The context for the first three problems is the following story. A company is making four types of concrete blocks. The

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The Context For The First Three Problems Is The Following Story A Company Is Making Four Types Of Concrete Blocks The 1 (116.39 KiB) Viewed 42 times

The Context For The First Three Problems Is The Following Story A Company Is Making Four Types Of Concrete Blocks The 2 (112.41 KiB) Viewed 42 times

The context for the first three problems is the following story. A company is making four types of concrete blocks. The manufacturing process for each of the products consists of three sub-processes, let us call them mixing, molding, and inspecting. For each sub-process there is an associated time constraint: no more than 800 hours for mixing, 1000 hours for molding, and 340 hours for inspection are available. The director wants to choose the optimal product mix to maximize the profit. The data for the problem was compiled by an OR specialist into the following tableau: z x1 X2 X3 X4 X5 X6 X7 RHS -1 8 14 30 50 0 0 0 0 0 1 2 10 16 1 0 0 800 0 1.5 2 4 5 0 1 0 1000 0 0.5 0.6 1 2 0 0 1 340 The variables x5, 6, and x7 are the slack variables, and the specialist used the maximization version of the simplex algorithm (i.e., choosing the pivot column based on the most positive cost coefficient of the non-basic variable, and so on). The specialist was hired away by another company, but left a partial print-out of the final tableau after the execution of the simplex method. There was a note on the print-out that read: "My replacement should be able to fill in the missing values on the spot, without running the simplex method." Z X1 X2 X3 X4 X5 X6 X7 RHS * * * * -5 0-6000 0 0 11 19 1.5 0 * 0 1 -2 2 * 0 0 0.1 -0.4 * -1 100 0-12 -22 0.4 1.6 -2 -1 0 1
Problem 1. Please write down the maximization problem coded by the first tableau (removing the slack variables and replacing them with the appropriate inequalities). Then write down the dual problem to the one you have stated. Problem 2. Using what you know about the revised simplex method, please determine the missing values, without executing the simplex algorithm. Please be sure to explain how you found the missing values. What is the optimal solution (both decision variables and optimal value)? If the above seems hard, for substantial partial credit please use the simplex algorithm on the problem coded by the first tableau to determine the optimal solution. Problem 3. Using what you know about the primal and dual variables and sensitivity analysis, please determine: (a) Are all the sub-processes (mixing, molding, inspecting) fully utilized in the optimal solution? If not, which of the workers would be "slacking"? (b) How would the optimal profit change if the company reallocated 10 hours from inspect- ing to mixing? (c) How big should be the profit of each unit of the third product (the one corresponding to 3) in order for the company to start producing it? Please be sure to explain your answers.