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<p>1 Report of Analytic Skill Name: Zhao Mingming Major: Marketing Analytics ID number: 24202207 Word count: 5,490 2 Introduction According to Anderson et al (2008), management science is an approach to managerial decision making based on extensive use of quantitative analysis. Linear programming plays a key role in management science. It is problem-solving approaches which can help managers make decisions (ibid). The programming needs to formulation the problem to develop a model including decision variables, constraints and objective function ( Ignizio, 1968). The applications of linear programming include production scheduling, media selection, financial planning, capital budgeting, transportation, product mix and staffing (Hillier and Lieberman, 1995). This report will utilize linear programming to firstly solve a financial planning problem for Nori & Leets Co. and t hen find a solution for Alabama Atlantic to solve its transportation problem. Moreover, all the results are accurate to 2 decimal digits. Finally, it will examine a report about application of linear programming for dormitory development plan at Petra Christian University. Q1 The Nori & Leets Co. needs to reduce the emissions of three main types of pollutants which are particulate, sculpture oxides and hydrocarbons. The policy standard of required reduction in annual emission rate is 60 million pounds particulates, 150 million pounds sulfur oxides, and 125 million pounds hydrocarbons respectively. The two sources of pollution are blast furnaces for making pig iron and opening hearth furnaces for changing iron into steel. There are three effective abatement methods: (1) increasing height of the smokestacks, (2) using filter devices, and (3) including better fuels for the furnaces. However, these methods have technological limit that emissions has a abatement capacity as Table 2.1 shows. 3 Pollutant Taller smokestacks Filters Better fuels Blast Furnaces Open-Hearth Furnaces Blast Furnaces Open-Hearth Furnaces Blast Furnaces Open-Hearth Furnaces Particulates 12 9 25 20 17 13 Sulfur oxides 35 42 18 31 56 49 Hydrocarbons 37 53 28 24 29 20 Table 1.1 (millions of pounds) Furthermore, emission reduction by each method is independent. Engineers also concluded that it should use some combination of methods. However, each method can lead to a huge cost such as opportunity revenue loss, operating and maintenance expenses and start-up cost. Therefore, the company has estimated total annual cost given in the table below for using the methods at full abatement capacities. Abatement Method Blast Furnaces Open-Hearth Furnace Taller Smokestacks $8 million 10 million Filters 7 million 6 million Better Fuels 11 million 9 million Table 1.2 The company intends to minimize the annual cost of achieving the required emission reduction rates for three poll utants by drawing up a plan which specifies which types of abatement methods will be used and at what fractions of their abatement capacities for the blast furnaces and the open-hearth furnaces. a It is a deterministic model and there is a linear relationship between total cost and fraction of capacities. Moreover, the variables are independent. Therefore, it will use 4 linear programming to solve the financial planning problem. The first step is to define variables. Letting Xij (i= taller smokestacks, filters, better fuels; j=blast furnaces, open-hearth furnaces) denotes the fraction of total annual cost from the maximum feasible use of each abatement method on both two sources of pollutant. The following table shows the variables clearly. Xij Blast Furnaces Open-Hearth Furnace Taller Smokestacks Xtb Xto Filters Xfb Xfo Better Fuels Xbb Xbo Table 1.3 The next step is to determine the coefficients of variables, Cij. As the total annual cost (Z) is the sum of the actual total annual cost of each method, Cij are the total annual cost from maximum feasible use of each abatement method as shown in Table 1.2. Hence, objective function is to minimize Z= 8Xtb + 10Xto + 7Xfb + 6Xfo + 11Xbb + 9Xbo. Furthermore, it needs to set constraints. There are two kinds of constraints: the total emission reduction of these three pollutants should be more than or equal to required reduction in annual emission rate (mentioned above), and the Xij should be less than or equal to 1 and more than 0 as it is fraction. Total emission reduction The total emission reduction of each pollutant should be the sum of the fraction of abatement capacity of every method timing the abatement capacity from maximum feasible use of