Please solve the question thoroughly with all the variables, equations,calculations and explanations etc. Thanks

Willy Wonka Chocolate Factory produces chocolate bars from different ingredients like cocoa, cocoa butter, milk, sugar, hazelnut, pistachio etc. Willie Wonka has two cocoa butter plants located in Balkesir and Mula. Cocoa butter is produced from cocoa at these two plants or it is imported at the zmir harbor. The produced and imported cocoa butter is transported to chocolate plants in stanbul and Adana, where chocolate bars are produced. Willy Wonka ships chocolate produced at the chocolate plants to its main distribution centers in stanbul, zmir, Ankara, Bolu, Trabzon, and to the zmir harbor for export. PEYS NKA Nutty Crunch Sumse The estimates on the transportation costs between the cocoa butter plants and the chocolate plants; and between the chocolate plants and the distribution centers are given in Table 1. Table 2 contains the chocolate demand forecast at the distribution centers for the next year. Table 3 shows the current cocoa butter and chocolate production capacities as well as cocoa butter yield. Table 4 summarizes the current distribution plan prepared for the next year. You have to figure out how many tons of cocoa butter to ship from each cocoa butter plant to each chocolate plant and how many tons of the chocolate to ship from each chocolate plant to each distributor. You have to make sure that none of the capacities of the plants is exceeded, and all the chocolate demand of the distributors is satisfied. The shipment plan that minimizes the total costs must be quite different from the current plant. Table 1: TRANSPORTATION COSTS: Annual cost of shipping 1000 tons of cocoa butter from cocoa butter plant i to chocolate plant i (Million $ /1000 ton) Destinations Chocolate Plants Cocoa butter plants Istanbul Adana Balkesir 0.026 0.017 Mula 0.037 0.017 Import (Izmir) 0.032 0.033 Annual cost of shipping 1000 ton of chocolate from chocolate plant to distribution center k (Million $/1000 ton) Destinations Distribution Centers Plants Istanbul | 0.000 0.032 Istanbul Izmir 0.040 0.041 Ankara | 0.052 0.039 Export Bolu Trabzon (Izmir) 0.017 | 0.055 | 0.042 0.027 | 0.023 | 0.043 Adana Table 2: DEMAND: Annual demand at distribution centers Dist. Centers Next Year 103 Istanbul Izmir Ankara Bolu Trabzon Export(Izmir) Total Table 3: CAPACITY: Annual capacity of chocolate plants (1000 tons/year) Annual capacity of cocoa butter plants (1000 tons/year) Balkesir Mula 68 Istanbul Adana 220 200 Import (Izmir) 20 CHOCOLATE YIELD: 1000 tons of chocolate produced from 1000 ton of cocoa butter Domestic 8.333 9.091 Import Table 4: Single Period - Current Shipment Plan Amount of cocoa butter shipped from plant i to chocolate plantj next year (1000 ton) Istanbul Adana Total Capacity Balkesir 0 24.00 30.00 2.42 68.00 Mula 200 20.00 20.00 Import Total cocoa butter 22.42 24 202 200 Total chocolate (1000 tons) 2.42 Total Capacity 220 Amount of chocolate shipped from chocolate plant to distribution center k next year (1000 ton) Export Istanbul Izmir Ankara Bolu Trabzon (Izmir) Istanbul 103 49 50 0 0 0 Adana 0 25 0 60 10213 Total 103 74 50 60 102 13 Demands 10374 50 60 10213 202 200.00 200 Total shipping cost 11.244 (Million dollars) Analyze Willy Wonkas distribution strategy and suggest possible improvements over the current distribution plan. You should answer the following questions when you work on the case: Question 1. How can the current distribution plan be improved? Are the savings in costs significant? (40 pts) Hint: Optimize the distribution system by formulating and solving a linear programming model. Develop a cocoa butter and chocolate shipment plan that minimizes total transportation costs while satisfying chocolate demand at the distribution centers. You need two different types of decision variables. The first type is for the shipment of cocoa butter from cocoa butter plants to chocolate plants and the second type is for the shipment of chocolates produced at chocolate plants to distribution centers. Accordingly, both type of decision variables have two indices. You should not forget that the amount of chocolate produced at a chocolate plant is equal to the yield obtained from cocoa butter shipped to that chocolate plant. Use GAMS to solve your