Stochastic Programming problem.

guidance on approach required to solve problem required please.

*CSV file summarised into one to make uploading easier and for comvenince

Questions 1. Best Brewers is a suall brewery that produces three craft beer types in South Africa. The profit earned from each (500ml) bottle of beer, the production hours required to produce each beer type (in hours/litre), and the annual demand (in bottles) for each type of beer are given in Table 1. Table 1: Best Brewers' input data The required production time (in hours/litre) per beer type (depicted by 7 -gamma) and the anmual demand (in 500ml bottles) per beer type (depicted by ( zeta) are random variables, determined through the 100 experimental observations included in the attached files ganma1.csv, gamma2.csv, gamma3.csv, zeta1.csv, zeta2. csv and zeta3.csv. (a) A maximum of 18000 production bours are available anuually. Formulate a stochastic progran that can be used to determine how many bottles of each type of beer should be producod annually to maximise Best Brewers' profit, whilst satisfying demand for different bext types. Show and discuss all your steps (including model formulation and (b) Solve the expected value model. Show and discuss all your steps (including adjusted model formulation, results and interpretation). (c) Apply the scenario approach if it is given that the available production hours constraint mast hold 95% of the time (with a 95% cotifdence), and the satisfy demand constraint must hold 90% of the time (with a 95% confidence). Show and discuss all your steps (including adjusted model formulation, solution code, results and interpretation). (d) Apply the approximation approuch if it is given that the arailable production hours constraint must hold 95% of the time, and the safisfy demand constraint must hold 90% of the time. Show and discuss all your steps (including adjusted model formulation, solution code, results and interptetation). (e) Apply the fixed recourse approach taking into account that the exceeding the production hour tusage is penalised by R5/ hour/lite and not satisying demand coets the business R4 per bottle that is shott. Show and discuss all your steps (including stochastic program with recourse, dual decomposition structure, solution code, results and interpretation). (f) Compare the reliability of the chance-constrained scenario approach, chance-constrained approximation approach, and recourse programming approach to that of the expectedvalue model. Which one do you recommesul? Why? Show and discuss all your steps. \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline & A & B & c & D & E & F & G & H \\ \hline 1 & & Gamma1 & Gamma2 & Gamma3 & Zetal & Zeta2 & Zeta3 & \\ \hline 2 & 1 & 0.38288 & 0.517799 & 0.653552 & 12845.13 & 13277.28 & 13718.77 & \\ \hline 3 & 2 & 0.509158 & 0.700721 & 1.349977 & 11233.02 & 