Management decision-making for transportation problems through goal programming

Rakesh K. Sharma

ABSTRACT

This paper presents a lexicographic goal programming (LGP) model for management decision-making in petroleum refinery industry for distribution of oil to the various depots. The model presented in the paper is designed to illustrate how LGP can be used as an aid for solving transportation problems with multiple objectives. The data for the study has been used from a petroleum refinery industry in India.

1. INTRODUCTION

The classical single objective transportation problems are a special type of linear programming (LP) problems. The sources may include plants and warehouses and destinations may include sales outlets and customers. The coefficients of the objective function represent transportation cost, delivery time, number of goods transported, unfulfilled demand, and many others. In operations research, several quantitative techniques have been used for solving transportation problems. The most commonly used techniques are linear programming (LP) and generalized minimum cost network (Hadley 1972, Hemaida & Kwak 1994). The decision-maker, public or private, may not have a utility function with a single argument (usually profits). The single objective optimization techniques presented by Romero in his article are examples (Romero, 1991). Businesses and industries are practically faced with both economic optimization such as cost minimization and non-economic items that are vital to the existence of their firms (Lee, 1972). Transportation problems involve multiple and conflicting goals such as the cost minimization, balancing work among the plants, transportation fleets, and many others. These multiple and conflicting goals can be achieved by using goal programming (GP) technique.

The goal programming (GP) technique has become a widely used approach in Operations Research (OR). GP model and its variants have been applied to solve large-scale multi-criteria decision-making problems. The GP technique was first used by Charnes and Cooper in 1960s. This solution approach has been extended by Ijiri (1965), Lee (1972), and others. For detailed research survey on GP, see Lee (1972), Ignizio (1976), Romero (1991), Romero (1986), Tamiz and Jones (1995), and Sharma, Alade and Vasishta (1999).

Lee and Moore (1973) used GP model for solving transportation problem with multiple and conflicting objectives. Arthur and Lawrence (1982) designed a GP model for production and shipping patterns in chemical and pharmaceutical industries. Kwak and Schniederjans (1985) applied GP to transportation problem with variable supply and demand requirements. Several other researchers (Sharma et al., 1999) have also used the GP model for solving the transportation problem.

In this paper, we present a lexicographic goal programming (LGP) model for management decision-making in petroleum refinery industry, involving the distribution of oil to the various depots. The model is designed to illustrate how LGP can be used as an aid for solving transportation problems with multiple objectives. The results of the study have utilized in decision making process at a petroleum refinery industry in India.

2. MODEL FORMULATION

The LGP model has been described in detail by Lee (1972), Ignizio (1976) and Olson (1984). The general model of the LGP can be written as follows:

Find [bar.X] ([X.sub.11],[X.sub.12],[X.sub.13], ..., [X.sub.mn]) so as to

Minimize [P.sub.1]([w.sub.i1.sup.+] [d.sub.ij.sup.+] + [w.sub.i1.sup.-][d.sub.i2.sup.-]),

Minimize [P.sub.2]([w.sub.i2.sup.+][d.sub.i2.sup.+] + [w.sub.i2.sup.-][d.sub.i2.sup.-]),

Minimize [P.sub.k]([w.sub.ik.sup.+] [d.sub.ik.sup.+] + [w.sub.ik.sup.-][d.sub.ik.sup.-]),

Minimize [P.sub.K]([w.sub.iK.sup.+] [d.sub.iK.sup.+] + [w.sub.iK.sup.-][d.sub.iK.sup.-]), where i = 1,2,3, ..., m

Subject to,

[f.sub.i]([bar.X]) + [d.sup.-.sub.i] - [d.sup.+.sub.i] = [b.sub.i],

[d.sup.-.sub.i], [d.sup.+.sub.i], X [greater than or equal to] 0 and [d.sup.-.sub.i] x [d.sup.+.sub.i] = 0 for i = 1,2,3, ..., K

where X = vector of [m.sup.*]n decision variables.

[P.sub.k] = [k.sup.th] priority factor as assigned to the set of goals, 1 [less than or equal to] k [less than or equal to] K [less than or equal to] m.

