LP-assignment.a: A real-life problem from the chemical industry

Practical assignment for Discrete Optimalization and Linear Programming (2V300)

Below is given a description of a practical problem arising in the chemical process-industry. Model this problem in terms of a Linear programming problem. The model should be EXPLICIT meaning that you should give appropriate names to variables and constraints. Then both model and solver reports will make sense and will be interpreted easily. Solve the LP-models with a solver of your choice: PC-Prog, SIMPLEX, or better LINDO or CPLEX. See Dealing with LP-problems for further explanations.

Description
In a chemical plant six kettles are used for the production of several types of oil and oil-products. In this market the prices of the produced goods are more or less fixed. For each unit of product the variable cost, the selling price, and upper and lower limits on possible sales are given in table 1.

The production process has the following features. For the production of one unit of OLIE one uses a quarter unit of the product TRAAN. And for the production of one unit of SM-08 one uses one unit of product SM-07. There are two ways to produce products EOR-44 and AMD-66. As they require different kettle capacities the distinct product types appear in the tables below separately. Each kettle is available for 1000 hours. There are 10 operators that each are available for 1000 hours for supervising the process. The number of hours one unit of product claims from a particular kettle is given in table 2.

The company hires you as a consultant to answer the following Het bedrijf stelt U als consulent de volgende MANAGERIAL QUESTIONS:

  1. What is the product mix with highest profit?
  2. One feels that the current number of operators employed (10) is too high. Can you confirm or deny?
  3. The sales manager claims that the sales prices given are accurate upto 5 percent. Is this accurate enough for your decision on the optimal mix. If not, for which products do you need more precise information?
  4. The company thinks about reconstructing for part of the time kettle two (K2) into a kettle with teh same characteristics as kettle one (K1). For how many hours would you recommend to use kettle as a copy of kettle one? What is the optimal production mix after reconstruction of kettle two, what is the new profit and what are the gains or loss? Is it better to choose another kettle for modification?
  5. From marketing efforts it shows that sales of TRAAN can be increased. How much are you prepared to pay for selling an extra unit of TRAAN?
  6. What happens with the amount of produced SM-07 if the sales price varies between 50 and 80?

    Next some more TECHNICAL QUESTIONS:

  7. For modelling the problem above and making the computations to answer these questions you have set up an LP-description of the problem in so-called LP-format. This format is very well fit for an approach oriented to the constraints describing the problem. An older format is the so-called MPS-format. Set up the chemaical-plant problem in MPS-format.
  8. Name advantages and disadvantages of the MPS-format over the LP-format. Are there particular problems for which the MPS-format makes more sense?

TABEL 1: product data: 

product name |  variable cost     |  sales price   |  minimum sales  |  maximum sales
-------------------------------------------------------------------------------------
 CB-75       |   20               |    40          |   10            |    50
 MO-80       |   96               |   100          |    0            |   100
 VET-b       |   75               |    80          |   10            |   100
 SM-07       |   60               |    70          |    0            |    50
 BAND        |   27               |    30          |    0            |   250
 OLIE        |   20               |    25          |    5            |   150
 PL-99       |   10               |    16          |    0            |   300
 TRAAN       |   73               |    81          |   10            |   400
 EOR-44      |    6               |    18          |    0            |    20
 AMD-66      |    9               |    23          |    0            |   500

TABLE 2: production-characteristics (times in hours per unit): 

product name |  K1  |  K2  |  K3  |  K4  |  K5  |  K6  |  Operators
--------------------------------------------------------------------
 CB-75       |  50  |   0  |   0  |   0  |   0  |   0  |   100
 MO-80       |   1  |   0  |  25  |   0  |   0  |   0  |    50
 VET-b       |   0  |   0  |   0  |   0  |  10  |   0  |    50
 SM-07       |   0  |   0  |   0  |   0  |   9  |   0  |    25
 BAND        |   0  |  15  |   0  |   0  |   0  |   0  |    10
 OLIE        |   0  |  10  |   0  |   0  |   0  |   0  |     5
 PL-99       |   1  |   0  |   0  |   1  |   0  |   0  |    10
 TRAAN       |   1  |   0  |   0  |   1  |   0  |   0  |     5
 EOR-44A     |   0  |   0  |   0  |   0  |   0  |  50  |    12
 EOR-44B     |  20  |   0  |   0  |  30  |   0  |   0  |    12
 AMD-66A     |   0  |   0  |   0  |   0  |   0  | 100  |    40
 AMD-66B     |   0  |   0  |   0  |  30  |  40  |   0  |    60