Linear programming applied to Aggregate Production Planning - Research Paper Example

? LINEAR PROGRAMMING APPLIED TO AGGREGATE PRODUCTION PLANNING OF FLAT SCREEN MONITOR This project applies Linear Programming model to an aggregate production planning of flat screen monitors of type A and type B at XYZ Company with the goal to minimize production costs. Aggregate planning problems for flat screen monitor can be resolved and production optimized by using linear programming to reduce costs. Linear programming can also be used in finding an optimal solution to problems for the purpose of minimizing total costs by affecting variables such as the workforce and the demand planning, as well as the minimization of the inventory balance and holding cost. Table of Contents This project applies Linear Programming model to an aggregate production planning of flat screen monitors of type A and type B at XYZ Company with the goal to minimize production costs. Aggregate planning problems for flat screen monitor can be resolved and production optimized by using linear programming to reduce costs. Linear programming can also be used in finding an optimal solution to problems for the purpose of minimizing total costs by affecting variables such as the workforce and the demand planning, as well as the minimization of the inventory balance and holding cost. 3 Table of Contents 4 2.1.Problem Definition and Translation 6 1. Introduction The objective of this paper is to construct a linear programming model applied to aggregate production planning of flat screen monitor that is aimed towards minimizing the overall cost related to various variables such as workforce planning, demand planning, inventory balance, inventory holding cost, etc. Linear programming can be used for solving aggregate planning problem, optimally. Linear programming determines the optimal solution, minimizing the total costs, to the problem provided that the production capabilities, constraints on requirements, overtime, permitted workforces changes, sub-contracting limits and all the other related costs are given. The fundamental purpose of employing linear programming here for the aggregate production planning is simply the minimization of the overall cost associated with the aspect of planning. This can be attained through the minimization of the inventory investment, the variations of the production rate and the changes in the level of the workforce. 2. Linear Programming Model Formulation In order to optimally allocate the scarce resources, the mathematical procedure called mathematical programming or constrained optimization or simply optimization can be used. Linear programming (LP) can be defined as the most special and famous form of constrained optimization, which has been found to be applicable, in practice, in almost all the aspects of contemporary business strategies that range from advertising to production planning. The most typical elements of the linear programming problem analysis comprise of the issues related to aggregate production planning and transportation. It is important to note here that the mathematical programming is totally different from compute programming however the computer program can help in the estimation of the optimal solution of the linear programming model in the mathematical programming. The computer programming refers to the development of the instructions for executing certain task or working out some calculations whereas the mathematical programming refers to organizing and planning of a task to achieve. Hence, knowing one of the above mentioned forms of programming does not directly relate to the other form quiet as much however the aptitude in one signifies the potential for the other. Mostly, there are two fundamental and important classes of objects for an optimization problem. The first class of objects is confined or limited resources that include the production capacity of the plant, land, size of the sales force, etc.; these examples have been given in relation to aggregate production planning. The second class of objects refers to the activities or tasks that include produce low quality or one type of the desired product, or produce the desired product that may the second type and produce high quality of the desired product that may be the third type of the product and so on; these examples are given in relation to aggregate production planning. Every task or activity, particularly related to production, requires or possibly contributes some extra resources. The objective of the problem is to find out the combination of the desired tasks or activity levels that utilize the available limited resources in the best manner yielding optimum results due to which the cost of production can be minimized and thereby, the profits can be maximized. 