretical background and the mathematical formulation used to represent the group of restrictions in this case study are found in Carvalho17.
An optimization model composed of both the OF and the group of restrictions after considering the simplifications detailed above was implemented using Excel®, using the educational version of the program 'What's Best® 7.0' as an optimizer (Lindo Systems18).
Results of optimization
The outputs generated by the optimization model have been analysed, taking into account the different production scenarios. The production scenarios correspond to the monthly production strategy adopted by the company in 10 different periods during the years 2004 and 2005.
The results are analysed in two steps: the first quantifies the earnings that would be obtained by using the optimization model, and the second verifies the risk associated with each production scenario, considering that the parameters corresponding to the yield (coal recovery at the washing plant) behave stochastically, not deterministically.
Analysis of the responses
To quantify the earnings that an optimization model can generate in the production process, a comparison of the financial results obtained from the optimization model with those obtained in the same period adopting the non-optimized production scheme was conducted. The production strategy proposed by the mine planning staff without using optimization tools was adopted as the benchmark.
The scenarios chosen for comparison correspond to the production strategy adopted by the company in 10 different situations during the years 2004 and 2005. These situations, denominated 'Scenario 1' to 'Scenario 10', represent conditions where the market or the operation caused a significant change in the baseline conditions that define the production strategy of the company.
The optimization process yields as the final result the OF and the production strategy that need to be adopted. These strategies correspond to the amount of coal that should be extracted from each mine, its destination in terms of the processing plant, and the seam that provides the highest value for the specific OF.
To quantify the earnings that the optimization process can generate in a non-optimized process, the final results of the OF obtained from the optimization model were compared with the final results obtained following a non-optimized practical planning process adopted by the company, subject to the same conditions. The relative differences in the values of the OF for each of the 10 tested scenarios are presented in Figure 4.
Figure 4 portrays the results for 10 production scenarios, with a minimum increase of 2.2 per cent and maximum of 8.7 per cent in the gross earnings, when the optimized scenarios are compared with the respective non-optimized procedures adopted by the company. An important practical aspect that needs to be pointed out is that this gain in earnings can be obtained by optimizing the production process within its operational limits, without the need for any additional investment in the process.
In addition to measurable gains as described previously, the use of an optimization model provides greater flexibility in the decision-making process. The method provides the resource to analyse the impact of possible changes in the production process, such as changes in the quantity of product sold and increase or decrease in the production costs.
Risk analysis of the optimization model
An optimization model as introduced above searches for the highest result, assuming that the input parameters are deterministically known. However, when the model is influenced by parameters that are not perfectly known, there is no guarantee that the optimized results will be observed in reality.
In this situation, three questions arise:
How different can the practical result be in relation to the optimized one?
What is the probability of the highest financial result being smaller than a certain expected minimum value?
Which parameter has the most significant impact on the OF?
To answer these questions, the yield parameter was stochastically varied, by which the deterministic values were replaced by simulated values. These values were randomly derived based on the probability-distribution function constructed from a historical series of results (Bratley et al.19).
For each group of simulated values, the OF generates a different outcome. A statistical analysis of the group of possible answers shows the space of uncertainty (risk) associated with the expected optimal financial result.
Additionally, the coefficient of linear correlation between the group of possible outcomes of the model and each group of parameters identifies which parameter is the most relevant to the final outcome of the model.
To illustrate the risk-analysis methodology, one of the optimized scenarios was chosen and the yield of different washed coals from different coal seams was considered as the parameter of stochastic behavior. The probability distributions of the yields were constructed using a historical series of results corresponding to the previous 12 months. The statistical summary for these variables is presented in Table II.
Two thousand draws from these probability distributions were retrieved and used to calculate the OF.
The variability obtained in the OF and in the coal yield are presented in Figure 5, which shows the relative variation between the results of the OF generated based on the simulated parameters and the results obtained by the optimization process considering coal yield as a fixed parameter. Analysis of the results obtained shows that:
The distribution of the results has a low asymmetry. This indicates that, in practice, there is no tendency for over- or underestimation of the OF value in relation to the optimum value obtained when the average yield was used as the input > The results present a high variability, ranging from -42 per cent to +39 per cent in relation to the optimum value of the OF, and 90 per cent of the values are between -22 per cent and +22 per cent.