that method which is given in table 1.1 above. It is determined that cost of a method is roughly proportional to the fraction of the abatement capacity. As a result, the fraction of abatement capacity is roughly equal to Xij. Hence, the 5 constraints can be displayed as: Particulates: 12Xtb + 9Xto + 25Xfb + 20Xfo + 17Xbb + 13Xbo 60 Sulfur oxides: 35Xtb + 42Xto + 18Xfb + 31Xfo + 56Xbb + 49Xbo 150 Hydrocarbons: 37Xtb + 53Xto + 28Xfb + 24Xfo + 29Xbb + 20Xbo 125 0Xij1 The problem is calculated by Excel Solver (The process is expressed in Appendix 1). The optimal solution is Z= $32.15 million and the plan is displayed in Table 1.4. Taller Smokestacks Filters Better Fuels Blast Furnaces Open-Hearth Furnaces Blast Furnaces Open-Hearth Furnaces Blast Furnaces Open-Hearth Furnaces Fraction 1 0.62 0.34 1 0.05 1 Table 1.4 Recommendation: It is recommended that: All the three types of abatement methods are used but the fractions of their abatement capacities are different. Among them, taller smokestacks for blast furnaces, Filters for open-hearth furnaces, and better fuels for open-hearth furnaces should be maximum feasible used. Taller Smokestacks for open-hearth furnaces should be used at 62% of its abatement capacity. Filt ers for blast furnaces should be 34% of its abatement capacity and Better Fuels for Blast Furnaces is used at 5% of its abatement capacity. Sensitivity analysis According to Fabrycky and Mize (1982), sensitivity analysis in linear programming is the study of how the changes in coefficients impact optimal solution. It is important 6 because in real-life, the problems will seldom satisfy all of these assumptions . If data used have errors, the system will be modified. For this case, after designing the plan, management needs to conduct sensitivity analysis. The cost in table 1.2 is estimated and each one may easily be off by as much as 10 percent in either direction. Moreover, values in table 1.1 are uncertain as well but the uncertainty is less than cost. The policy standards are constants. However, company and government achieve the agreement that each 10% increase in the policy standards over the current values will result in $3.5 million tax reduction for the company. Furthermore, there is a debate between relative values of the policy standards for three pollutants as well. The next part is to make senility analysis based on the sensitivity report in Appendix 1. b Objective function coefficients (OFC) and right-hand side values of constraints are sensitive parameters in linear programming. As mentioned above, OFCs are cost parameters and RHS values are required reduction of pollutant. However, for this case, the constraints Right-Hand Sides values of constraints are constant because they are policy standards. Hence, only cost parameters which are given in Table 1.2 are sensitive parameters in this case. To find which parameters should be estimated more closely, it needs to identify the range of optimality for each coefficient of variables. Range of optimality supplies the range of values over which the current optimal solution will not change. Moreover, it is assumed that one coefficient change but others are constant. The formula of range of optimality is shown as: cuiient coeficient value allowable ueciease C cuiient coeficient value allowable inciease Allowable increase means that the maximized value the coefficient can increase to 7 remain the optimal solution. Conversely, allowable decrease refers to the maximized value the coefficient can decrease to keep optimal solution. Allowable increase and allowable decrease for each objective function coefficient is given in sensitivity report in Appendix 1. In terms of allowable decrease, the data of three coefficients is 1E+30 which means the number is nearly infinite. This applies that no matter how much this coefficient decrease, the optimal solution will not change. However, because the coefficient refers to cost in real world, it cannot be negative and the lowest value is 0. Hence, the range of optimality (in million dollars) for each OFC is: Taller Smokestacks for Blast Furnaces: 0 C 8.34 Taller Smokestacks for Open-Hearth furnaces: 9.33 C 10.43 Filters for Blast Furnaces: 5 C 7.38 Filters for Open-Hearth furnaces: 0 C 7.82 Better Fuels for Blast Furnaces: 10.96 C 13.98 Better Fuels for Open-Hearth furnaces: 0 C 9.04 The area value of the range (upper value minus lower value) displays the probability that estimated value can remain the optimal solution. If the range area are is great, this implies that the estimate value has a large probability to be located in the range