#3 Case Problem Julia's Food Booth A: Formulation of the LP Model X1(Pizza), X2(hotdogs), X3(barbecue sandwiches) Constraints: Cost: Maximum fund available for the purchase = $1500 Cost per pizza slice = $6 (get 8 slices) =6/8 = $0.75 Cost for a hotdog = $.45 Cost for a barbecue sandwich = $.90 Constraint: 0.75X1 + 0.45X2+ 0.90(X3) 1500 Oven space: Space available = 3 x 4 x 16 = 192 sq. feet = 192 x 12 x 12 =27648 sq. inches The oven will be refilled before half time- 27648 x 2 = 55296 Space required for pizza = 14 x 14 = 196 sq. inches Space required for pizza slice = 196/ 8 = 24.50 sq. inches Space required for a hotdog=16 Space required for a barbecue sandwich = 25 Constraint: 24.50 (X1) + 16 (X2) + 25 (X3) 55296 Constraint: Julia can sell at least as many slices of pizza(X1) as hot dogs(x2) and barbecue sandwiches (X3) combined Constraint: X1 X2 + X3 = X1 - X2 - X3 0 Julia can sell at least twice as many hot dogs as barbecue sandwiches X2/X3 2 = X2 2 X3 =X2 - 2 X3 0 X1, X2, X3 >= 0 (Non negativity constraint) Objective Function (Maximize Profit): Products: Pizza (X1) Hotdog s (X2) Barbecue Sandwich(X 3) Sell 1.5 1.5 2.25 Cost 0.75 0.45 0.9 Profit 0.75 1.05 1.35 Profit =Sell- Cost Profit function: Z = 0.75 X1 + 1.05 X2 + 1.35 X3 LPP Model: Maximize Z = 0.75 X1 + 1.05 X2 + 1.35 X3 Subject to 24.5 X1 + 16 X2 + 25 X3 55296 0.75 X1 + 0.45 X2 + 0.90 X3 1500 X1 - X2 - X3 0 X2 - 2 X3 0 X1 0, X2 0 and X3 0 Solve the LPM -answer in excel solution Based on the excel solution the optimum solution: Pizza (X1) = 1250; Hotdogs(X2) = 1250 and Barbecue sandwiches (X3) = 0 Maximum value of Z = $2250 Julia should stock 1250 slices of pizza, 1250 hot dogs and no barbecue sandwiches. Maximum Profit = $2250. Maximum Profit $ 2,250.00 Booth Rent per game $ (1,000.00) Warming Oven 600 for total of 6 home games 600/6 =100 $ Profit for the 1st Game $ 1,150.00 (100.00) Profit for the 1st Game $ 1,150.00 Wants to clear $1,000 Difference $ 150.00 Julia would clear $1000. She would make a small profit of $150, she should lease the booth. B: Should Julia borrow more money? The sensitivity report shows the shadow price for the budget constraint is 1.5 and the allowable increase is 138.4. So for each dollar added to the budget it would increase a profit of $1.5. The maximum allowable increase is 138.4 which is the maximum amount that Julia can borrow to make a profit. The additional profit would be 138.4 x 1.5 = $ 207.60. Based on the shadow price and the allowable increase for the space factor constraints, Julia should not borrow more money, there would be no profit. C: Should Julia hire a friend to help at $100 per game? Maximum Profit $ 2,250.00 Booth Rent per game $ (1,000.00) Warming Oven 600 for total of 6 home games 600/6 =100 $ (100.00) Profit for the 1st Game $ 1,150.00 Profit for the 1st Game $ 1,150.00 Wants to clear $1,000 Difference $ 150.00 Hires a Friend $ 100.00 Difference $ 50.00 If Julia hires a friend she would still make a profit of $50 and she is still clearing $1000, so it is worth leasing the booth. If it is too much work for her, I think she should hire her friend. D: What kind of things could go wrong? The 1st uncertainty is her assumption that she will sell everything and create a customer base for the season. If this does not happen she would be out the $1500, cost of the warmer and the booth rent (each game). In A, she can earn a profit of $150, so she does not want her lose to be below $150. 2nd uncertainty she is relying on a friend to help her because it's too much work. If her friend is unable to help her she would have to find someone else to help at the same price or cheaper. This may not be possible. 3rd uncertainty, what about the cost of things like, napkins, ketchup etc.... How much would these items cost her? If you don't have what a customer wants to put on their food, they could go elsewhere, she could lose some profits. 