2.1. Problem Definition and Translation The XYZ Company manufactures two types of flat screen monitors, which are the “type A” and the “type B”. Each one type of the flat screen monitors has a different production line. The production line of ‘type A’ monitors has the potential to manufacture 60 sets each day while, on the other hand, the production line of ‘type B’ monitors has the potential to manufacture just 50 sets each day. The ‘type A’ monitor has the requirement of the labor resource equal to one worker per hour and the ‘type B’ monitor has the requirement of the labor resource equal to two workers per hour. At the moment, 120 man hours of the labor resource each day can be allocated at maximum for the production of the two types of flat screen monitors. What should be the optimum daily production for each type provided that the profit contributions are USD 20 for each of the type A monitors and USD 30 for each of the type B monitors. An explanation of the objective of the task in a structured way is given below: Minimize the overall cost of production of the two types flat screen monitors in order to optimize the profit contribution related to the production of type A monitors less than or equal to its total production capacity, the production of type B monitors less than or equal to its total production capacity and the labor utilized less than or equal to the maximum availability of the labor resource. The precision can be obtained in the solution of our problem, in the absence of the expert system software, by defining the following variables: A refers to the units of type A flat screen monitors to be produced each day. B refers to the units of type B flat screen monitors to be produced each day. Moreover, it is important to measure the following elements in order to determine the optimum solution for the aggregate production problem: Profit contribution in USD. Usage of type A in units produced. Usage of type B in units produced. Labor in worker-hours. Thus, a precise mathematical expression of our problem can be written as: Optimize or maximize the objective function 20A + 30B (USD) Provided that: A ? 60 (capacity of type A production) B ? 50 (capacity of type B production) (A + 2B) ? 120 (Labor in worker-hours) The above stated three mathematical expressions represent the three constraints of the requirements. Many optimization program that are often referred as ‘solvers’ consider that all the variables are compelled to be non-negative, hence it is unnecessary to state that the constraints A and are greater than or equal to zero (A ? 0 and B ? 0). There are three resources in our aggregate production problem in accordance to the definition of the terminologies, activities and resources, which are: production capacity of type A monitors, production capacity of type B monitors and labor capacity. On the other hand, the activities in our aggregate production problem are the production of type A monitors and the production of type B monitors. The general fact is that some resources can be associated with each constraint in the linear programming optimization problem. And there is always a physical activity related to each decision variable. 2.2. Gather Data The data used in this linear programming optimization model have been acquired from the XYZ Company that manufactures flat screen monitors. It is useful to assess the dual prices in the solution to this problem. The constraint A ? 60 has the dual price equal to USD 5 per unit. Initial, this quantity might be suspected to be equal to USD 20 per unit since the simple profit contribution of type A monitor is USD 20 with the production of its each unit. However, an additional unit of type A monitor will require sacrifices at some other point. Since the entire labor force is being utilized, the production of more type A monitors would require to free up the labor by reducing the production number of type B monitors. The labor trade-off rate of type A and type monitors is ?. This infers that the production of one more type A monitor requires the production of type B monitor to be reduced by ? of a unit. The net rise in profits is 20 – (?) 30 = 5 USD since the profit contribution of type B monitor is equal to USD 30 per unit. Now, consider the labor constraint with the dual price of USD 15 per hour. If we have one more labor hour then it will completely utilized for the production of more type B monitors since one type B monitor has a profit contribution of USD 30 per unit. As we know that one hour of labor is only enough to produce one half of the type B monitor, thus, the values of the additional labor hour is USD 15. If the XYZ Company start producing another type of flat screen monitors known as type C. The profit contribution of type C flat screen monitors is equal to USD 47 and the XYZ Company has the potential of producing maximum 50 units of type C product daily. The new type C monitors will be produced on the product line of type A monitors and will take 3 hours of labor. Now, consider that we want or produce three different types of flat screen monitors, type A, type B and type C, by utilizing the manufacturing resources of the XYZ Company that is currently manufacturing type A, type B and type C of flat screen monitors. First we should be able to estimate the initial minimum hourly rate to offer to the XYZ Company for using each of its three resources, which are: type A product-line capacity, type B product-line capacity and labor. Our decision variables are the hourly rates of these three resources. Indeed, we would prefer to rent the entire available capacity for each one of the required resources, mentioned above. Hence, we would like to minimize the overall rental fee for the all the available capacities that are type A capacity = 60, type B capacity = 50 and labor capacity = 120. If our offer price is to be accepted then our hourly rate for each of the available resources must be just high enough so that all of desired product types are worth of producing, as for instance, the rental charges conceded with the production of a type A flat screen monitor has to be above USD 20. These conditions ensuring that the resources are being rented out basically represent the three constraints for the objective function. 