Also P, >>[P.sub.i+1]; 1 [less than or equal to] i [less than or equal to] K

[f.sub.i]([bar.X]) = linear function for [i.sup.th] goal

[d.sub.ik.sup.-] = under-achievement from the [i.sup.th] goal level [b.sub.i]. at the priority level [P.sub.k]

[d.sub.ik.sup.+] = under-achievement from the [i.sup.th] goal level [b.sub.i]. at the priority level [P.sub.k]

[w.sub.ik.sup.+] and [w.sub.ik.sup.-] ([greater than or equal to] 0) are numerical weights associated with the deviational variables [d.sub.ik.sup.-] and [d.sub.ik.sup.+] respectively where [d.sub.ik.sup.-] and [d.sub.ik.sup.+] are the renamed for the convenience of actual deviational variables [d.sup.-.sub.j] and [d.sup.+.sub.j] respectively.

Variables and Constants

The decision variables, deviational variables, and constants for model formulation are defined as below:

Decision Variables [X.sub.ij] = the amount of oil to be transported from the i-th refinery to the [j.sup.th] depot, i = 1, ..., m, j = 1 ..., n

Constants and Co-efficient

[S.sub.i] = the production capacity of the refinery i

[R.sub.i] = minimum amount of oil to be supplied by the refinery i, at the crisis period

[D.sub.j] = the demand at the depot j

[L.sub.j] = minimum amount of oil to be transported to depot j

[C.sub.ij] = the unit transportation cost from the [i.sup.th] refinery to the [j.sup.th] depot

TC = total transportation cost.

Constraints

(i) a) Refineries have installed production capacity. The refineries can not supply more than the production capacity. The LGP constraints for supply are in the form:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

b) In crisis period, to ensure the minimum supply from the refineries the goal constraints can be developed as follows:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(ii) The oil transported from refineries to the depots should not exceed the demand of individual depots. The goal constraints for demand are

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iii) There should be a minimum amount of oil to be transported from refinery i to depot j. The goal constraints are

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iv) The budgetary constraint of total transportation cost is:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3. APPLICATION

The oil company (name withheld for security reasons) used in this study uses crude oil to produce petroleum products in the northern India. The company has two oil refineries and fifteen depots in the northern region. The supply chain begins with the refineries. The crude oil is refined to obtain petroleum as finished products. The petroleum is then transported to different depot locations by rail, road & pipelines. Since all three modes of transportation are not available for all the depots, the minimum transportation cost for the available modes is taken into consideration in the model formulation. The monthly production capacities of oil product and the monthly demand of each depot and cost per Metric Ton at the two refineries are given in Table 1.

The Goals

The following goals are set by the management in order of their importance:

[P.sub.1] Achieve the minimum amount to be supplied by refineries and the minimum demand of depots.

[P.sub.2] Achieve the installed production capacity of the refinery and maximum demand of depots.

[P.sub.3] Minimize the total transportation cost.

Goal Constraints

The LGP model constraints for the transportation problem are formulated as follows:

Supply Constraints

a) The refineries have installed production capacities. The refineries can not supply more than their capacities. The LGP constraints for supply can be given as follows:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

b) In crisis period, to ensure the minimum supply from the refineries the goal constraints can be presented as follows:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Demand Constraints

a) The refined oil, shipped to the depots from the refineries, should not exceed the depots-demand individually. The LGP constrains for demand can be given as follows:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(16) [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

b) The refined oil, shipped to the depots from the refineries, should not below the depots' minimum demand. The LGP constrains for demand can be given as follows:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

c) The total transportation cost should not be greater than the budgeted amount, Rs. 19,254,710.