4th uncertainty, what if something happens to the warmer doing the game? She may or may not be able to recoup her money from the people she is leasing the warmer from. I think Julia should do a booth rent for the 1st game. Just to see how things sell. She may decide to change her menu. She could look at how the other vendors are doing in what they are selling. LP_max Linear, Integer and Mixed Integer Programming Signs < = > less than or equal to equals (You need to enter an apostrophe first.) greater than or equal to Data Objective Constraint 1 Constraint 2 Constraint 3 Constraint 4 Results Variables Objective Pizza SliceHot Dogs BBQ 0.75 1.05 1.35 sign 0.75 0.45 0.9 < 24.5 16 25 < 0 1 -2 > 1 -1 -1 > 1250 1250 RHS 1500 55296 0 0 0 2250 Page 1 Results LHS Slack/Surplus 2250 1500 0 50625 4671 1250 -1250 0 0 1 Julia's Food Booth Quantitative Methods - MAT540 Case Analysis Paper Julia's Food Booth Chapter 3, page 109 Student Name Date Professor ____________________ Strayer University Parameters/Background The case study involving Julia's food booth .... (provide background and parameters very similar to an Executive Summary in a Business Report). Julia is considering leasing a food booth outside Tech Stadium at home (6) football games. If she clears $1000 in profit for each game she believes it will be worth leasing the booth. $1000 per game to lease the booth $600 to lease a warming oven She has $1500 to purchase food for first game and will for remaining 5 games she will purchase her ingredients with money made from previous game. Each pizza costs $6 for 8 slices which is ? per slice, and she will sell it for $1.50 2 Each hot dog costs 0.45, and she will sell it for $150 Each BBQ Sandwich costs 0.90, and she will sell it for $2.25 There are Food Cost, Oven and Ratio Constraints that include: QM assessment (Describe the Excel Solver and/or QM for Windows tool input) Pizza Slices x1 Hot Dogs x2 BBQ x3 Maximize Food Costs Oven Space Hot Dog to BBQ ratio demand Pizza to Hot Dog and BBQ ratio demand RHS <= <= >= >= 55296 0 0 Equation form (fill in coefficients, amounts, etc.) Maximize Z = 0.75Pizza Slices x1 + _Hot Dogs x2 + _BBQ x3 Food Cost Constraint: _Pizza Slices x1 + _Hot Dogs x2 + _BBQ x3 <= ___ Oven Space Constraint: _Pizza Slices x1 + _Hot Dogs x2 + _BBQ x3 <= 55296 Hot Dog to BBZ ratio Constraint: _Hot Dogs x2 + _BBQ x3 >= 0 Pizza to Hot Dog and BBQ Constraint: _Pizza Slices x1 - _Hot Dogs x2 - _BBQ x3 >= 0 Linear Programming Results (from Excel Solver and/or QM for Windows): Optimal Value (Z) = 3 Julia's Food Booth Ranging Case Study Questions A. Formulate and solve a linear programming model for Julia that will help you advise her if she should lease the booth. Answer: Conclusion: If Julia were to open a food booth at her college's home football games, her optimal value would be _______with Pizza x1 value _____ Hot dogs x2 value of ____ and BBQ x3 value of ______ B. If Julia were to borrow some more money from a friend before the first game to purchase more ingredients, could she increase her profit? If so, how much should she borrow and how much additional profit would she make? What factor constrains her from borrowing even more money than this amount (indicated in your answer to the previous question)? Answer: After solving the linear program in QM and utilizing the ranging function (see ranging function in QM assessment) the upper bound for food costs is ________. Since Julia already is starting with $1500 for food cost, she could increase her profit and the most she should borrow from her friend is $_________ If she borrowed money from her friend the additional amount of profit she could generate is _________. 4 This is determined because when looking in the ranging section of the solution, the dual value is ______. This means it is worth _______ to Julia for each additional dollar that she receives. So with this is mind, we can conclude that ....... The factor that constrains her from borrowing even more money is .... Conclusion: C. When Julia looked at the solution in (A), she realized that it would be physically difficult for her to prepare all the hot dogs and barbecue sandwiches indicated in this solution. She believes she can hire a friend of hers to help her for $100 per game. Based on the results in (A) and (B), is this something you think she could reasonably do and should do? Answer: In solution A, In solution B, Conclusion: D. Julia seems to be basing her analysis on the assumption that everything will go as she plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Julia's analysis? Given these uncertainties and the results in (A), (B), and (C), what do you recommend that Julia do? 5 Julia's Food Booth Answer/Conclusion: Conclusions/Final thoughts