2.3. Setting Parameters and Selecting Decision Variables The variables defined for the formulation of the model for this problem are provided below: Cit = Cost per unit for producing type ‘i’ flat screen monitors to its available capacity ‘t’. Lit = Cost per unit of type ‘i’ flat screen monitors for using the available labor capacity ‘t’. Xit = Units of producing type ‘i’ flat screen monitors to its available capacity ‘t’. 2.4. Linear Programming Model Hence, the suitable linear programming model is given by: N T N T Minimize ? ? Cit Xit + ? ? Lit Xit i=1 t=1 i=1 t=1 Subject to: Type A Flat Screen Monitor: Cit + Lit > 20 i = 1, t Type B Flat Screen monitor: Cit + 2 * Lit > 30 i = 2, t Type B Flat Screen monitor: Cit + 3 * Lit > 47 i = 3, t The above stated three constraints compel the rent to be adequately high so that it is not feasible for the XYZ Company to manufacture any of its products. 2.5. Optimize Solution The optimal solution to this problem is: Objective value: 2100.000 Variable Value Reduced Cost PA 5.000000 0.0000000 PB 0.0000000 20.00000 PL 15.00000 0.0000000 Row Slack or Surplus Dual Price 1 2100.000 1.000000 2 0.0000000 -60.00000 3 0.0000000 -30.00000 4 0.0000000 0.0000000 The original profit contribution of the XYZ Company is exactly same as the price renter should pay for using the XYZ’s resources. The linear programming optimization model for minimizing the cost of the resources in relation to all activities being unprofitable is regarded as the dual problem of the original LP model for XYZ that maximized the overall profit confined by the available resources. It is apparent from the above solution that the dual formulation is basically the primal or original formulation of model but it is being transposed. 2.6. Validate & Test Model The reduced cost is a quantity is associated with each variable in any solution. If the units of the variables used in the requirements or constrains of the problem are gallon and the units of the variables for the objective function of the problem are dollars then the units for the reduced cost becomes dollars per gallon. Hence, in our problem of XYZ Company, the unit of the reduced cost of type A and type B flat screen monitors would be dollars per unit. The reduced cost associated with a variable can be defined as the amount or unit price through which the profit contribution of that variable can be increased, as for example by reducing the cost of the variable, prior to the variable used in the problem would have a non-negative value in the solution of the optimization problem. It is apparent that a variable that has been already included in the optimal solution will have value of zero for the reduced cost. The reduced can also be defined or interpreted as the rate of deterioration of the objective function when a variable that has current value of zero is arbitrarily compelled to increase by a small value. Now, suppose that the reduced cost of type A product is USD 2 per unit. This signifies that if the profitability of A were increased by USD 2 per unit then one unit of A could be introduced in to the solution without influencing the overall profit. Obvious that the total profit would be decreased by USD 2 if the value of A were increased by one unit without changing its original profit contribution. 2.7. Implement Results The XYZ implemented the results obtained through the above optimization of the linear programming problem of the aggregate production planning in order to maximize their profit for each day of production. 3. Conclusion Using linear program to solve optimization problems tends to be substantially simpler and easier in comparison to solving them through more general mathematical programs. Hence, it is worth dwelling on the linearity function for a bit. Linear programming is directly applicable only to situations where the affects of the various activities that can be engaged are linear in nature. References Barish, N.N. (1962). Economic Analysis for Engineering and Managerial Decision-Making. NY: McGraw-Hill. Balakrishnan, V. and Boyd, S. (1992). Global optimization in control system analysis and design. NY: Academic Press. Fletcher, R. (1987). Practical Methods of Optimization. NY: Wiley, New York. Gill, F., Murray, W., and Wright, M. (1981). Practical Optimization. London: Academic Press. Giordano, R.F., Weir, D.M., and Fox, P.W. (2002). A First Course in Mathematical Modeling. Brooks: Cole. Hansen, E. (1997). Preconditioning linearized equations, Computing, 58:187,196. Tuy, D.C. (1995). Optimization: Theory, methods and algorithms. Handbook of Global Optimization, pages 149, 216. Dantzign B.G. (1960). On the significance of solving linear programming problems with some integer variables. Econometrica, 28:30, 44. Dantzig, B.G., Johnson, S. and White W. (1958). A linear programming approach, Management Science, 5:38, 43. Dantzig, B.G. (1963). Linear Programming and Extensions. US: Princeton University Press. Gale, D. (1960). The theory of Economic Models. US: McGraw Hill. Karlin, S. (1959). Mathematical Methods and Theory, Programming and Economics, Vol. 1. Williams, P.H. (1993). Model Building in Mathematical Programming. Chichester: John Wiley and Sons. Chvatal, V. (1983). Linear Programming. W.H. Freeman & Co. Strayer, K.S. (1989). Linear Programming and Applications. US: Springer-Verlag. Read More