(40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The Objective Function

The priority structure of the problem is as follows:

Minimize [P.sub.1] [[2d.sub.3.sup.-] + [2d.sub.4.sup.-] + [d.sub.20.sup.-] + [d.sub.21.sup.-] + [d.sub.22.sup.-] + [d.sub.23.sup.-] + [2d.sub.24.sup.-] + [d.sub.25.sup.-] + [d.sub.26.sup.-] + [2d.sub.27.sup.-] + [d.sub.28.sup.-] + [d.sub.29.sup.-] + [d.sub.30.sup.-] + [d.sub.31.sup.-] + [d.sub.32.sup.-] + [d.sub.33.sup.-] + [d.sub.34.sup.-]]

Minimize [P.sub.2] [[d.sub.1.sup.+] + [d.sub.2.sup.+] + [d.sub.5.sup.+] + [d.sub.6.sup.+] + [d.sub.7.sup.+] + [d.sub.8.sup.+] + [d.sub.9.sup.+] + [d.sub.10.sup.+] + [d.sub.11.sup.+] + [d.sub.12.sup.+] + [d.sub.13.sup.+] + [d.sub.14.sup.+] + [d.sub.15.sup.+] + [d.sub.16.sup.+] + [d.sub.17.sup.+] + [d.sub.18.sup.+] + [d.sub.19.sup.+]]

Minimize [P.sub.3] [[d.sub.35.sup.+]]

4. RESULTS

The LGP transportation problem contains 30 variables, 35 constraints, 3 priorities, and an objective function. A summary of results is as follows:

TABLE 2: DECISION VARIABLE ANALYSIS

  Decision     Value
  Variable

[X.sub.1,1]     2267
[X.sub.1,2]    103.25
[X.sub.1,3]    3199.6
[X.sub.1,4]    3139.5
[X.sub.1,5]     6805
[X.sub.1,6]     4110
[X.sub.1,7]    16724
[X.sub.1,8]     5650
[X.sub.1,9]     2088
[X.sub.1,10]    2413
[X.sub.1,11]    4503
[X.sub.1,12]   10634
[X.sub.1,13]     0
[X.sub.1,14]     0
[X.sub.1,15]     0
[X.sub.2,1]      0
[X.sub.2,2]    166.75
[X.sub.2,3]    2152.4
[X.sub.2,4]    3393.5
[X.sub.2,5]     5090
[X.sub.2,6]     2030
[X.sub.2,7]      0
[X.sub.2,8]     2055
[X.sub.2,9]      0
[X.sub.2,10]     0
[X.sub.2,11]     0
[X.sub.2,12]   14200
[X.sub.2,13]    1113
[X.sub.2,14]    2129
[X.sub.2,15]    2005

TABLE 3: ANALYSIS OF OBJECTIVE FUNCTION

Priority   Achievements  Deviational Values

[P.sub.1]    Achieved    All associated deviational variables are zero.
[P.sub.2]    Achieved    All associated deviational variables are zero.
[P.sub.3]    Achieved    All associated deviational variables are zero.

The solution of the problem indicates that all three priorities are fully achieved.

5. CONCLUSION

In this study, we have been able to demonstrate that LGP approach is a better technique than the single objective criterion when multiple conflicting objectives are involved. There are several practical applications of the technique proposed in this paper in the petroleum industry. Other constraints may be included in the model based on the situation surrounding the decision processes on the business. The model is general enough to incorporate many of the incommensurable and incompatible economic and operational goals of industries. Some of the practical aspects of the case study have not been studied thoroughly.

TABLE 1: MONTHLY DEMAND OF EACH DEPOT AND COST
PER TON FROM EACH REFINERY

                            To Depots

               1       2       3       4       5

From
Refineries

1             41      14      14      12     272.
                      5.1     7.1      4       3

2            398.2   58.8    247.3   140.9   276.4

Min          1260     165    3052    4220    8600
Demand

Max           226     27      53      65      118
Demand         7       0      52      33      95

                           To Depots

               6       7       8       9      10

From
Refineries

1             19     255.     15      46.     10
              6.4      8      3.8      1      7.6

2            276.4   410.4   175.1   310.5    390

Min          4030    10720   5605    2015    2240
Demand

Max           61      167     77      65      66
Demand        40      24      05      88      43

                                 To Depots

              11      12      13      14      15     Capacity

From
Refineries

1            119.    72.8     40      37      41      100000
               8              8.6     0.6     4.5

2            415.9   53.2    200.6   158.2   119.8    85000

Min          4500    15600   1050    2018    1998
Demand

Max           125     248     36      61      52
Demand        43      34      63      49      44

REFERENCES

Arthur, J.L & Lawrence, K.D., "Multiple goal production and logistics planning in a chemical and pharmaceutical company," Computers & Operations Research, 9(2), 1982, 127-137.