model. Discuss the resulting benefits of the suggested plan. Willy Wonka Chocolate Factory produces chocolate bars from different ingredients like cocoa, cocoa butter, milk, sugar, hazelnut, pistachio etc. Willie Wonka has two cocoa butter plants located in Balkesir and Mula. Cocoa butter is produced from cocoa at these two plants or it is imported at the zmir harbor. The produced and imported cocoa butter is transported to chocolate plants in stanbul and Adana, where chocolate bars are produced. Willy Wonka ships chocolate produced at the chocolate plants to its main distribution centers in stanbul, zmir, Ankara, Bolu, Trabzon, and to the zmir harbor for export. PEYS NKA Nutty Crunch Sumse The estimates on the transportation costs between the cocoa butter plants and the chocolate plants; and between the chocolate plants and the distribution centers are given in Table 1. Table 2 contains the chocolate demand forecast at the distribution centers for the next year. Table 3 shows the current cocoa butter and chocolate production capacities as well as cocoa butter yield. Table 4 summarizes the current distribution plan prepared for the next year. You have to figure out how many tons of cocoa butter to ship from each cocoa butter plant to each chocolate plant and how many tons of the chocolate to ship from each chocolate plant to each distributor. You have to make sure that none of the capacities of the plants is exceeded, and all the chocolate demand of the distributors is satisfied. The shipment plan that minimizes the total costs must be quite different from the current plant. Table 1: TRANSPORTATION COSTS: Annual cost of shipping 1000 tons of cocoa butter from cocoa butter plant i to chocolate plant i (Million $ /1000 ton) Destinations Chocolate Plants Cocoa butter plants Istanbul Adana Balkesir 0.026 0.017 Mula 0.037 0.017 Import (Izmir) 0.032 0.033 Annual cost of shipping 1000 ton of chocolate from chocolate plant to distribution center k (Million $/1000 ton) Destinations Distribution Centers Plants Istanbul | 0.000 0.032 Istanbul Izmir 0.040 0.041 Ankara | 0.052 0.039 Export Bolu Trabzon (Izmir) 0.017 | 0.055 | 0.042 0.027 | 0.023 | 0.043 Adana Table 2: DEMAND: Annual demand at distribution centers Dist. Centers Next Year 103 Istanbul Izmir Ankara Bolu Trabzon Export(Izmir) Total Table 3: CAPACITY: Annual capacity of chocolate plants (1000 tons/year) Annual capacity of cocoa butter plants (1000 tons/year) Balkesir Mula 68 Istanbul Adana 220 200 Import (Izmir) 20 CHOCOLATE YIELD: 1000 tons of chocolate produced from 1000 ton of cocoa butter Domestic 8.333 9.091 Import Table 4: Single Period - Current Shipment Plan Amount of cocoa butter shipped from plant i to chocolate plantj next year (1000 ton) Istanbul Adana Total Capacity Balkesir 0 24.00 30.00 2.42 68.00 Mula 200 20.00 20.00 Import Total cocoa butter 22.42 24 202 200 Total chocolate (1000 tons) 2.42 Total Capacity 220 Amount of chocolate shipped from chocolate plant to distribution center k next year (1000 ton) Export Istanbul Izmir Ankara Bolu Trabzon (Izmir) Istanbul 103 49 50 0 0 0 Adana 0 25 0 60 10213 Total 103 74 50 60 102 13 Demands 10374 50 60 10213 202 200.00 200 Total shipping cost 11.244 (Million dollars) Analyze Willy Wonkas distribution strategy and suggest possible improvements over the current distribution plan. You should answer the following questions when you work on the case: Question 1. How can the current distribution plan be improved? Are the savings in costs significant? (40 pts) Hint: Optimize the distribution system by formulating and solving a linear programming model. Develop a cocoa butter and chocolate shipment plan that minimizes total transportation costs while satisfying chocolate demand at the distribution centers. You need two different types of decision variables. The first type is for the shipment of cocoa butter from cocoa butter plants to chocolate plants and the second type is for the shipment of chocolates produced at chocolate plants to distribution centers. Accordingly, both type of decision variables have two indices. You should not forget that the amount of chocolate produced at a chocolate plant is equal to the yield obtained from cocoa butter shipped to that chocolate plant. Use GAMS to solve your model. Discuss the resulting benefits of the suggested plan