To quantify the influence of each stochastic parameter in the results of the OF, the coefficient of linear correlation was calculated for the simulated values with respect to each parameter, and, subsequently, the OF values were calculated. The results of these calculations are shown in Figure 6.
From Figure 6, it is noted that all the simulated parameters show a positive correlation with the results of the OF.
Discussion and conclusions
The results obtained fom mathematical modelling tools, optimization using LP, and risk analysis, suggest that financial gains can be obtained by using these techniques in mine planning. With these tools, the decisionmaker can evaluate a series of choices and hypotheses regarding the production process and thereafter conveniently propose structural changes in the process to generate a higher profit. Among the 10 scenarios analysed, the results of the optimization process indicate that the economic gains could increase by an average of 5.8 per cent, compared with the results obtained by the production strategies originally adopted by the company without any optimization process.
The results also show a large variability in the OF when the yields for the various coal products are used as stochastic inputs. This variability in the input is probably overestimated due to the use of historical data in the simulation process. The historical data used corresponds to a period of 12 months, during which mining occurred in different locations within the deposit. This fact can lead to significant variations in the product quality and yield. Hence, geostatistical simulation should be used to build coal-yield models and, after these simulations of the models are completed, the results obtained should be used to feed the yield values into the optimization process.
The use of a representative mathematical model for the production process generates other benefits, in addition to the tangible gains described previously. It allows a comprehensive understanding, transparency, and interaction of the production process, helping to quickly determine the influence of various factors on the response by different parameters related to this process. Consequently, the model can be used not only for the optimization of the current production process, but also to test alternatives such as the evaluation of new mine areas or the commercialization of other products. The correlation coefficient relating each stochastic parameter and the answers obtained by the OF can be used to rank the parameters that generate the greatest impact on the final value of this function. After the definition of the most relevant parameters, their behaviour can be more closely followed, and whenever possible, be directed toward values that improve the result of the OF.
The use of simulation techniques provides the opportunity to quantify the impact that the stochastic parameters produce in the optimum response. The stochastic approach enlarges the range of information available to the decisionmaker, allowing him or her to evaluate the risk of a given result, instead of depending on a single deterministic value of the OF.
1. Prado, D.S. Programajao Linear. Belo Horizonte, MG, Desenvolvimento Gerencial, 1999. [ Links ]
2. Ravindran, A., Philips, D.T. and Solberg, J.J. Operations Research Principles and Practice. 2nd edn. John Wiley & Sons, New York, 1987. [ Links ]
3. Loesch, C. and Hein, N. Pesquisa Operacional. FURB, Blumenau, 1999. [ Links ]
4. Montevevechi, J.A.B. Pesquisa operacional (Programação linear). MSc. thesis, Escola Federal de Engenharia de Itajubá, 2000. [ Links ]
5. Goldbarg, M.C. Otimização combinatória e programação linear: modelos e algoritmos. Rio de Janeiro, Campus, 2000. [ Links ]
6. Dantzig, G.B. Linear Programming and Extensions. Princeton University Press, New Jersey, 1963. [ Links ]
7. Fogliatto, F. Pesquisa operacional I - Modelos determinísticos. MSc.thesis, Universidade Federal do Rio Grande do Sul, Porto Alegre, 2004. [ Links ]
8. BAZARAA, J. Linear Programming and Network Flows. 4th edn. Hoboken, New Jersey, John Wiley & Sons, 1999. [ Links ]
9. Bronson, R. Pesquisa Operacional. Sao Paulo. MacGraw-Hill, 1985. [ Links ]
10. Motta, R.R. Análise de investimentos - Tomada de decisão em projetos industriais. Sao Paulo, Atlas, 2002. [ Links ]
11. Menner, W.A. Introduction to modeling and simulation. JohnsHopkins APL Technical Digest, vol. 16, no. 1, 1995. [ Links ]
12 Freitas, P.J. Introdução à Modelagem e Simulação de Sistemas. Visual Books, Florianópolis, 2001. [ Links ]
13. Schriber, T.J. Simulation using GPSS. New York, Wiley, 1974. [ Links ]
14. Pegden, C.D., Shannon, R.E., and Sadowski, R.P. Introduction to simulation using SIMAN. New York, McGraw-Hill, 1990. [ Links ]
15. Banks, J. and Carson, J.S. Discrete-event System Simulation. Prentice Hall College Div, New Jersey, 1984. [ Links ]
16. Hartman, H.L. SME Mining Engineering Handbook., SME, Littleton Colorado, ,1992. [ Links ]
