Please solve part a and b only. For part b asks for the optimal solution, but also clearly state what the optimal value is.

6. The Sixth Sin Chocolate Company has come out with a new chocolate soft drink that they named Gula Cola. Sarah Hopkinson, Director of Marketing Research has indicated that test market sales of Gula have been brisk and she projects demand for the next 3 months in thousands of gallons to be 100, 140, and 190. They expect their production costs to go down as they gain experience and economies of scale. The costs are reflected in the table below. They currently have production capacity of 160 thousand gallons per month. PERIOD PRODUCTION COST ($/1,000 GAL) INVENTORY HOLDING COST CHARGED AT THE END OF THE MONTH (S/1,000 GAL) 260 20 250 25 240 The company has an initial inventory of 10 thousand gallons at the beginning of period 1, and would like to hold a minimum inventory of at least 25 thousand gal- lons at the end of month 3 to handle any variation in demand. a. Formulate a linear program, clearly defining all variables used to determine the production and inventory in each period that minimizes the total cost for The Sixth Sin over the 3 periods, while meeting its demand requirements, b. Use Solver to find the optimal solution to the linear program from part a. C. The company has the option to use another production line to expand capac- ity. There would be a one-time setup cost of $3,000 to refit the production line if they use it. As the line uses older technology, production costs will be 10% higher than the normal manufacturing cost per gallon, for each month. This production line can produce a maximum of 20 thousand gallons per month. Formulate a linear program, to minimize the total cost for The Sixth Sin over the 3 months, while meeting its demand requirements. Modify your formula- tion in part a to reflect this change. 6. The Sixth Sin Chocolate Company has come out with a new chocolate soft drink that they named Gula Cola. Sarah Hopkinson, Director of Marketing Research has indicated that test market sales of Gula have been brisk and she projects demand for the next 3 months in thousands of gallons to be 100, 140, and 190. They expect their production costs to go down as they gain experience and economies of scale. The costs are reflected in the table below. They currently have production capacity of 160 thousand gallons per month. PERIOD PRODUCTION COST ($/1,000 GAL) INVENTORY HOLDING COST CHARGED AT THE END OF THE MONTH (S/1,000 GAL) 260 20 250 25 240 The company has an initial inventory of 10 thousand gallons at the beginning of period 1, and would like to hold a minimum inventory of at least 25 thousand gal- lons at the end of month 3 to handle any variation in demand. a. Formulate a linear program, clearly defining all variables used to determine the production and inventory in each period that minimizes the total cost for The Sixth Sin over the 3 periods, while meeting its demand requirements, b. Use Solver to find the optimal solution to the linear program from part a. C. The company has the option to use another production line to expand capac- ity. There would be a one-time setup cost of $3,000 to refit the production line if they use it. As the line uses older technology, production costs will be 10% higher than the normal manufacturing cost per gallon, for each month. This production line can produce a maximum of 20 thousand gallons per month. Formulate a linear program, to minimize the total cost for The Sixth Sin over the 3 months, while meeting its demand requirements. Modify your formula- tion in part a to reflect this change