answer:Given that the first goal (G1) has the same importance as the third goal (G3), and G3 is twice as important as the second goal (G2), we can assign the importance weights as follows: - Importance weight of G1 = 1 - Importance weight of G2 = 0.5 - Importance weight of G3 = 2 Using these weights, the objective function for the goal programming formulation is: Minimize (1*deviation from G1 + 0.5*deviation from G2 + 2*deviation from G3) Where "deviation from Gi" represents the degree to which the ith goal is not met. Expanding the above expression, we get: Minimize (deviation from G1 + 0.5*deviation from G2 + 2*deviation from G3) Multiplying both sides by 2, we get: Minimize (2*deviation from G1 + deviation from G2 + 4*deviation from G3) Therefore, the correct answer is "Minimize (2G1 + G2 + 4G3)" as per the weights method in goal programming.

answer:In a transportation problem with N supply points and M demand points, the number of constraints required in its linear programming (LP) formulation is N + M - 1. A transportation problem involves shipping a quantity of goods from several supply points to several demand points while minimizing the total transportation cost. In an LP formulation of a transportation problem, the objective is to minimize the total cost of transportation subject to supply and demand constraints. The constraints in LP formulation of a transportation problem arise from the supply and demand availability at the supply and demand points. Both supply and demand must be balanced, meaning the total supply from all sources should equal the total demand at all the destinations. The supply and demand constraints for a transportation problem are as follows: - Supply constraint: The total quantity of a commodity supplied from a single supply point cannot exceed its capacity. - Demand constraint: The total quantity of a commodity demanded to a particular destination must be fulfilled by various supply points. We need N supply constraints (one for each supply point) and M demand constraints (one for each demand point) to ensure the supply and demand are met. However, the sum of the supply constraints gives N equations and the sum of demand constraints gives M equations. Since one of the equations can be derived from the other equations (i.e., the sum of supply must equal the sum of demand), we can eliminate one constraint and write N + M - 1 constraints. Therefore, the number of constraints required in the LP formulation of a transportation problem with N supply points and M demand points is N + M - 1. Therefore, the correct answer is "N + M - 1".

answer:Dynamic programming divides problems into a number of decision stages. Dynamic programming is a problem-solving method that is used for optimization problems with a well-defined structure. In dynamic programming, a complex problem is broken down into smaller sub-problems that follow a recursive sequence. These sub-problems can be solved using a "divide and conquer" approach where the solution to the current sub-problem is dependent on the solutions to one or more previous sub-problems. In dynamic programming, a problem is divided into a number of decision stages that are sequenced in a particular way. Each stage consists of a set of decisions that must be made, and the optimal decision for each stage depends on the state of the problem at that stage and the optimal decisions made in previous stages. The optimal solution to the problem is then obtained by recursively computing the optimal solution to each sub-problem and combining these solutions to obtain the overall optimal solution. Therefore, the correct answer is "decision stages".

answer:The minimal cost network flows are a generalization of transportation problems. Transportation problem is a special case of the minimum cost network flow when supply points are equal to demand points, and supply or demand is conserved. That is, in a transportation problem, the objective is to transport goods from sources (supply points) to destinations (demand points) at a minimum cost, subject to supply and demand constraints. In contrast, the minimum cost network flow problem generalizes the transportation problem by allowing the network to have multiple sources and/or destinations, each with its own supply or demand. The objective is to find the optimal flow that satisfies all the supply and demand constraints while minimizing the total cost of transporting goods. Therefore, the correct statement is "The transportation problems are special cases of the minimal cost network flows".