17. Carvalho, J.A. Jr. Abordagem probabilistica em um modelo de programação linear aplicado ao planejamento mineiro. MSc. Thesis, Universidade Federal do Rio Grande do Sul, 2006. [ Links ]
18. Lindo Systems Inc. Premier optimization modeling tools. http://www.lindo.com Accessed Jan. 2005. [ Links ]
19. Bratley, P.,Fox, B.L., and Schrage, L.E. A Guide to Simulation. Springer-Verlag, New York, 1987. [ Links ]
Paper received Aug. 2010
Revised paper received Dec. 2011
Profit Optimization Using Linear Programming Model:
A Case Study of Ethiopian Chemical Company
Vishwa Nath Maurya1, Ram Bilas Misra2, Peter K Anderson3, Kamlesh Kumar Shukla4
1Department of Applied Mathematics and Statistics, School of Science & Technology, The University of Fiji, Lautoka, Fiji
2Department of Mathematics & Computing Science, Divine Word University, Madang, Papua New Guinea
3Dept. of Information Systems, and Dept. of Mathematics & Computing Science, Divine Word University, Madang, Papua New Guinea
4Department of Management, Adama Science and Technology University, Adama, Ethiopia
To cite this article:
Vishwa Nath Maurya, Ram Bilas Misra, Peter K Anderson, Kamlesh Kumar Shukla. Profit Optimization Using Linear Programming Model: A Case Study of Ethiopian Chemical Company. American Journal of Biological and Environmental Statistics. Vol. 1, No. 2, 2015, pp. 51-57.doi: 10.11648/j.ajbes.20150102.12
Received: October 12, 2015; Accepted: April 15, 2016; Published: June 16, 2016
Abstract: This paper aims for profit optimization of an Ethiopian chemical company located in Adama (Ethiopia) using linear programming model. Particularly, our present study brings out clearly the necessity of using quantitative techniques for utilization in Ethiopian company; a factory situated within Adama about 90 kms. from Addis Ababa (Capital of Ethiopia). The first step comprises data generation. A questionnaire is prepared and circulated amongst company staff both executive and technical to determine the production, sales and profit during a few months of 2014. The profits varied considerably owing to subjective approach. It was established that the decisions are undertaken by experienced people without use of quantitative people and quantitative method. Whole approach applied here is seemingly subjective. A theoretical perspective undertaken for the present study is review of various different applications of linear programming. The characteristics of base assumptions of linear programming and its advantages and disadvantages towards establishing its need for optimization are briefly outlined in terms of its application to the factory. Survey data is analyzed to determine the style of decision making and the problem is defined. An objective function is created in terms of decision variables of production, sales and profit over a period of time using the quantitatively available data of these parameters. A linear programming model for company is developed for profit optimization. The model equations with adequate restraints taking into account manufacturing limitations are solved using MS-Excel solver. Finally, some conclusive observations have been drawn and recommendations have been suggested.
Keywords: Profit optimization, linear programming model, simplex method, manufacturing limitation, service industries
Linear Programming (LP) is a problem-solving approach developed to help managers make decisions. Numerous applications of linear programming can be found in today’s competitive business environment Anderson . The term linear programming was first used by G.B. Dantzig  in 1947 to refer to specific problems of optimization which assume that both constraints and objective function are linear. As with other branches of Operations Research, the first applications of LP are found in military planning activities, how to distribute men, weapons, and supplies efficiently to the various fronts during World War II. (Here, the word programming means creating a plan that solves a problem; it is not a reference to computer programming.) Soon after that, LP came into wide use in industry, with the most fruitful utilization in the petroleum, petrochemical and food industries (extensive references can be found in Dantzig  and . Linear Programming is not a new modeling technique: it has been used routinely for over forty years to describe different productive and economic systems, and also problems in scheduling and distribution. The mathematics of linear programming are well established and presented in number of books (,, , ) while computer packages for solving large LP models are well developed and widely available, e.g. (Dash Assoc., 1993). Linear programming is one specialized mathematical decision-making aid. It can be applied to many problems in the real world, not because the world is linear but because it is a powerful problem-solving technique. Here, the researches carried out by Kim  and Mehdipoor et al.  are also worth mentioning.
