Linear Model For Emporium Coffee †Click to get free solution

Question: Discuss about theLinear Model for Emporium Coffee. Answer: Introduction According to Lucey (1994), the first step is to define the variables so that we can be able to develop the linear model. In order to develop the linear model we are given the following information; The company produces 2 coffee blends; Blend O and blend F Each blend requires an exact number of caffeine units for each 250g Blend O requires 7.5 caffeine units/ 250g Blend F requires 6 caffeine units/ 250g Selling price/250g of blend O is $ 190 Selling price/250g of blend F is $ 165 Production cost/250g of blend O is $ 25 Production cost/250g of blend F is $ 22 Overhead cost/250g of both blends is $ 10 To produce both blends requires 2 varieties of raw coffee beans; Arusha beans and Boyo beans Arusha beans contain 9.5 caffeine units/ 250g Boyo beans contain 5 caffeine units/ 250g Company capacity as inventory is Arusha beans 1,250kg; Boyo beans 3,750 kg The next step is to convert the conditions into symbolic form. We first identify the decision variables thus; Let x = Blend O y = Blend F Express the constraints as a system of inequalities x 0, y 0, where x and y are whole numbers The objective is to maximize the monthly profit thus the objective function is; Maximize 190x + 165y Based on the information above, the linear model can now be expressed as; Maximize 190x + 165y (profit) Subject to: 7.5x + 6y 250 (precision constraint) 25x + 22y 250 (production constraint) 10x + 10y 250 (overhead constraint) x + y 250 (Arusha beans constraint) x + y 250 (Boyo beans constraint) x + y 1,250 (Arusha beans inventory constraint) x + y 3,750(Boyo beans inventory constraint) The model works by first designing the Linear programming equation. Since the objective is to maximize profit, the objective function is designed in such a way to achieve optimality given the resources and constraints available. The method used is the simplex method of linear programming and the calculations of optimality are calculated using the solver function of Microsoft excel 2007. The LP model was designed that way to determine the maximum achievable profit given the seven constraints and recommend the best way to utilize the available resources at minimum cost The Solver answer report below shows the maximum Boyo beans required per 250g to produce blend O and F has been achieved. This means that if we were to utilize Boyo beans only, optimum profit is attainable. Table 1: Solver Answer Report The answer report also had non-binding variables which indicates there is a difference between the LHS and RHS. The slack values indicated that there were some unused resources at optimal solution. For the precision constraint, 37.5g is used for a total of 250g available. Likewise, in the production,overhead, Arusha beans, Arusha beans inventory and Boyo beans inventory constraints they were underutilized by 125,200,4.5, 1245 and 3745 respectively of the maximum allocated. The answer report shows that the optimum value of $ 950. See the table below. Table 2: The LP Model with Optimized Solution Target Cell (Max) Cell Name Original Value Final Value $B$6 Z Blend O (x) 0 950 Optimum value Adjustable Cells Cell Name Original Value Final Value $B$5 Solutions Blend O (x) 0 5 $C$5 Solutions Blend F (y) 0 0 Constraints Cell Name Cell Value Formula Status Slack $B$17 precision constraint LHS 37.5 $B$17=$C$17 Not Binding 212.5 $B$18 production constraint LHS 125 $B$18=$C$18 Not Binding 125 $B$19 overhead constraint LHS 50 $B$19=$C$19 Not Binding 200 $B$20 Arusha beans constraint LHS 5 $B$20=$C$20 Not Binding 4.5 $B$21 Boyo beans constraint LHS 5 $B$21=$C$21 Binding 0 $B$22 Arusha beans inventory constraint LHS 5 $B$22=$C$22 Not Binding 1245 $B$23 Boyo beans inventory constraint LHS 5 $B$23=$C$23 Not Binding 3745 $B$5 Solutions Blend O (x) 5 $B$5=0 Not Binding 5 $C$5 Solutions Blend F (y) 0 $C$5=0 Binding 0 The sensitivity report below includes the adjustable cells and constraints section. In the range of optimality represents the allowable increase and decrease of the original objective coefficient that is, the profit of blend O and blend F. using the constraints of the objective function, blend O can be reduced to $ 165 and no maximum increase limit without affecting the final value. In addition, blend F can be reduced to zero and increased to $ 190 The range of feasibility relates to the constraints available and the same case as in the optimality check. Precision constraint can be reduced by 37.5 with no upper limit, production constraint reduced to 125 with no upper limit, overhead constraint can be reduced to 50 with no upper limit, Arusha beans constraint can be reduced to 4.5with an upper limit of 5; Boyo beans constraint cane be reduced to zero and a maximum of 9.5; Arusha beans inventory constraint can be reduced to 5 with no upper limit and; Boyo beans inventory constraint to a low of 5 with no upper limit. All these adjustments can be done without affecting the final values . Constraints with no upper limits are those that were non-binding in the model and should be handled with caution. The sensitivity report also produced a shadow price. The shadow price means that if we were to increase the value of Boyo beans by one, assuming the range of feasibility of between zero and 9.5,then instead of getting the optimum value (z) of $950 we would get $ 1,140. All the other constraints were not binding since the shadow prices were zero. Therefore, the critical constraint is the Boyo beans constraint. Table 3: Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Range of Optimality Cell Name Value Cost Coefficient Increase Decrease Lower limit Upper limit $B$5 Solutions Blend O (x) 5 0 190 infinity 25 165 infinity $C$5 Solutions Blend F (y) 0 -25 165 25 infinity 0 190 Constraints Final Shadow Constraint Allowable Allowable Range of feasibility Cell Name Value Price R.H. Side Increase Decrease Lower limit Upper limit $B$17 precision constraint LHS 37.5 0 250 infinity 212.5 37.5 infinity $B$18 production constraint LHS 125 0 250 infinity 125 125 infinity $B$19 overhead constraint LHS 50 0 250 infinity 200 50 infinity $B$20 Arusha beans constraint LHS 5 0 9.5 infinity 4.5 5 infinity $B$21 Boyo beans constraint LHS 5 190 5 4.5 5 0 9.5 Critical constraint $B$22 Arusha beans inventory constraint LHS 5 0 1250 infinity 1245 5 infinity $B$23 Boyo beans inventory constraint LHS 5 0 3750 infinity 3745 5 infinity Conclusion Emporium Coffee produces two blends of coffee, blend O and blend F. The model was designed to illustrate how the company can maximize profits given the constraints of each blend. The LP model found a maximum profit of $ 950 per month while considering the constraints. It can be concluded that Boyo beans are more profitable than Arusha beans given the available constraints. The profit can be maximized to $1,140. It is therefore recommended that Emporium Coffee should consider using only Boyo beans to maximize profit. It is also recommended that the inventory be reduced for both varieties of beans to improve cash flow. References Lucey, T. (1996)Quantitative techniques. United Kingdom: Cengage Learning EMEA.