9386.873 & 8636.748 & \\ \hline 4 & 3 & 0.408746 & 0.522833 & 0.911782 & 11316.25 & 9926.141 & 13411.2 & \\ \hline 5 & 4 & 0.339872 & 0.716318 & 0.709614 & 11523.8 & 10937.32 & 11924.83 & \\ \hline 6 & 5 & 0.566369 & 0.701867 & 1.197495 & 11821.43 & 11146.13 & 11209 & \\ \hline 7 & 6 & 0.159473 & 0.578382 & 0.918769 & 12033.1 & 12132.84 & 11557.74 & \\ \hline 8 & 7 & 0.306706 & 0.464022 & 1.085684 & 12815.6 & 10338.89 & 11537.67 & \\ \hline 9 & 8 & 0.634904 & 0.710748 & 0.992854 & 11459.53 & 10188.78 & 12451.7 & \\ \hline 10 & 9 & 0.461834 & 0.793049 & 0.970999 & 11486.04 & 10163.2 & 14366.06 & \\ \hline 11 & 10 & 0.283588 & 0.491598 & 0.83228 & 11263.72 & 10578.7 & 15994.88 & \\ \hline 12 & 11 & 0.243194 & 0.399724 & 1.134813 & 12768.49 & 7157.391 & 12766.56 & \\ \hline 13 & 12 & 0.46545 & 0.790417 & 0.458753 & 12498.78 & 11924.88 & 11276.9 & \\ \hline 14 & 13 & 0.126434 & 0.450239 & 0.698285 & 11166.66 & 11005.9 & 11537.85 & \\ \hline 15 & 14 & 0.440891 & 0.630538 & 0.629483 & 9378.969 & 9280.13 & 12624.45 & \\ \hline 16 & 15 & -0.053317 & 0.561447 & 1.093437 & 10190.96 & 12674.48 & 13429.64 & \\ \hline 17 & 16 & 0.283778 & 0.559918 & 1.213427 & 11650.57 & 8977.218 & 13521.27 & \\ \hline 18 & 17 & 0.553948 & 0.692005 & 1.393601 & 12573.02 & 13396.41 & 13953.26 & \\ \hline 19 & 18 & 0.085711 & 0.572699 & 1.413201 & 9834.941 & 11323.94 & 12701.07 & \\ \hline 20 & 19 & 0.047947 & 0.495666 & 0.93304 & 10944.48 & 9609.555 & 16144.06 & \\ \hline 21 & 20 & 0.229122 & 0.523937 & 0.889059 & 13644.99 & 11857.22 & 13891.07 & \\ \hline 22 & 21 & 0.424441 & 0.715896 & 0.893099 & 11253.74 & 12051.92 & 12355.06 & \\ \hline 23 & 22 & 0.420684 & 0.59356 & 0.729693 & 12153.17 & 11835.17 & 15901.85 & \\ \hline 24 & 23 & 0.291477 & 0.441334 & 0.648499 & 10495.83 & 10529.1 & 16200.51 & \\ \hline 25 & 24 & 0.674596 & 0.743383 & 1.500565 & 10420.18 & 11933.58 & 14028.55 & \\ \hline 26 & 25 & 0.187145 & 0.664592 & 1.168808 & 13488.64 & 11631.47 & 14404.67 & \\ \hline 27 & 26 & 0.796887 & 0.472137 & 1.476426 & 12210.54 & 12230.59 & 12456.32 & \\ \hline 28 & 27 & 0.162432 & 0.485336 & 0.435255 & 10794.98 & 9206.473 & 17189.87 & \\ \hline 29 & 28 & 0.425722 & 0.47253 & 1.08872 & 13589.47 & 11500.13 & 14712.83 & \\ \hline 30 & 29 & 0.411965 & 0.715098 & 0.846134 & 9297.953 & 11311.15 & 13701.63 & \\ \hline 31 & 30 & 0.335677 & 0.393351 & 0.668731 & 12234.52 & 9760.194 & 10395.37 & \\ \hline 32 & 31 & 0.295703 & 0.658424 & 1.158369 & 9667.879 & 9229.562 & 16977.19 & \\ \hline 33 & 32 & 0.18757 & 0.696939 & 1.025612 & 12349.97 & 11948.13 & 15679.81 & \\ \hline 34 & 33 & 0.500441 & 0.746138 & 0.833653 & 12791.88 & 10247.79 & 12593.25 & \\ \hline 35 & 34 & 0.582835 & 0.56577 & 1.125041 & 10548.39 & 11614.89 & 12527.37 & \\ \hline 36 & 35 & 0.448343 & 0.498256 & 0.985584 & 11532.14 & 11067.29 & 13432.14 & \\ \hline 37 & 36 & 0.024077 & 0.739442 & 1.474602 & 11364.91 & 10464.08 & 9596.874 & \\ \hline 38 & 37 & 0.575561 & 0.701873 & 0.922789 & 10990.55 & 11261.22 & 11074.26 & \\ \hline 39 & 38 & 0.885463 & 0.632158 & 0.770094 & 12712.3 & 9937.201 & 10715.58 & \\ \hline 40 & 39 & 0.355026 & 0.62665 & 1.233592 & 11029.29 & 11506.54 & 11544.94 & \\ \hline 41 & 40 & 0.223179 & 0.608227 & 1.104626 & 11703.32 & 9849.759 & 12304.01 & \\ \hline 42 & 41 & 0.106185 & 0.700257 & 1.181769 & 13108.45 & 10154.82 & 15450.72 & \\ \hline 43 & 42 & 0.501866 & 0.536305 & 0.762102 & 11551.3 & 12260.89 & 10037.07 & \\ \hline 44 & 43 & -0.054908 & 0.596659 & 0.600259 & 11408.79 & 10538.84 & 12690.73 & \\ \hline 45 & 44 & 0.097423 & 0.458745 & 0.995945 & 11499.7 & 11261.31 & 14216.31 & \\ \hline 46 & 45 & 0.477324 & 0.669229 & 0.987846 & 14733.55 & 11194.77 & 12827.32 & \\ \hline 47 & 46 & 0.425245 & 0.624542 & 0.871471 & 13994.5 & 9989.581 & 13298.41 & \\ \hline 48 & 47 & -0.020065 & 0.723194 & 0.641102 & 13198.19 & 11430.48 & 10007.84 & \\ \hline 49 & 48 & 0.384444 & 0.658827 & 1.250826 & 12470.08 & 11571.14 & 11599.12 & \\ \hline 50 & 49 & 0.520934 & 0.597462 & 1.301489 & 10523.59 & 10796.4 & 13276.6 & \\ \hline 51 & 50 & 0.239942 & 0.648226 & 1.366085 & 17364.03 & 11245.55 & 10120.18 & \\ \hline 52 & 51 & 0.104535 & 0.635385 & 0.431339 & 10921.36 & 11222.19 & 15606.7 & \\ \hline 53 & 52 & 0.212849 & 0.407908 & 1.262086 & 9734.849 & 12964.59 & 14216.51 & \\ \hline 54 & 53 & 0.423562 & 0.53865 & 1.734822 & 10134.29 & 11861.64 & 16508.68 & \\ \hline 5 & & & & & & & & \\ \hline \end{tabular} Questions 1. Best Brewers is a suall brewery that produces three craft beer types in South Africa. The profit earned from each (500ml) bottle of beer, the production hours required to produce each beer type (in hours/litre), and the annual demand (in bottles) for each type of beer are given in Table 1. Table 1: Best Brewers' input data The required production time (in hours/litre) per beer type (depicted by 7 -gamma) and the anmual demand (in 500ml bottles) per beer type (depicted by ( zeta) are random variables, determined through the 100 experimental observations included in the attached files ganma1.csv, gamma2.csv, gamma3.csv, zeta1.csv, zeta2. csv and zeta3.csv. (a) A maximum of 18000 production bours are available anuually. Formulate a stochastic progran that can be used to determine how many bottles of each type of beer should be producod annually to maximise Best Brewers' profit, whilst satisfying demand for different bext types. Show and discuss all your steps (including model formulation and (b) Solve the expected value model. Show and discuss all your steps (including adjusted model formulation, results and interpretation). (c) Apply the scenario approach if it is given that the available production hours constraint mast hold 95% of the time (with a 95% cotifdence), and the satisfy demand constraint must hold 90% of the time (with a 95% confidence). Show and discuss all your steps (including adjusted model formulation, solution code, results and interpretation). (d) Apply the approximation approuch if it is given that the arailable production hours constraint must hold 95% of the time, and the safisfy demand constraint must hold 90% of the time. Show and discuss all your steps (including adjusted model formulation, solution code, results and interptetation). (e) Apply the fixed recourse approach taking into account that the exceeding the production hour tusage is penalised by R5/ hour/lite and not satisying demand coets the business R4 per bottle that is shott. Show and discuss all your steps (including stochastic program with recourse, dual decomposition structure, solution code, results and interpretation). (f) Compare the reliability of the chance-constrained scenario approach, chance-constrained approximation approach, and recourse programming approach to that of the expectedvalue model. Which one do you recommesul? Why? Show and discuss all your steps. \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline & A & B & c & D & E & F & G & H \\ \hline 1 & & Gamma1 & Gamma2 & Gamma3 & Zetal & Zeta2 & Zeta3 & \\ \hline 2 & 1 & 0.38288 & 0.517799 & 0.653552 & 12845.13 & 13277.28 & 13718.77 & \\ \hline 3 & 2 & 0.509158 & 0.700721 & 1.349977 & 11233.02 & 9386.873 & 8636.748 & \\ \hline 4 & 3 & 0.408746 & 0.522833 & 0.911782 & 11316.25 & 9926.141 & 13411.2 & \\ \hline 5 & 4 & 0.339872 & 0.716318 & 0.709614 & 11523.8 & 10937.32 & 11924.83 & \\ \hline 6 & 5 & 0.566369 & 0.701867 & 1.197495 & 11821.43 & 11146.13 & 11209 & \\ \hline 7 & 6 & 0.159473 & 0.578382 & 0.918769 & 12033.1 & 12132.84 & 11557.74 & \\ \hline 8 & 7 & 0.306706 & 0.464022 & 1.085684 & 12815.6 & 10338.89 & 11537.67 & \\ \hline 9 & 8 & 0.634904 & 0.710748 & 0.992854 & 11459.53 & 10188.78 & 12451.7 & \\ \hline 10 & 9 & 0.461834 & 0.793049 & 0.970999 & 11486.04 & 10163.2 & 14366.06 & \\ \hline 11 & 10 & 0.283588 & 0.491598 & 0.83228 & 11263.72 & 10578.7 & 15994.88 & \\ \hline 12 & 11 & 0.243194 & 0.399724 & 1.134813 & 12768.49 & 7157.391 & 12766.56 & \\ \hline 13 & 12 & 0.46545 & 0.790417 & 0.458753 & 12498.78 & 11924.88 & 11276.9 & \\ \hline 14 & 13 & 0.126434 & 0.450239 & 0.698285 & 11166.66 & 11005.9 & 11537.85 & \\ \hline 15 & 14 & 0.440891 & 0.630538 & 0.629483 & 9378.969 & 9280.13 & 12624.45 & \\ \hline 16 & 15 & -0.053317 & 0.561447 & 1.093437 & 10190.96 & 12674.48 & 13429.64 & \\ \hline 17 & 16 & 0.283778 & 0.559918 & 1.213427 & 11650.57 & 8977.218 & 13521.27 & \\ \hline 18 & 17 & 0.553948 & 0.692005 & 1.393601 & 12573.02 & 13396.41 & 13953.26 & \\ \hline 19 & 18 & 0.085711 & 0.572699 & 1.413201 & 9834.941 & 11323.94 & 12701.07 & \\ \hline 20 & 19 & 0.047947 & 0.495666 & 0.93304 & 10944.48 & 9609.555 & 16144.06 & \\ \hline 21 & 20 & 0.229122 & 0.523937 & 0.889059 & 13644.99 & 11857.22 & 