See chapter 8 in the textbook. Dr Muhammad Al Salamah, Industrial Engineering, KFUPM

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1 Goal programming Objective of the topic: Indentify indutrial baed ituation where two or more objective function are required. Write a multi objective function model dla a goal LP Ue weighting um and preemptive method to olve goal LP See chapter 8 in the tetbook. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

2 There are ituation in indutrial planning that require finding the optimal parameter for two or more objective function: Reduce labor cot and improve efficiency Increae production and reduce energy uage Goal programming provide a mechanim to olving thee ituation whoe objective function might be conflicting. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

3 Production mi A company i producing indutrial detergent with four grade. The revenue (price cot) of the four grade are Grad Revenue (SR per m 3 ) The management of the company ha decided to collect SR 60,000 in revenue for the net month. But, they aked that the revenue from grade hould not eceed 0% of the total revenue. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

4 Alo, revenue from grade 3 hould not eceed 0% of the total revenue. Alo, the production of grade hould not eceed 0 m 3. The indutrial engineer ha been aked to find the production mi that will cloely atify all the requirement of management. Define m of grade produced m of grade produced 3 m of grade 3 produced m of grade produced Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

5 The goal of the management can be epreed mathematically a , ( , (5500, 3, ) ) Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

6 The goal can be written a , , 70 3, , The indutrial engineer find difficult to meet all goal of the management with a ingle production mi. Hence, he i looking for a compromie olution that may not necearily atify all the contraint. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

7 The fleible goal can be written a , 3, 3, 3, 60, The variable - and are called deviational variable, and by their definition, only one of them, not both, can be baic (> 0). The deviational variable - and reemble the lack and urplu variable in their interpretation. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

8 For eample, for the firt goal, if i > 0, then the contraint left ide i greater than the right ide, and merely aborb the ece to make the left ide equal the right ide. When - i > 0, then the left ide i le than the right ide, and - add to the left ide to make the two ide equal. Therefore, any olution for which 0, the olution i feaible for the firt contraint. The violation of the contraint i meaured by, which hould be minimized. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

9 A compromie olution to the production mi will be to minimize the deviation: minimize G - minimize G - minimize G 3-3 minimize G Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

10 Goal programming algorithm Two algorithm are dicued:. Weighted um method. Preemptive method Both method repreent the goal programming problem with a ingle objective optimization problem. The two method do not generally produce the ame olution becaue of the ubjectivity involved. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

11 Weighted um method The weighted um method aign weight to the deviation and minimize the um of the weighted deviation. The weight aigned to a pecific deviation ignifie the importance of that deviation and how important to reduce it in comparion with the other deviation. Aume there are n goal: minimize G i, i,, n Then, the combined objective function i defined a minimize z w G w G w n G n w i > 0 i the weight given to goal i. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

12 Eample In a ingle-tage aembly line, there are 0 worker available. The plant produce two model of a product. The revenue, torage pace requirement, and worker utilization per model are a: Model Model Revenue (00) SR SR 8 Storage pace requirement 5 m m Worker utilization worker worker The available torage pace i 00 m. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

13 Becaue of the nature of the demand, production of model hould not eceed 6 part. It ha been decided that the total revenue i not to be le than SR,500. The indutrial engineer ha to plan the aembly that will increae the company revenue and reduce inventory, ince it i a common torage. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

14 Define : unit of model aembled : unit of model aembled The production mi problem can be written in math a Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

15 Add deviational variable to both goal: ,, 6 0 Through etenive meeting market tudie and 0,, Through etenive meeting, market tudie, and inventory ize, it ha been decided that the total revenue i twice a important a overtocking. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

16 Hence, the objective function i min z t. min z 6 0 0,, Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

17 The problem i then olved uing the imple method: i t min z Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

18 The optimal olution i z ret of the variable 0 Thi i to ay, with thi production mi, the revenue will be in hort by SR 500 and torage utilization will be le than the maimum preferred by m. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

19 Notice that there are other olution for which z 0: Hence, we call the olution efficient i olution rather than optimal olution. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

20 Preemptive method The modeler might have a preference for the mot important goal and the leat important goal. With the preemptive method, the goal are ranked in the order of importance, o that any olution will not atify low priority goal on the epene of high priority goal. Given n goal, the objective of the problem are Minimize G (highet priority) Minimize G n (lowet priority) The method conider one goal at a time, tarting with the highet priority, and the proce i carried out uch that the olution obtained from a lower priority goal never degrade any higher-priority olution. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

21 There are two way to olve a goal problem uing preemptive method: Succeive addition of contraint Goal imple Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

22 Succeive addition of contraint The ucceive addition of contraint work bet if implemented in an optimization oftware. The method work by olving for highet priority goal firt, then the value of the goal i put a a retriction (contraint) for the olution of the problem for the econd-highet priority goal, and o on. Take the general goal model with the objective function being the highet priority goal: Min G.t. contraint t Let the olution be G *. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

23 Then, the econd problem i to olve Min G.t. G G * contraint Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

24 Eample Conider the production mi problem: i G 5 8 t min min G G t. 0 6 Aume that meeting the revenue goal i more important than the inventory goal. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

25 The firt LP that ha to be olved i min G.t A olution to thi LP i G * 5. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

26 The econd LP that ha to be olved i i G t. min G A olution i 5,.5. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

27 Goal imple The goal imple optimize, rather than atifie, the goal. It add an objective row to every goal in the goal model, and the goal are optimized equentially one by one. When the proce of optimizing one goal reache the end, before the net optimization i tarted, any nonbaic variable having a reduced cot in the row of the goal jut been optimized different from zero mut be removed. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

28 Eample Conider the production mi problem to maimize revenue and reduce inventory: ma min.t Aume that maimizing revenue i more important than minimizing inventory. Hence, the firt objective will be optimized firt. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

29 Adding lack variable, the imple tableau with two objective i: Baic Solution P P To improve P, ha to be baic, o it will be conidered a the entering variable. By applying the minimum ratio, ha to leave the bai. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

30 Apply the row operation to eliminate from all row ecept it row: Baic Solution P P ½ Thi tableau i optimal, all reduced cot in P are 0. Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

31 The nonbaic variable ha reduced cot of in P. Hence, hould be eliminated before P can be optimized. The tableau become: Baic Solution P P ½ Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

32 The row of P can be eliminated a it ha no effect on the olution. To improve P, hould be baic (> 0). By applying the minimum ratio, hould become nonbaic: Baic Solution P 0 P ½ 0 6 Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM

33 The tableau i optimal and the olution ha been reached: 6 Dr Muhammad Al Salamah, Indutrial Engineering, KFUPM