Download Goal Programming-Operation Research-Lecture Handout and more Exercises Operational Research in PDF only on Docsity!
[Mohammad Ali Jinnah University, Islamabad Campus] Page 1
Topic 9: Goal Programming
Goal programming (GP): an introduction It has been already discussed in our previous section that, in a general linear programming (LP) model, the following three basic conditions are fulfilled: o The decision variables Xi are allowed to have fractional values/continuous (like 4.33, 120.69). o There is a unique objective function (not more than one). o All mathematical expressions (objective function, constraints) have to be linear (not non-linear). And whereas these conditions are relaxed, we rely on the following models. Integer programming (non-fractional/non-continuous/discrete) Goal programming (with multiple goals) Nonlinear programming Goal programming (GP) is an extension of the LP model wherein, unlike the LP model which has only one/unique objective, the objective function consists of more than one objectives. In LP, and even in an IP model, we try to optimize a single measure, while in GP, we set multiple objectives. In most decision modeling situations, some of these objectives are conflicting in nature, and are achieved at the expense of each other. We therefore establish a hierarchy or rank of importance among these goals so that lower- ranked goals are given less prominence than higher-ranked goals. Based on this hierarchy, GP then attempts to reach a ‘satisfactory’ level for each goal. It is therefore usually said that LP tries to ‘optimize’ and GP tries to ‘satisfice’ the multiple objectives; and this means, coming as close as possible to their respective goals, rather than to optimize them. How does GP satisfice the goals? Instead of minimizing or maximizing the objective functions directly, GP tries to minimize deviations between the specified goals and what we can actually achieve for the multiple objective functions within the given constraints. Deviations can be either positive or negative, depending on whether we overachieve or underachieve a specific goal. These deviations are not only real decision variables in the
[Mohammad Ali Jinnah University, Islamabad Campus] Page 2
GP model, but they are also the only terms in the objective function; the objective is to minimize some function of these deviation variables.
Goal programming (GP): an example Let’s take the example of a door manufacturing company to illustrate the formulation of a GP problem. The company manufactures three styles of doors – exterior, interior, and commercial. Each door requires certain amount of steel for its formation and certain amount of labour hours for its forming and assembly, as reported in the following table; table also provides other required data on total availability of the resources (steel and labour hours) and selling price per unit of door. Production Process
Types of Doors Availability Exterior (E = X 1 )
Interior (I = X 2 )
Commercial (C = X 3 ) Steel (pounds/door)
4 3 7 9000 pounds Forming (hours/door)
2 4 3 6000 hours Assembly (hours/door)
2 3 4 5200 hours Selling price/door($)
Formulating as LP model: Maximize Z = 70X 1 + 110X 2 + 110X 3 (9.1a) Subject to 4X 1 + 3X 2 + 7X 3 ≤ 9000 (9.1b) 2X 1 + 4X 2 + 3X 3 ≤ 6000 (9.1c) 2X 1 + 3X 2 + 4X 3 ≤ 5200 (9.1d) Xi ≥ 0 (9.1c) Solution set: Z*^ = 186000 $ X 1 (E) = 1400 X 2 (I) = 800 X 3 (C) = 0 Since LP solution suggests: Total sale = 186000, including: Sale of X 1 = 1400 x 70 = 98000 $ Sale of X 2 = 800 x 110 = 88000 $
[Mohammad Ali Jinnah University, Islamabad Campus] Page 4
variables set for underachievement, namely: DTM, DEM, DIM & DCM; this sets our objective function, like: Minimize total underachievement = DTM + DEM + DIM + DCM (9.3) Note that this objective function (9.3) specifies that the targets should be at least met; this does not prevent if one or all targets are overachieved. If we are interested in exactly achieving all goals, we will need to minimize all under-achievement and over- achievement variables. Weighted GP model: If we are interested to specify that total sale goal is five times as important as each of other goals, we will assign numeric weights to each deviation in the objective function, like: Minimize total weighted underachievement = 5DTM + DEM + DIM + DCM (9.4) Formulating the Weighted GP model: Objective function: Minimize total underachievement = 5DTM + DEM + DIM + DCM (9.5a) Constraints: Subject to 70E + 110I + 110C +DTM – DTP = 180000 (9.5b) 70E + DIM – DIP = 70000 (9.5c) 110I + DIM – DIP = 60000 (9.5d) 110C + DCM – DCP = 35000 (9.5e) 4E + 3I + 7C ≤ 9000 (9.5f) 2E + 4I + 3C ≤ 6000 (9.5g) 2E + 3I + 4C ≤ 5200 (9.5h) E, I, C, DTM, DTP, DEM, DEP, DIM, DIP, DCM, DCP ≥ 0 (9.5i) Solving the Weighted GP model: The Weighted GP model formulated in 9.5 can be solved as an LP model; so putting the data in to general LP format o TORA: Solution set: X 1 (E) = 1000 (Valuing 1000x70 = 70000, as per target) X 2 (I) = 800 (Valuing 800x110 = 88000; overachieved = 28000)
[Mohammad Ali Jinnah University, Islamabad Campus] Page 5
X 3 (C) = 200 (valuing 200x110 = 22000; underachieved = 13000) Total revenue = {(1000x70) + (800x110) + (200x110)} = 180000 $ DTM = 0 DTP = 0 DEM = 0 DEP = 0 DIM = 0 DIP = 28000 (deviation in Interior doors overachieved) DCM = 13000 (deviation in Commercial doors underachieved) DCP = 0
Take-home Assignment
(Wait for my Email message)