Linear programming, or linear optimization, is a mathematical method to achieve the minimum or maximum value of a linear function on a convex polyhedron. This convex polyhedron is, in fact, a graphical representation of some constraints as inequalities on/off functional variables. To put simply, one can achieve the best outcome (e.g. maximum profit or minimum cost) by using linear programming under specific settings and constraints. While linear programming is mainly used in management and economics, it can also be utilized for some engineering problems. Linear programming uses a mathematical model to describe the problem of concern. Thus, linear programming involves the planning of activities to obtain an optimal result, i.e. a result that reaches the specified goal best (according to the mathematical model) among all feasible alternatives. Although allocating resources to activities is the most common type of application, linear programming has numerous other important applications as well. In fact, any problem whose mathematical model fits the very general format for the linear programming model is a linear programming problem. Furthermore, a remarkably efficient solution procedure, called the simplex method, is available for solving linear programming problems of even enormous size . These are some of the reasons for the tremendous impact of linear programming in recent decades .
As stated above, linear programming was developed as a mathematical pattern during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. The method was kept secret until 1947. After the war, many industries began using it. The founders of linear programming are: G.B. Dantzig who published the simplex method in 1947, John von Neumann who developed the theory of duality, and Leonid Kantorovich - the Russian mathematician who applied similar techniques before Dantzig, and won the Noble Prize in 1957. The Babcock and Wilcox applied the linear programming to help plan a major expansion of the company’s Tubular Products Division (TPD) in Pennsylvania. Owen  has also used the linear programming method to design antenna array patterns that suppress interference from certain directions.
2. Statement of the Problem
Linear programming is a set of techniques and methods inferred from mathematics and other sciences which can play an efficient role in improving the management decisions. Although it is still regarded as a new science, but it has well proved to be capable of solving problems such as production planning, allocating resources, inventory control, and advertising. Those managers who care about the best outcomes for their decisions cannot be indifferent to this.
Wijeratne and Harris  proposed that the linear programming model is used by the managers to determine the most economical arrangement of finance, to arrange the best times to start and finish projects, and to select projects to minimize the total net present cost of capital. Linear programming optimizes (maximizing or minimizing) a dependent variable subject to a set of independent variables in a linear relationship, given a number of linear constraints of independent variables. The value of dependent variables which is the value obtained from solving the problem, is subject to the independent variables set by the decision maker (or determined by solving the problem). The dependent variables are usually set as objective function which may be one of the economic concepts such as profit, cost, income, production, sales, distance and time, etc. The independent variables in linear programming are known as variables of unknown value, and the decision maker has to calculate the value of such variables by solving the problem . In Ethiopia, most decisions were taken by government or non-governmental organizations, whether it is profitable or not for companies, manufacturing or service industries are based on trial and error. Qualitative decisions, like intuition, judgmental approaches are more dominant and the application of model based decision making like optimization techniques, e.g. linear programming models have little or no application. Hence, it is initiated by author to conduct an assessment of the application of linear programming in this particular company as a case study.
3. Objectives and Research Questions
3.1. Objectives of the Study
The main objectives of this paper are:
•To formulate a linear programming model that would suggest a viable product-mix to ensure optimum profit for Company.
•To highlight the peculiarities of using linear programming technique for the Company and prove that despite obstacles, the application of the technique in determining the product-mix of the company would be more profitable than otherwise.
•To know about the constraints of the company regarding cost, resources.
3.2. Research Questions
•How is the company currently making decision on the allocation of resources for the production of its
•products to maximize its sales and profit?
•Whether the company uses qualitative decisions or employs mathematical/statistical models or both?
•Whether the current method adopted in making decision is effective or not?
•Whether the resources available are sufficient for production of the products to satisfy the demands of the customers?
4. Significance of the Study
It helps to understand the best way of making decisions using quantitative models in order to determine its optimal product-mixes that can maximize its profits subject to the scarce resources it has. The study will provide a deep understanding and insight of the applications of linear programming models in industries and how to apply such models in practical and real world experience. To other researchers of similar interest who are willing to undertake further investigation on the topic, this research document can be used as a secondary information source.