of optimality. In contrast, if the range area is small, the estimated coefficient will have a large probability to be out of range of optimality and then the optimal solution changes. The area value of the range for each coefficient is given as in table 1.5. Coefficient Area of range of optimality C 8.34 (8.34) C 1.1 (10.43) C 2.38 (7.38) C 7.82 (7.82) C 3.02 (13.98) C 9.04 (9.04) 8 Table 1.5 Recommendation As the Table 1.5 show, Cto (Taller Smokestacks for Open-Hearth furnaces) has the lowest area value and it is the parameter that the company should be estimated more closely to true value without changing optimal solution. In addition, Cfb (Filter for Blast Furnaces) and Cbb (Better fuels for Blast Furnaces) should also be estimated a little accurately as their area of range of optimality is not very large. c As mentioned above, the true cost parameter may be off by 10% in either direction. To analyze the effect of inaccuracy in estimating each cost parameter, it needs to assume that if one parameter change, others are constant as well. The assumption is valid for every parameter. In terms of the situation when true value is less than estimated value, the true value and range of optimality is displayed in Table 1.6. Coefficient Estimated value True value (10% decrease) Range of optimality C 8 7.2 [0, 8.34] C 10 9 [9.33, 10.43] C 7 6.3 [5, 7.38] C 6 5.4 [0, 7.82] C 11 9.9 [10.96, 13.98] C 9 8.1 [0,9.04] Table 1.6 (million dollars) It is clear that the true value of Ctb, Cfb, Cfo and Cbo is located in its range of optimality respectively. Therefore, true value of these four coefficients does not change the optimal solution. However, for Cto (Taller Smokestacks for Open-Hearth Furnaces) and 9 Cbb (Better Fuels for Blast Furnaces), its true value is out of their range of optimality and the optimal solution will change. As for the situation when the true value is 10% more than estimated value, the process is similar to the situation when true value is 10% less than estimated value. It also assumes that when one parameter changes, other parameters do not change. The following table shows the true value and range of optimality. Coefficient Estimated value True value (10% decrease) Range of optimality C 8 8.8 [0, 8.34] C 10 11 [9.33, 10.43] C 7 7.7 [5, 7.38] C 6 6.6 [0, 7.82] C 11 12.1 [10.96, 13.98] C 9 9.9 [0,9.04] Table 1.7 (million dollars) It is explicit that if true value of cost parameter is 10% more than estimated value, nearly all of them is not in their range of optimality except Cbb. This displays that except Cbb, the optimal solution will c hange when other cost parameters true value is 10% more than estimated value. Recommendation: The company should conduct much effort to make its true cost value lower than its estimated value. The reason is that the probability of changing optimal solution when the true value is more than estimated value is really high. As the table 1.7 shows, five of six parameters will change optimal solution. However, the probability of changing optimal solution when true value is less than estimated value is relatively low. Two of six parameters will modify optimal solution. 10 Moreover, the company should put more focus on Cbb and Cbo to estimate more closely. In perspective of Cbb, even the estimated value is in the optimality, but 11 is so close to lower limit which is 10.96 in range of optimality. If the true value of Cbb is a little less than 11, the optimal solution will change probably. Similar to Cbb, Cbos estimated value is too close to upper limit which is 9.04 as well. If the true value of Cbo is a bit more than 9, the optimal may solution change in a large probability. Thereforethe company should revise Cbb (better fuels for blast furnaces) and Cbo (better fuels for open-hearth furnaces). d Shadow price in the constraints part of sensitivity report refers to the change in the value of the solution because of per unit increase in the right-hand side of the constraints. However, shadow price is only valid when there is fairly small change in right-hand side. Allowable increase and allowable decrease in sensitivity report specify how much the right-hand side value can be changed without changing the value of optimal solution. As mentioned above, the required reduction in the annual emission rate of pollutant is right-hand side value and the total cost is the value of optimal solution. Hence, shadow price is a good approach to specify the rate at which the total cost of an optimal solution would change with any small change in the required reduction in annual emission rate of each pollutan...</p>