In order to pose some questions about dual linear programs and sensitivity analysis, we turn to a production transportation model. A variety of products must be mamifactured at given origins, and then shipped to giveu destinations, that the overall cost is minimal. We thus define the sets 0, D and P to represent origins, destinations, and products. The relevant data values include capacitica and requirements Te tons of product that can be mamfactured per hour at origin , for each of O and PEP productim bours available at origin o, for each of O terms of product p that must be delivered to destination de fer each de and PEP and also costs: dollars to make a ton of product p at origin o, for each o O nad p e P dap dollars to ship a ton of product p from origin o to destinatiou d. for cach o Ode Dand DEP The decision variables are: Xof tons of product p to be manufactured at origin o, for each oe and PEP Yode tons of procuct p to be shipped from origin a to destination d, for each oe 0, de D and DEP A lincer programming formulation for this situation can be written as follows: Miriniso Eco SPED CpXop + E.co Led ERED dep Yato Subject to Exp(1/top)Xop Sho: for each od Ede Yonly = X for each and PEP Ede Yody = Delp for each de D and PEP Xo20, for each o O and PEP Yap 20, for each o 0, de D and PEP You have seen that, for every linear program, there is a Qurespancing dual linear program that has the same optimal objective value. In the case of this prxduction- transportation model, the dul can be written as follows: Maximize + Subject to Tellop - Hop Cup, for each oE O and PEP Aap tap I dodao for cache O, DED and pep T. SO, for each OO Let X., and Yally represent the autimal values for the original primal" linear program, and let and represent optimal values for the dual LP. (You should use this notation when Answering the questions below.) a: You know that there is a one-to-one correspondence between constraints in the primal LP, and variables in the dual LP. Which dual variable corresponds to the construint EEP(1/ 5X54? Which dual variable responds to the construint ved Yoxy - Xg? Which dunl variable Sponds to the comistanit Eno Yodp-bdp? b: If boty 15 decreased by 1 ton so that fewer tons of p are required to be shipped to destination dthen do you expect the total est of the optimal shipping plan to go up or down? By how much would you expect the cost to change when bois decreased by 1 tont Give en Allwer in terins of the optimal dual values. c: If you could have an independent contractor deliver sunne tons of product p to origin , then you would effectively reduce the amount of product p that you would have to supply through your own production. What is the greatest delivery charge per ton that you would be willing to pay the contractor? Give an lywer in terms of the optimal dual values. d: If of top-top