This research paper is structured in seven sections.§1 is introduction and it concerns with background of the study and the company. § 2 – 4 deal with the statement of the problem, objectives of the study, research question, and significance of the study. §5 deals with data and methodology. Linear programming and its applications are discussed in §6. Presentation of data, analysis and results are also discussed in §6. Major findings and conclusions are given in§ 7. Finally, some recommendations are illustrated in the end.
This study was conducted for three reasons. Firstly, there was no reliable and comprehensive study to examine the role of linear programming in companies in Ethiopia. Secondly, it might pave the way forward for the company, policy makers, and financial institutions to understand the different roles of institutions in the development process. Finally, this study advances the knowledge of linear programming in decision making.
5. Data and Methodology
5.1. Method of Data Collection
In order to gather the relevant data from the company producing sulfuric acid, oleum and Aluminum sulfate utilizing partially locally available raw materials. This paper was based on primary and secondary sources of data. Especially to get information on the decision making practice of the company under study, an interview was conducted with the manufacturing manager and the sales managers of the company and data collected on the unit costs of production of the products, the unit selling price of the products and also contacted the sales and the purchasing, production and marketing managers of the company. And additionally secondary data sources were used to get accurate information.
5.2. Method of Data Analysis
Collected data were presented in a narrative form and analyzed using linear programming model. Since the purpose of this study was to develop linear programming model for the collected data from the company, the authors tried to transform the data into a linear programming model and solved the model (problem) using simplex algorithm using by applying MS-Excel solver in order to determine the optimal combination of the products of the company that can maximize its profit within the available scarce resources. Simplex algorithm was preferred over graphical approach because of this method can help to solve linear programming problems of any number of decision variables. Excel solver was preferred for accuracy purpose.
6. LPP Model and its Application
Linear Programming is a mathematical technique for generating and selecting the optimal or the best solution for a given objective function. Technically, linear programming may be formally defined as a method of optimizing (i.e. maximizing or minimizing) a linear function for a number of constraints stated in the form of linear inequalities. According to Fagoyinbo , Williams  and Wood and Dantzig  the problem of LP stated as that of the optimization of linear objective function of the following form:
Z = c1x1 + c2x2 (Objective function),
subject to the linear constraints:
a11 x1+ a12 x2(≤ or≥) b1,a21 x1+ a22 x2(≤ or≥) b2,…… ,
am1 x1+ am2x2(≤ or≥)bm,x1, x2(≤ or≥) 0.
6.1. The Proposed Model
The objective function and constraints were used in the following general form:
Maximize Z = P1X1 + P2X2 subject to constraints:
C11X1 + C12X2≤B1, C21X1 + C22X2≤B2, C31X1 + C32X2≤B3,
C41X1 + C42X2≤B4, C51X1 + C52X2≤ B5, X1, X2≥ 0,
Z = total profit contribution of Aluminum sulphate and Sulphuric acid,
Pi = profit contribution coefficients expressing per unit contribution to the profit equation,
X1, X2 = the set of unknowns to be determined, i.e. the tons of Aluminum sulphate
And Sulphuric acid produced by company,
C’s = the numerical values that expressed in per unit usage of the right hand side,
B1, B2,….,B5 = the resource values which were fully utilized.
Estimates of the variables are presented in tables. The optimum values of the different brands produced by the firm shows the combination (product mix) obtained through the application of linear programming model.
6.2. Assumptions of the Model
For applying the proposed model to case of profit optimization, following assumptions were taken into consideration:
(i)Proportionality assumption: The contribution of each activity to the value of the objective function Z is proportional to the level of the activity xj, as represented by the terms cjxj in the objective function. Similarly, the contribution of each activity to the left-hand side of each functional constraint is proportional to the level of the activity xj, as represented by the terms aijxj in the constraints.
(ii)Additive assumption: Every function in a linear programming model (whether the objectives function or the function on the left-hand side of a functional constraint) is the sum of the individual contributions of the respective activities.
(iii)Divisibility assumption: Decision variables in a linear programming model are allowed to have any (real) values, including non-integer values that satisfy the functional and non-negativity constraints. Since each decision variable represents the level of some activity, it is being assumed that the activities can be run at fractional levels.
(iv)Certainty assumption: The value assigned to each parameter of a linear programming model is assumed to be a known constant.