Charnes, A. and Cooper, W. W., Management Models and Industrial Applications of Linear Programming, 1, John Wiley & Sons, New York, 1961.

Hadley, G., Linear Programming, Addition-Wesley Publishing Company, Massachusetts, 1972.

Hemaida, R. & Kwak, N. K., "A linear goal programming model for transshipment problems with flexible supply and demand constraints," Journal of Operational Research Society, 45(2), 1994, 215-224.

Ijiri, Y., Management Goals and Accounting for Control, Amsterdam, North-Holland, 1965.

Ignizio, J.P., Goal Programming and Extensions, Lexington Books, Massachusetts, 1976.

Kvanli, A., "Financial planning using goal programming," Omega, 8, 1980, 207-218.

Kwak, N.K. and Schniederjans, M.J., "A goal programming model for improved transportation problem solutions," Omega, 12, 1979, 367-370.

Kwak N.K. and Schniederjans, M.J., "Goal programming solutions to transportation problems with variable supply and demand requirements," Socio-Econ. Planning Science, 19(2), 1985, 95-100.

Lee, S.M., Goal Programming for Decision Analysis, Auerbach, Philadelphia, 1972.

Moore, L.J., Taylor III, B.W. & Lee, S.M., "Analysis of a transshipment problem with multiple conflicting objectives," Computers & Operations Research, 5, 1978, 39-46.

Olson, D.L., "Comparison for four Goal Programming Algorithm", Journal of Operational Research Society, 35(4), 1984, 347-354.

Romero, C., Handbook of Critical Issues in Goal Programming, Pergamon Press, Oxford, 1991.

Romero, C., "A survey of generalized goal programming (1970-1982)," European Journal of Operational Research, 25, 1986, 188-191.

Sharma, Dinesh K., Alade, J. A. and Vasishta, "Applications of Multiobjective Programming in MS/OR," Acta Ciencia Indica, XXV M(2), 1999, 225-228.

Tamiz, M. and Jones, D.F., "A Review of Goal Programming and its Applications," Annals of Operations Research, 58, 1995, 39-53.

Author Profiles:

Mr. Rakesh K. Sharma is a lecturer in the Department of Mathematics and Computer Science at the University of Maryland Eastern Shore. He received his Master in Information Sciences (MIS) degree from the North Carolina Central University. He has worked for Electronic Data Systems (EDS) and held consulting positions in the IT department of Department of Defense, American Airlines, and United States Postal Service. He has presented several papers in International conferences. His area of research includes Mathematical programming and information system design.

Mr. Avinash Gaur earned his M.S. from the Chaudhary Charan Singh University in 1996. He is a lecturer in the department of applied mathematics at RKGIT Engineering College, Ghaziabad and Ph.D. student in the department of Mathematics at Meerut College, Meerut, U.P. (India). His research has focused on mathematical programming and applied business studies.

Dr. Daniel Okunbor is an associate professor in the Department of Mathematics and Computer Science at the University of Maryland Eastern Shore. He received Ph.D. in Computer Science from the University of Illinois at Urbana-Champaign. His research interests are in mathematical modeling, numerical scientific computing, parallel processing, object-oriented software design and analysis. Dr. Okunbor is a member of the Society of Industrial and Applied Mathematics (SIAM), IEEE Computer Society, American Mathematical Society and Association of Computing Machinery. He received National Science Foundation Research Initiation Award and has given many invited research presentations worldwide.

Rakesh K. Sharma, University of Maryland Eastern Shore, Princess Anne, Maryland, USA Avinash Gaur, RKGIT Engineering College, Ghaziabad, INDIA Daniel Okunbor, University of Maryland Eastern Shore, Princess Anne, Maryland, USA

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