13891.07 & \\ \hline 22 & 21 & 0.424441 & 0.715896 & 0.893099 & 11253.74 & 12051.92 & 12355.06 & \\ \hline 23 & 22 & 0.420684 & 0.59356 & 0.729693 & 12153.17 & 11835.17 & 15901.85 & \\ \hline 24 & 23 & 0.291477 & 0.441334 & 0.648499 & 10495.83 & 10529.1 & 16200.51 & \\ \hline 25 & 24 & 0.674596 & 0.743383 & 1.500565 & 10420.18 & 11933.58 & 14028.55 & \\ \hline 26 & 25 & 0.187145 & 0.664592 & 1.168808 & 13488.64 & 11631.47 & 14404.67 & \\ \hline 27 & 26 & 0.796887 & 0.472137 & 1.476426 & 12210.54 & 12230.59 & 12456.32 & \\ \hline 28 & 27 & 0.162432 & 0.485336 & 0.435255 & 10794.98 & 9206.473 & 17189.87 & \\ \hline 29 & 28 & 0.425722 & 0.47253 & 1.08872 & 13589.47 & 11500.13 & 14712.83 & \\ \hline 30 & 29 & 0.411965 & 0.715098 & 0.846134 & 9297.953 & 11311.15 & 13701.63 & \\ \hline 31 & 30 & 0.335677 & 0.393351 & 0.668731 & 12234.52 & 9760.194 & 10395.37 & \\ \hline 32 & 31 & 0.295703 & 0.658424 & 1.158369 & 9667.879 & 9229.562 & 16977.19 & \\ \hline 33 & 32 & 0.18757 & 0.696939 & 1.025612 & 12349.97 & 11948.13 & 15679.81 & \\ \hline 34 & 33 & 0.500441 & 0.746138 & 0.833653 & 12791.88 & 10247.79 & 12593.25 & \\ \hline 35 & 34 & 0.582835 & 0.56577 & 1.125041 & 10548.39 & 11614.89 & 12527.37 & \\ \hline 36 & 35 & 0.448343 & 0.498256 & 0.985584 & 11532.14 & 11067.29 & 13432.14 & \\ \hline 37 & 36 & 0.024077 & 0.739442 & 1.474602 & 11364.91 & 10464.08 & 9596.874 & \\ \hline 38 & 37 & 0.575561 & 0.701873 & 0.922789 & 10990.55 & 11261.22 & 11074.26 & \\ \hline 39 & 38 & 0.885463 & 0.632158 & 0.770094 & 12712.3 & 9937.201 & 10715.58 & \\ \hline 40 & 39 & 0.355026 & 0.62665 & 1.233592 & 11029.29 & 11506.54 & 11544.94 & \\ \hline 41 & 40 & 0.223179 & 0.608227 & 1.104626 & 11703.32 & 9849.759 & 12304.01 & \\ \hline 42 & 41 & 0.106185 & 0.700257 & 1.181769 & 13108.45 & 10154.82 & 15450.72 & \\ \hline 43 & 42 & 0.501866 & 0.536305 & 0.762102 & 11551.3 & 12260.89 & 10037.07 & \\ \hline 44 & 43 & -0.054908 & 0.596659 & 0.600259 & 11408.79 & 10538.84 & 12690.73 & \\ \hline 45 & 44 & 0.097423 & 0.458745 & 0.995945 & 11499.7 & 11261.31 & 14216.31 & \\ \hline 46 & 45 & 0.477324 & 0.669229 & 0.987846 & 14733.55 & 11194.77 & 12827.32 & \\ \hline 47 & 46 & 0.425245 & 0.624542 & 0.871471 & 13994.5 & 9989.581 & 13298.41 & \\ \hline 48 & 47 & -0.020065 & 0.723194 & 0.641102 & 13198.19 & 11430.48 & 10007.84 & \\ \hline 49 & 48 & 0.384444 & 0.658827 & 1.250826 & 12470.08 & 11571.14 & 11599.12 & \\ \hline 50 & 49 & 0.520934 & 0.597462 & 1.301489 & 10523.59 & 10796.4 & 13276.6 & \\ \hline 51 & 50 & 0.239942 & 0.648226 & 1.366085 & 17364.03 & 11245.55 & 10120.18 & \\ \hline 52 & 51 & 0.104535 & 0.635385 & 0.431339 & 10921.36 & 11222.19 & 15606.7 & \\ \hline 53 & 52 & 0.212849 & 0.407908 & 1.262086 & 9734.849 & 12964.59 & 14216.51 & \\ \hline 54 & 53 & 0.423562 & 0.53865 & 1.734822 & 10134.29 & 11861.64 & 16508.68 & \\ \hline 5 & & & & & & & & \\ \hline \end{tabular}