6.3. Application and Analysis
This section deals with the presentation and analysis of the data gathered from the company. For collection of the necessary data, the last author prepared a guide questions based on its objectives and research questions. He made an effort to know whether the company applies any model for their decision making especially whether linear programming models are applicable for decision making. If decision is not based on mathematical model, then researchers tried to investigate the style of decision making of this company. Through this research it is identified that the company manufactures two major products namely: aluminum sulphate and sulphuric acid.
6.4. The Decision Approach of the Company
According to collected data of the Company authors investigated that the company has annual based sales records. In order to determine how much to manufacture for the next year the company do not use any sound mathematical or statistical decision making models. Instead it decides the quantity of its products on a simple trial and error or simple individual’s judgment. For this reason their annual sales plan and the actual sales they achieve varies every year. When it comes to the case of the applicability of linear programming model for the optimal decision of the product mixes.
Table1. Production, sales and gross profit trends for the years 2006-2012.
(Source: Annual bulletin of company, 2012/13)
According to the Annual Bulletin of the company (2012/13), it has been described as below: As compared to the design (production) capacity of the company, the actual production achievement of aluminum sulphate and sulphuric acid is on average 21.51% and 19.56% only. Even though there is an improvement in the actual sales of these items\o some degree over the past four years, compared to the capacity of the company, it did not exceed annual average sales of 20%. Though the reason for such low sales is assumed to be the decline in domestic market the major problems are:
Some users of the products were importing substitute products for their purpose. However, considering the current industry development strategy of the country and the attention given by the government for industry, the company was expecting more demand for its products in the near future. Even though the company states the above factors as causes for low demand and sales of its products, the last author identified that the annual demand for aluminum sulphate was on average 6,000 tons per annum and for sulphuric acid it was 5,000 tons per annum. This indicates that the company was not fully utilizing the currently available demand for its product. One of the cause for underutilizing the demand was the poor decision making approach used by the company. Had it used Quantitative (model based) decision making especially linear programming, it might have achieved more than what it did. According to the information gathered from the company using linear programming model (MS-Excel Solver) it is identified below that the company should produce 20tons of aluminum sulphate per day (20×300=6,000tons per year) and 6.98tons of sulphuric acid per day (6.98×300=2,094 tons per year) in order to get a gross profit of Birr107,353.17 per day (107,353.17×300= 32,205,951 per year).
Figure 1. Year wise grass profit in Birr.
6.5. LPP of the Chemical Company
A linear programming model consists of certain common components and characteristics. The model components include decision variables, an objective function, and model constraints, which consist of decision variables and parameters. Decision variables are mathematical symbols that represent levels of activity by the firm.
Data gathered from the company into a linear programming model as follows:
A chemical industry owned by the Ethiopian Government manufactures two major products, aluminum sulphate and sulphuric acid each of which passes through three different processes: reaction, filtration, and evaporation. Aluminum sulphate is a coagulant used for purification of drinking water, industrial waste water treatment, paper sizing, dyeing, pharmaceuticals, soap modification, tanning cellulose , etc. so it is highly demanded by companies like: water supply and sewerage authorities, pulp and paper factories, leather industries, textiles, food and beverages, metals, chip woods, plywood and others. Sulphuric acid is a chemical used in the production of aluminum sulphate as a raw material or as medium of production for oleum, phosphates, car batteries, pulp & paper, steel picking and textiles (brochures of the company).
According to data collection, the company was manufacturing 51.5 tons of sulphuric acid per day (51.5×300=15,450 tons per year) and only 40 tons of aluminum sulphate per day (40×300 = 12,000 tons per year). The major constraint the company states for this is demand which is at most 6,000tons for aluminum sulphate per annum and utmost 5,000 tons per annum for sulphuric acid. But in addition, availability of machine hours on 3 processes of manufacture was also constraining manufacturing of the products as given below:
Table2. Constraint of machine hours on 3 processes of manufacture.
|Products||Machine hours on||Demand in tons/annum|
|Aluminum sulphate (for 40tons per day)||18||8||4||6,000|
|Sulphuric acid (produced 51.5 tons daily)||24||24||24||5,000|
|Total available time (assuming 300 working days per year)||7,200||7,200||7,200|
According to the production / sales departments of the company, the production of Aluminum sulphate costing Birr 5,237.69 per ton is sold for Birr8,998.35 per ton, whereas the manufacturing of sulphuric acid costing Birr 5,708.87 per ton is sold for Birr 10,313.45 per ton. Thus, the profit earned per ton of these items is Birr 3,760.66 and Birr 4,604.58 respectively. The objective is to determine how many tons of these items should be manufactured per day to maximize the daily profit of the company operating on average of 300 days in a year.
