Linear Programming

1. Linear Programming last topic of the semester • What is linear programming (LP)? • Not about computer programming • “Programming” means “planning” • “Linear” refers to the nature of the mathematical relationships involved, i.e., all relationships are defined by lines So linear programming is about planning, or making decisions, when the mathematical relationships are linear

2. Linear Programming (cont’d) • What sets LP apart from other topics we’ve discussed? • There is no probability in LP; it is “deterministic” • There is a definitive way to judge the best decision (based on your model) • It actually tells you the best decision (based on your model) • More than any other topic so far, LP is about you making good models of business situations

3. Linear Programming (cont’d) A little history… LP was first recognized and solved by George Dantzig using the “simplex method” in 1947 at the Pentagon based on his planning during WWII LP was not really useful at the time because computers were too slow; it took until 1955 before benefits were realized Since then, the use of LP has grown tremendously along with the growth of computers All sorts of industries (manufacturing, financial, service, health, etc.) now use LP to plan their operations…

4. First Example The furniture company you work for manufactures two types of beds: Your company currently has 1200 and 700 units of wood and metal available, respectively Producing how many of each bed yields the most profit, assuming sufficient demand?

5. right-hand sides objective function decision variables constraints nonnegativity constraints First Example (cont’d) The furniture company you work for manufactures two specific collections of bedroom furniture: Your company currently has 1200 and 700 units of wood and metal available, respectively Producing how many of each bed yields the most profit, assuming sufficient demand? maximize 20*A + 45*B subject to 12*A + 12*B  1200 6*A + 10*B  700 A  0, B  0 A = # produced of Bed A B = # produced of Bed B

6. LP Definitions Decision Variables • The quantities that are unknown but we wish to determine • All other quantities in the situation depend on the decision variables (e.g., profit, resources used) • Every linear program has decision variables, and they must be identified ASAP in the modeling process Objective Function • The quantity that we would like to maximize or minimize (e.g., profit or cost) • Written as a linear expression of the decision variables

7. LP Definitions (cont’d) Constraints • The limitations imposed on the decision variables • Written as a linear expression of decision variables that is , =, or  some number; that “some number” is called the right-hand side Nonnegativity Constraints • Special constraints that say the decision variables must be restricted to taking on nonnegative values (0 or positive, not negative) • A very natural constraint that occurs in most LP problems • Since nonnegativity constraints are very common, we will assume them unless otherwise stated

8. First Example in Excel (see Excel) • I strongly suggest to setup your LP in Excel like I have • Top left section has decision variables in columns (colored cells) • Bottom left section has objective function and constraints in columns that match the decision variables, plus right-hand sides for constraints • Bottom right section has actual results for decision variables (colored cells), lined up with objective function and constraints; use the SUMPRODUCT Excel command to do these calculations • All labels in the same places

9. First Example in Excel (cont’d) • Then go to Tools > Solver and fill in information • Target Cell: the colored cell that contains actual objective function • Max or Min: depending on how you want to optimize • By Changing Cells: array of colored decision variables • Subject to the Constraints: the colored cells , =, or  the corresponding rhs, as appropriate; constraints can be done in groups or one-by-one • Options > Assume Linear Model: check this box • Options > Assume Non-Negative: check this box • Then click Solve and save Answer Report

10. Bank Cashy Bank Cashy has only two types of assets, loans (X) and investments (Y). A total of $700 million is to be allocated between X and Y. Bank Cashy wishes its loans to equal at least $300 million and its investments to be at least 30% of the total of X + Y. The bank earns 12% on its loans and 14% on its investments. Formulate the LP to maximize earnings, and then setup and solve the model in Excel (see whiteboard and Excel)

11. Y  0.3 X + 0.3 Y -0.3 X + 0.7 Y  0 Bank Cashy (cont’d) This model brings up an important point For your Excel model, it is necessary to transfer each of your constraints to a “standard form”: “all the decision variables on one side” Y  0.3 ( X + Y )

12. Some Other Things to Watch Out For • When solver finishes, it gives you one of three messages: • “Solver found a solution.” (good!) • “Solver could not find a feasible solution.” (bad) • “The Set Cells values do not converge.” (bad) When either of the last two occur, most likely something is wrong with your model. Double-check it for errors!

13. c11 D1 d1 s1 S1 D2 d2 s2 S2 D3 d3 s3 S3 c34 D4 d4 Transportation LP Problems A transportation LP problem is a special type of LP that involves the transportation of goods from supply nodes to demand nodes Decision vars are the arrows – how much Si gives to Dj cij is the cost of shipping one unit from Si to Dj Objective is to minimize total cost of meeting all demands

14. Transportation LP Problems (cont’d) min c11x11 + c12x12 + c13x13 + c14x14 + c21x21 + c22x22 + c23x23 + c24x24 + c31x31 + c32x32 + c33x33 + c34x34 + c41x41 + c42x42 + c43x43 + c44x44 s.t. x11 + x12 + x13 + x14 s1 x21 + x22 + x23 + x24 s2 x31 + x32 + x33 + x34 s3 x11 + x21 + x31  d1 x12 + x22 + x32  d2 x13 + x23 + x33  d3 x14 + x24 + x34  d4 (nonnegativity) Supply constraints Demand constraints Note: Feasible only if total supply exceeds total demand!

15. Pluckett Company The J.F. Pluckett Company manufactures furnaces at three different plants and sells these furnaces at four different sales outlets. The number of furnaces constructed at each plant is 10, 15, and 11. Company policy requires that nine furnaces per week be sent to each sales outlet. The problem is to determine a shipping schedule for the furnaces (from plants to outlets) that minimizes total transportation costs. The transportation cost for shipping a single furnace from each of the three plants to each of the four outlets is shown here (in dollars). Find the shipping schedule that minimizes costs per week. (see whiteboard and Excel)

16. Pluckett Company (cont’d) This problem brings up another good point Even if your hand-written model doesn’t include every decision variable in each constraint, your Excel model has to. Those “missing” variables will just have a 0 in Excel

17. Refrigerators Two models of GE side-by-side refrigerators are produced at the GE plant in Bloomington: Model A and Model B. No more than 100 of these two models can be produced (i.e., the sum of A and B cannon exceed 100). Company policy is to produce at least as many of Model A as Model B. You are given the following information. Formulate and solve an LP to maximize profit. (see whiteboard and Excel)

18. Fertilizer The Grow-Tall Fertilizer Company is considering mixing its four current fertilizer products in various amounts to form a new fertilizer with some desired properties. The current products are: (The remaining percentage of each product is a neutral base.) The new fertilizer should be a minimum 11% composed of each of the three main ingredients, but should meet these requirements at minimum cost. Formulate and solve an LP to determine how the four current products should be combined to form the new product. (see whiteboard and Excel)