T and c 0 nonnegativity results of the original

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T and C ≥ 0 (nonnegativity)Results of the Original ScenarioThe optimal solution is to produce 320 tables and 360 chairs for a maximum profit of $4,040. The objective function coefficient (OFC) in the furniture storeproblem is $7 for T and $5 for C. To determine the impact of a change to an OFC, consider a demand for the chairs. The profit for the chairs is increased from $5 to $6 due to high demand. The optimal level profit line changes, but the corner point is still the optimal solution. The only change to the optimal solution is the profit. Optimal profit = $7T + $6C = $7(320) + $6(360) = $4,400.A reduction in the demand for chairs could lower the profit from $5 to $4, which would also simply change the optimal profit to $7(320) + $4(360) = $3,680.Revised ScenarioThey are considering producing a shelving unit. The shelving unit will require 2 carpentry hours and 1 painting hour. The current resources for the carpentry and painting hours will decrease.Step 1: Check the Validity of the 100% RuleCheck the 100% rule. The allowable decrease for carpentry hours is 900, and the allowable decrease for painting hours is 150. The new product requires 2 carpentry hours and 1 painting hour.Sum of ratios = (2 / 900) + (1 / 150) = 0.008890.00889 ≤ 1Because the sum of the ratios is less than 1, the shadow price is valid.Step 2: Calculate the Worth of the Resources Needed by the NewProductThe worth of the resources needed by the new product is the sum of the required resource times the shadow price of each constraint. The shadow price for carpentry hours is $0.60 and for the painting hours is $2.60.
Worth of resources = (2 * 0.60) + (1 * 2.60) = $3.80.For the new product to be viable, the profit contribution of the new product must cover the shortfall in the profit. In other words, the OFC for the product

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