6.6. Mathematical Formulation of LPP for the Company
The profit maximization objective of the company is mathematically expressed as:
Zmax = 3,760.66x1 + 4,604.58x2.
Decision variables are mathematical symbols that represent levels of activity.
Letx1 = the tons of aluminum sulphate manufactured per day, and
x2 = the tons of sulphuric acid manufactured per day.
The model constraints are also linear relationships of the decision variables; they represent the restrictions placed on the firm by the operating environment. Four constraints relate to the number of hours of machine time available on three processes (reaction, filtration, and evaporation); and demand also restricts the number of tons of the items that can be manufactured as presented below:
Table3. Number of hours of machine time on three processes and the demand of items.
|Products||Machine hours per ton per day*||Produced/day (in ton)||Profit /per ton (in Birr)|
*The machine hours are roughly converted to per day basis for simplicity of model formulation and calculation while company officials suggested that it is sometimes not logical to make such conversion.
The constraints of the company are expressed as follows:
Maximize Zmax = 3,760.66x1 + 4,604.58x2,
0.45x1 + 2.15x2≤24 hrs. (machine hrs. on reaction),
0.2x1 +2.15x2≤24 hrs. (machine hrs. on filtration),
0.1x1 + 2.15x2≤24 hrs. (machine hrs. on evaporation),
x1≤20 tons (demand for aluminum sulphate per day),
x2≤51.5 (sulphuric acid produced per day),
x1, x2≥0 (non-negativity).
6.7. Solution of the LPP Using MS-Excel Solver
To apply the MS-Excel Solver, first we translate the linear programming model into its standard form. Thus, the above model, written in the standard form, becomes.
Zmax = 3,760.66x1+ 4,604.58x2 + 0s1 +0s2 +0s3 +0s4 +0s5,
0.45x1 + 2.15x2 +s1 =24 hrs., 0.2x1 +2.15x2 +s2=24hrs.,
0.1x1+ 2.15x2+s3 =24 hrs.,
x1+s4=20 tons, x2+s5= 51.5 tons,
x1, x2,s1, s2,…,s5≥0 (non-negativity).
Eliminating x2 from the first three equations there also result the equations:
0.25 x1 +s1 =s2, 0.35 x1 + s1 =s3, 0.1 x1 + s2= s3.
There are five equations involving seven unknown variables. So, their basic solutions can be obtained by assigning zero values to any (additional) 2unknowns. Setting s1=s2 = 0 a basic and feasible solution is
x1= 0, x2=24/2.15 = 11.16, s3 = 0,s4= 20, s5 = 40.34;
that determines Z = 51,400.08. In the following, we workout all basic solutions and compute the values of Z:
Table 4. Solution of the LPP for the Chemical Company by MS-Excel Solver.
|x1||x2||s1||s2||s3||s4||s5||Is soln. feasible||Z|
|— 192.72||51.5||"||< 0||<0||212.72||0||No|
|20||20/2.15 = 9.3||— 5||"||2||0||42.2||No|
|— 433.2||51.5||<0||"||< 0||453.2||0||"|
|20||22/2.15 = 10.23||— 7||— 2||0||0||41.27||"|
Thus, the optimal feasible solution is x1= 20 tons of aluminum sulphate per day, x2 = 6.98 tons of sulphuric acid per day, s1=0, s2=5 hrs., s3=7 hrs., s4=0, s5=44.52 tons, yielding the maximum profit Zmax= Birr 107,353.17 per day.
6.8. Result Analysis
Company produces 20tons of aluminum sulphateperday (20tons/day×300working days = 6,000 tons/annum) and 6.98 tons of sulphurc acid per day (6.98tons/day×300 working days = 2,094tons/annum) in order to get a maximum daily profit of Birr107,353.17 per day (Birr107,353.17 per day ×300working days =Birr32,205,951per annum). In this way, the company is left with an idle filtration and evaporation times of 5 and 7 hours per day and unutilized demand for sulphuric acid of 44.52tons per day but the demand for aluminum sulphate is fully utilized.
