David Palma EGR 546 By David Palma Problem number 3 on page 71 Source of the problem 3 page 71 Wayne L Winston Operations Research Applications and Algorithms fourth edition Thomson Brooks Cole Problem statement Peg and Al Fundy have a limited food budget so Peg is trying to feed the family as cheaply as possible However she still wants to make sure her family members meet their daily nutritional requirements Peg can buy two foods Food 1 sells for 7 per pound and each pound contains 3 units of vitamin A and 1 unit of vitamin C Food 2 sells for 1 per pound and each pound contains 1 unit of each vitamin Each day the family needs at least 12 units of vitamin A and 6 units of vitamin C 1 Verify that Peg should purchase 12 units of food 2 each day and thus over satisfy the vitamin C requirements by 6 units 2 Al has put his foot down and demanded that Peg fulfill the family s daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C The optimal solution to the new problem will involve ingesting less vitamin C but it will be more expensive Why X1 X2 Food 1 Food 2 Table 1 Summary of input data Price per pound Units vitamin A 7 3 1 1 At least 12 Units vitamin C 1 1 at least 6 Objective Function MinZ 7 x1 1x 2 The decision variables must satisfy the following 2 constrains Constrain 1 Minimum of vitamin A 3 x1 1x 2 12 Constrain 2 Minimum of vitamin C 1x1 1x2 6 Sign constraint x1 x 2 0 1 David Palma EGR 546 Create the first simplex tableau do not solve manually by Ms Mia Cornelius Z X1 X2 e1 e2 a1 a2 rhs 1 7 1 0 0 0 0 0 0 3 1 1 0 1 0 12 0 1 1 0 1 0 1 6 BV Z a1 a2 Table 2 Input Data WinQSB Original Problem Primal Table 3 Solution WinQSB Primal This solution shows that to minimize the cost considering our constraints the family should buy 12 units of food 1 Then the minimum cost will be 12 The following tables and graphs show the sensitivity analysis over the objective function variable coefficients and the right hand side values of the constraints Sensitivity Analysis Table for OF coefficient for X2 Cost of Food 2 per pound 2 David Palma EGR 546 Sensitivity Analysis Graph for OF coefficient for X2 Cost of Food 2 per pound Sensitivity Analysis Table for RHS of C1 Required Vitamin A per day 3 David Palma EGR 546 Sensitivity Analysis Graph for RHS of C1 Required Vitamin A per day 4 David Palma EGR 546 The following is the report to a hypothetical manager Dear hypothetical Manager Per your request the following is the result of my research on this problem In order to find out the best purchasing plan which minimizes the overall cost of weekly food purchases I used a simple linear programming approach and with help of a user friendly software WinQSB I came up with the followings We need to purchase 12 pounds of food 2 in each day It is basically not financially feasible to purchase any of food 1 By this plan we can easily minimize our daily cost to 12 00 I also wanted to mention that this plan can maintain the required amounts of Vitamin A and C for the family This will provide 12 units of Vitamin A and 6 units of Vitamin C In order to find out the sensitivity of the above mentioned plan on uncertainties over the costs of different foods and also the required minimum amount of each of the vitamins I used the same software and here is the analysis In order to be able to have food 1 in the plan they need to give us at least 4 00 discount But if their price is 4 00 or more we would have the same optimal plan If the price they offer would be 3 00 we will have the same daily cost of 12 00 but from that point for each 1 00 dollar discount not only the plan will change but also the daily cost will be reduced by 3 00 In order to have the same plan as the optimum one the food 2 supplier should not increase their price more than 2 33 otherwise the plan will change I should also add that even though the plan is the same but the daily cost will be increased 12 00 for each 1 00 increase in the price 5 David Palma EGR 546 If the requirement of Vitamin A would be more than 6 we will have the same plan but the daily cost will decrease 1 00 for each 1 00 unit reduction in Vitamin A requirement Of course if the requirement would be less than 6 the plan will be completely changed along with the daily cost The only way change in required Vitamin C per day will change the optimum solution is having it more than 12 00 units per day In that situation the plan will be changed also 6