1. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India

2. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India GOAL PROGRAMMING 7 CHAPTER

3. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Learning Objectives Solve problems with single goal Solve problems with equally ranked multiple goals Solve problems with priority ranked multiple goals.

4. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Goal Programming LP deals with optimisation of a single quantifiable objective function. In case of a number of objectives, which my or may not be prioritised and which may or may not be conflicting, goal programming is used.

5. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Goal Programming Goals can be originally ranked without the requirement of a common yardstick. Prioritisation must be done carefully, as the achievement of a lower priority goal is addressed only after the higher priority goals have been achieved to the fullest extent possible. Goals may not be fully achieved. Provides a satisficing solution rather than an optimal solution.

6. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Example – Goal Programming TablesChairsTotal Available (per week) Carpentry (Hours) 84120 Finishing (Hours) 4896 ProfitsRs 80Rs 60

7. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 Single Goal Model Goal is to to earn Rs 1400 profit per week. Let T be the number of tables produced and C be the number of chairs Let d + be the over achievement of the profit beyond Rs 1400 and d - be the under of achievement of the profit goal. The objective now is to minimise the under achievement of the goal.

8. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 Single Goal Model The problem can now be formulated as: Solve this problem by simplex. d + and d - are called deviation variables

9. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 Initial Simplex Tableau is: CjCj 001000 CbCb TCd-d- d+d+ S1S1 S2S2 Quantity d-d- 180601001400 S1S1 0840010120 S2S2 048000196 ZjZj 80601000 C j - Z j -80-600100

10. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 Final Simplex Tableau is: CjCj 001000 CbCb TCd-d- d+d+ S1S1 S2S2 Quantity d-d- 1001-8.33-3.3380 T010000.167-0.08312 C00100-0.0830.1676 ZjZj 001-8.33-3.3380 C j - Z j 00018.333.33 The optimal solution is to make 12 tables and 6 chairs yielding a profit of Rs 1320. There is an under achievement of the goal by Rs 80, as indicated by the value of d - in the table.

11. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 Equally Ranked Multiple Goals The firm has two equally ranked goals –Achieve a profit of Rs 1000 –Meet the demand of producing 10 tables. Let T be the number of tables produced and C be the number of chairs Let and be the over and under achievement of the profit goal Let and be the over and under achievement of the target for producing tables

12. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Equally Ranked Multiple Goals Since both goals are equally important, the problem can be formulated as

13. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 Solving by Simplex the final tableau is: CjCj 00101000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ S1S1 S2S2 Quantity dt+dt+ 000.750.0125-0.01251002.5 T010.750.0125-0.0125000012.5 S1S1 00-2-0.10.1001020 S2S2 005-0.050.05000146 ZjZj 000000000 C j - Z j 00101000 The target of 10 tables is over achieved by 2.5 tables. The profit is exactly Rs 1000 as both and are not in the solution.

14. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 Priority Ranked Multiple Goals The firm has priority ranked goals –Priority 1. Produce 13 tables in the week. –Priority 2. Achieve a profit of Rs 1350. –Priority 3. Produce 5 chairs in the week. Using the same notations as before, let and be the over and under achievement of the goal of producing 5 chairs The objective function includes the priorities allotted as and, and. These are called pre-emptive priority factors

15. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Priority Ranked Multiple Goals The model can be formulated as

16. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 The initial Simplex Tableau is: CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 806010000001350 dt-dt- P1P1 10001000013 dc-dc- P3P3 0100001005 S1S1 08400000010120 S2S2 0480000000196 P3P3 ZjZj 0100001000 C j - Z j 000000100 P2P2 ZjZj 80601000000 C j - Z j -80-6001000000 P1P1 ZjZj 100010000 C j - Z j 000010000

17. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 There is a separate row for each priority. Separate rows are required as we do not add deviations from one goal to the other. For example we do not add the deviations from the chairs goal to the tables goal as these do not have the same units. While calculating rows for a particular priority the other priorities are ignored. For example while calculating rows for Priority, only rows are considered. Generally the P rows are shown from bottom to top in order of priority. value for any column is shown in the priority rows at the bottom of the tableau. For example the ………row for the column T is contained in the and … rows and can be read as

18. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 In selecting the variable to enter the solution, we start with the most important priority and choose the most negative value; in this case T. The out going variable is calculated by computing replacement ratio and selecting the row with the least positive value. Once all the negative values in the row in priority have been exhausted, we consider the next priority. Negative values in the row are ignored if there is a positive value in the row in the same column in a higher priority (a lower row in the table). This is because a positive value in a row of higher priority would imply that its introduction will only increase the deviations in the higher priority goal.

19. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 Identifying incoming and out going variable CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 806010000001350 dt-dt- P1P1 10001000013 dc-dc- P3P3 0100001005 S1S1 08400000010120 S2S2 0480000000196 P3P3 ZjZj 0100001000 C j - Z j 000000100 P2P2 ZjZj 80601000000 C j - Z j -80-6001000000 P1P1 ZjZj 100010000 C j - Z j 000010000 In coming Variable Outgoing Variable

20. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 20 CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 0601-80800000310 T010001000013 dc-dc- P3P3 0100001005 S1S1 00400-88001016 S2S2 00800-44000144 P3P3 ZjZj 010000100 C j - Z j 000000100 P2P2 ZjZj 0601-80800000 C j - Z j 0-600180-800000 P1P1 ZjZj 0000000000 C j - Z j 0000100000 Second Table. has been optimised. Consider incoming variable is and outgoing variable is

21. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 21 CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 02010000-100150 T010.50000000.125015 dc-dc- P3P3 0100001005 dt+dt+ 000.5001000.12502 S2S2 006000000-0.5136 P3P3 ZjZj 0100001000 C j - Z j 000000100 P2P2 ZjZj 02010000-100 C j - Z j 0-20010000100 P1P1 ZjZj 0000000000 C j - Z j 0000100000 Third Table. C is incoming variable and outgoing variable is

22. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 22 CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 00140-4000-15070 T010001000013 dc-dc- P3P3 00002-21-0.2501 C00100-22000.2504 S2S2 0000012-1200-2112 P3P3 ZjZj 00002-21-0.2501 C j - Z j 0000-22010.250 P2P2 ZjZj 00140-4000-15070 C j - Z j 0001-404000150 P1P1 ZjZj 00000000000 C j - Z j 0000100000 Fourth Table. The value of in the row of is still negative. But this is ignored as the value in the row of is positive.

23. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 23 CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 00140-4000-15070 T010001000013 dc-dc- P3P3 00002-21-0.2501 C00100-22000.2504 S2S2 0000012-1200-2112 P3P3 ZjZj 00002-21-0.2501 C j - Z j 0000-22010.250 P2P2 ZjZj 00140-4000-15070 C j - Z j 0001-404000150 P1P1 ZjZj 00000000000 C j - Z j 0000100000 Now consider The value of in the row is negative. But this is ignored as the value in the row of is positive.

24. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 24 CjCj 00P2P2 0P1P1 0P3P3 000 CbCb TCdp-dp- dp+dp+ dt-dt- dt+dt+ dc-dc- dc+dc+ S1S1 S2S2 Qty dp-dp- P2P2 00140-4000-15070 T010001000013 dc-dc- P3P3 00002-21-0.2501 C00100-22000.2504 S2S2 0000012-1200-2112 P3P3 ZjZj 00002-21-0.2501 C j - Z j 0000-22010.250 P2P2 ZjZj 00140-4000-15070 C j - Z j 0001-404000150 P1P1 ZjZj 00000000000 C j - Z j 0000100000 An optimal solution is reached. The profit goal is under achieved by Rs 70 and the chair goal is under achieved by 1. Make 13 tables and 4 chairs

25. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 25 Summary Goal programming deals with problems involving multiple prioritised or equal priority goals. Deviational variables are added and goals converted to structural constraints during problem formulation. The objective function is spelt out in terms of minimising or maximising deviational variables Solve by the simplex method calculating net evaluation row for each priority separately. Higher priority goals are fulfilled before lower priority goals Fulfilment of lower priority goals is sacrificed if it causes deviations to occur in the higher priority goals.