Linear Programming problem Diet Mix:  Computer Exercise 3

(problem B-42 from the heizer Render web site)
http://cwx.prenhall.com/bookbind/pubbooks/heizer2/chapter24/deluxe.html
goto additional problems, Module B, B.42

Set this up in EXCEL and solve using Solver
set up the same problem in POM-QM and compare the results. This comparison should help you interpret the sensitivity and range analyses from the solver output.  Everyone will have slightly different numbers as you will use the last 4 digits of your student number to replace the cents in the prices for ground meat and chicken.

Rachel Yang, campus dietitian for a small Illinois college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal-content requirements should be met: (1) between 900 and 1,500 calories (you will need two rows for calories, one to set the upper limit, one to set the lower limit); (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates.
On a particular day, Rachelís food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and its contribution to each of the five nutritional requirements are given in the accompanying table:

Procedure:

Data can be put into EXCEL in one of three ways:
  1. Use the text tool in Acrobat to copy it from the Heizer-Render website. Paste into EXCEL. use data/text-to columns (space delimiter) to parse it, make necessary adjustments. See the Video
  2. Copy the data from the bottom of this page and paste into EXCEL. use data/text-to-columns (space delimiter) to parse it, make necessary adjustments.
  3. Type the data in, being careful to avoid transcription errors.
Substitute the last 4 digits of your STUDENT number to replace the cents (shown as A,B,C,D below) in the prices for ground meat and chicken. There is no credit given for submissions that are not individualized in this way. The screen captures below use 4567 as an example student number entry.

Use copy, paste special /transpose (video on paste special transpose) to put the data in a more familiar configuration (columns for decision variables, rows for objective function and constraints). Duplicate the calories row to allow two (upper and lower) constraints on calories. Move the cost coefficients row (objective function) to the top data row. Add a row for Values (which Solver will manipulate), and rows for Upper and Lower Limits on the ranges of optimality. Add columns for direction of constraint, RHS, amount provided, and Shadow Price for each ingredient.

The formula for each 'amount provided' will consist of the sum of each of the (variable values * amount provided per pound)  The easiest way to do this is through the Sumproduct Function. In cell L6, enter   =sumproduct($C13:$I13,C6:I6) you can do this by entering "=sumproduct(" then select the ranges, use F4 to absolute references to the value row and put in the close parenthesis before pushing enter. You will only have to enter the calculation once, then you can just copy it down to subsequent rows.
In the event solver tells you there is "no feasible solution found" check your formulas and make sure you have the right directionality on the constraints. (The selections for sensitivity reports will be grayed out if there is no feasible solution.) Note that the amount provided column for the cost row will give the cost.
Use Solver to solve this problem and give sensitivity analysis. See the videos on the X-Y problem to see how to add in and use Solver and how to interpret the sensitivity analysis.

Tips for using POMQM

Use POM-QM's LP module to solve this same problem and print out the results to help you interpret the EXCEL output.
It is possible to copy and paste data one part at a time using the POM-QM menu item, (edit/paste-from-clipboard doesn't work for the constraint directions from EXCEL to POM-QM).  It's a little tricky. You could also simply type in the data using your EXCEL sheet layout as a guide. POM-QM should give the same solutions as SOLVER. Be aware the sensitivity numbers are expressed differently, but the solutions (values and objective value) should be the same.
If EXCEL and POM-QM don't agree on the basic values, see the list of things that can go wrong, below.

Enter values for the Upper and Lower Limits on the Range of Optimality and the Shadow Prices on this sheet and put it first to facilitate grading. It's easiest to understand these on the POM-QM sensitivity analysis (Ranging) but you should compare these to the SOLVER sensitivity analysis just to understand how they are expressed differently.

Printouts due: (one legible, complete page gets a bonus 5 points. Example of what needs to be in it. Excess paper loses 10 points)

1) the EXCEL problem formulation sheet, Adding in  results to summarize shadow prices and ranges of optimality. Printouts of the range and sensitivity analyses  to support the summary numbers put in the initial sheet.
2) the same for the POM-QM solution and ranges. No need to add summaries as you will show these on the EXCEL sheet. The results should be the same as the EXCEL Solver solution. Please don't print out iterations.
3) answer these questions:
What combination and amounts of food items will provide the nutrition Rachel requires at the least total food cost?
a) How much could the price of milk increase without changing the solution?
b) How low would the price of ground meat have to be before you would use more?
c) How much would the cost increase if you needed one more gram of iron?
d) how high could the price of beans go before you would change the diet?

Table of Food Values* and Costs
Food Calories/ Iron Fat Protein Carbohydrates Cost/
Item Pound (Mg/Lb) (Gm/Lb) (Gm/Lb) (Gm/Lb) Pound ($)
Milk 295 0.2 16 16 22 0.60
Ground meat 1216 0.2 96 81 0 2.AB
Chicken 394 4.3 9 74 0 1.CD
Fish 358 3.2 0.5 83 0 2.25
Beans 128 3.2 0.8 7 28 0.58
Spinach 118 14.1 1.4 14 19 1.17
Potatoes 279 2.2 0.5 8 63 0.33
Source: C. F. Church and H. N. Church, Bowes and Churchís, Food Values of Portions Commonly Used, 12th ed. Philadelphia, J. B.
Lippincott, 1975.

Things that can go wrong:

If POM-QM and EXCEL solutions are different:
try putting in the $2.45  for ground meat and $1.67 for chicken and compare to the screen captures above, this will tell you whether things are set up right and will identify which one is off.
Proofread the directionalities of the constraints and make sure you MINIMIZE cost!.