Question #3 The polytope {x|Ax = b, x E R > 0} has only integer vertices if A is totally unimodular and b is integer. 1. Formulate the assignment problem from lecture 12 as a linear program in which you assume that the variables take on only integer values. Show that the constraint matrix is totally unimodular, and hence solving this linear program give a legitimate solution to the assignment problem. 2. Suppose a car company has m storage lots and n retail showrooms, and it needs to ship cars from stroage lots to showrooms. Each stroage lot has a supply a and each showroom has a demand b. We are given costs Cij between every storage lot and showroom. Shipping x units from storage lot i to showroom j costs Cijx. You want to determine an optimal shipment scheme subject to the supply and demand constraints, that minimizes total cost. Formulate this as a linear program and then show that the constraint matrix is totally unimodular. Question #3 The polytope {x|Ax = b, x E R > 0} has only integer vertices if A is totally unimodular and b is integer. 1. Formulate the assignment problem from lecture 12 as a linear program in which you assume that the variables take on only integer values. Show that the constraint matrix is totally unimodular, and hence solving this linear program give a legitimate solution to the assignment problem. 2. Suppose a car company has m storage lots and n retail showrooms, and it needs to ship cars from stroage lots to showrooms. Each stroage lot has a supply a and each showroom has a demand b. We are given costs Cij between every storage lot and showroom. Shipping x units from storage lot i to showroom j costs Cijx. You want to determine an optimal shipment scheme subject to the supply and demand constraints, that minimizes total cost. Formulate this as a linear program and then show that the constraint matrix is totally unimodular