7. Conclusive Observations
Based on the data gathered and the major findings of the research, the following conclusions were drawn:
(i)Linear programming model is applied for profit optimization of the Ethiopian chemical company located in Adama (Ethiopia) and the maximum profit Birr 107,353.17 per day for company was found.
(ii)The company was left with an idle filtration and evaporation time of 5 and 7 hours per day respectively and un-utilizing demand for sulphuric acid of 44.52 tons per day but the demand for aluminum sulphate was fully utilized.
(iii)Even though the company has an annual record of its production and sales volume it was not employing any quantitative model to forecast for their production volume and sales; rather the company was employing a simple trial and error and subjective estimation. It led the company to misunderstand the existing demand for their products and to produce less than its capacity and even than the existing local demand.
(iv)The LPP model can be used for other companies for its profit maximization.
Based on the findings and the conclusions of the study, we suggest the following recommendations:
(i)For companies to forecast their production capacity and sales volume different quantitative models like: linear regression, linear trend, nonlinear regression, nonlinear trend, etc. available. Company has annual production and sales records but the company was not employing any mathematical or statistical models for its production or sales forecast rather it depends on a simple trial and error. The company should employ linear regression, linear trend and/or nonlinear regression and trend models to forecast its production capacity and sales.
(ii)On the major constraint that is considered by the company was demand for its products. In the 2012/13 annual bulletin of the company, it was stated that some customers of company were importing substitute products. This indicates that there was a quality and supply problem. Therefore, company should work on improving the qualities of its products to meet customer expectations and to search for other customer groups who can use the products beyond the current target markets.
(iii)It is clear that model based decision is important for its accuracy and objectivity. But such decision making approach was not widely used in practice. Qualitative decisions like subjective estimation, intuition and trial and error were commonly used by company. It is eye opening concern to the policy makers of company to shift the model based decision making styles in general.
- Anderson, D.R.; Sweeney, D.J.; Williams, T.A.; Camm, J.D.; and Kipp, Martin:An Introduction to Management Science: Quantitative Approaches to Decision Making, Revised 13th ed., South-Western Cengage Learning, 2012.
- Dantzig G.B., Programming of interdependent activities: II Mathematical Model, Econometrica, 17 (3), pp. 200–211, 1949. doi:10.2307/1905523.
- Dantzig, G.B.: Compact basis triangularization for the simplex method, R.L. Graves and P. Wolfe (eds.), Recent Advances in Mathematical Programming, McGraw-Hill, New York, pp.125–132, 1963.
- Drayer, W. and Seabury, S.: Linear programming - A case example, strategy & leadership, 3(5), pp.24-26, 1975.
- Fagoyinbo, I.S.: Compendious text on quantitative techniques for professionals, Ilaro, Nigeria, Jombright Productions, 2008.
- Frederick, H.S. and Lieberman, J.G.: Introduction to Operations Research, McGraw-Hill, Operations research - 1214 pages, 2001.
- Kim, C.: Parametrizing an activity vector in linear programming, Operations Research, 19, pp.1632-1646, 1971.
- Mehdipoor, E.; Sadr-ol-ashraafi, S.M. and Karbaasi, A.: A comparison of canonical linear programming techniques, Meaty Chicken’ Feed Farming with Linear Programming Models, Scientific-Research Magazine of Agriculture, 12(3), 2006.
- Misra, R.B.: Numerical Analysis for solution of ordinary differential equations, Lambert Academic Publishers, Saarbrücken (Germany), 2010, ISBN 978-3-8433-8489-6.
- Owen, P. and Mason, J.C.: The use of linear programming the design of antenna pattern with prescribed nulls and other constraints, compel: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 3(4), pp.201-215, 1984.
- Wijeratne, N. and Harris, F.C.: Capital budgeting using a linear programming model, International Journal of Operations & Production Management, 4(2), pp.49-64, 1984.
- Williams, N.: Linear and non-linear programming in industry, Sir Isaac Pitman & Sons, Ltd., London.Garifinkel (1963). A solution of the Goddard problem, Journal of SIAM Control, 1(3), pp.349–368, 1963.
- Wood, M.K. and Dantzig, G.B.: Programming of interdependent activities: I General Discussion. Econometrica, 17 (3/4): 193–91949. JSTOR1905522.