An integrated reliable four-level supply chain with multi-stage products under shortage and stochastic constraints

ABSTRACT

In this paper, we design and optimise an integrated four-level Supply Chain (SC), which contains a supplier, a producer, a wholesaler and multiple retailers. The levels cooperate to make an Integrated SC (ISC) so that the inventory cost is minimised and the reliability is maximised, simultaneously. The model is constrained by real stochastic constraints on total space, number of orders, procurement cost, shortage cost, setup cost and production capacity. An Lp-Metric function converts the reliability function and cost function into a single-objective function to optimise the number of stockpiles and period lengths. The designed ISC is a large-scale Nonlinear Programming (NLP) and hard to solve by generic methods. Accordingly, two algorithms, entitled ‘Sequential Quadratic Programming (SQP)’ and ‘Interior Point (IP)’ with super-linear convergence rates are applied for finding the optimum solution. The performance of proposed algorithms is compared based on optimality criteria. Findings showed that the obtained solutions by SQP algorithm have better performance than IP algorithm in terms of optimality error and solution quality. However, the number of taken iterations by IP is less than SQP algorithm. Finally, the result of sensitivity analyses confirmed the excellent performance of the presented methods for solving the large-scale NLP models.

KEYWORDS:

• Reliability
• supply chain integration
• inventory
• Nonlinear Programming
• Sequential Quadratic Programming (SQP)
• Interior Point (IP)

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix A: The SAS code to solve the integrated function of numerical example in Table 9.

 proc optmodel;

  var L {1..2} > = 0;

  var T {1..2} > = 0;

minimise obj =

   .5*(((10/(T[1]*L[1]3) + 12

/(T[2]*L[2]3) + 138*T[1]*(L[1]−1)*

L[1]2 + 232.5*T[2]*(L[2]−1)*

L[2]2 + 24*T[1] + 36*T[2] + 175 + 53

/(T[1]*L[1]2) + 58/(T[2]*L[2]2) + 315

*T[1]*(L[1]−1)*

L[1] + 300*T[2]*(L[2]−1)*

L[2] + 2006*T[1] + 1590*T[2] + 222 + 21

/(T[1]*L[1]) + 25/(T[2]*L[2]) + 114

*T[1]*(L[1]−1) + 120*T[2]*(L[2]−1)

+ 60*T[1]+36*T[2]+238 + 51/T[1] +

62/T[2] + 577*T[1] + 702*T[2] + 225) -

3045.5761201)/ 3045.5761201)

+ .5*(((((0.13*((T[1]-.009)-.012)/1.30

)*(0.22*((T[2]-.012)-.016)/1.28)

*(0.13*((T[1]-.012)-.005)/1.74)*(0.18

*((T[2]-.014)-.014)/1.50)*(0.20*((T[1]

-.015)-.007)/2)*(0.14*((T[2]-.012)-.

018)/2)*(0.74*((T[1]-.020)-.009)/1.44

)*(0.22*((T[2]-.015)-.023)/1.45)*(0.10

*((T[1]-.015)-.025)/1.30)*(0.11*((T[2]

-.018)-.011)/1.53))).25)−0.9665131612

)/0.9665131612);

  con cons1:

  4.8*T[1] + 7.5*T[2]< = 310.573;

  con cons2:

  7.2*T[1] + 7.5*T[2] < = 305.651;

  con cons3:

  7.6*T[1] + 8*T[2] < = 310.099;

  con cons4:

  13.6*T[1] + 22.5*T[2] < = 297.718;

  con cons5:

  14.4*T[1] + 19*T[2] < = 292.119;

  con cons6:

  1/T[1] + 1/T[2] < = 205.337;

  con cons7:

  1/T[1] + 1/T[2] < = 228.736;

  con cons8:

  1/T[1] + 1/T[2] < = 242.822;

  con cons9:

  1/T[1] + 1/T[2] < = 484.706;

  con cons10:

  1/T[1] + 1/T[2] < = 515.678;

  con cons11:

  180*T[1] + 285*T[2] < = 11731.659;

  con cons12:

  684*T[1] + 540*T[2] < = 25140.209;

  con cons13:

  855*T[1] + 800*T[2] < = 31935.731;

  con cons14:

  1802*T[1]+2835*T[2]<=38372.578;

  con cons15:

  1980*T[1]+2280*T[2]<=37334.188;

  con cons16:

  12*(T[1]*L[1])+15*(T[2]*L[2])<

=3.09*(((0.09*L[1]2)+(0.08*L[2]2)).5)

+350*L[1]+340*L[2];

  con cons17:

  18*(T[1]*L[1])+15*(T[2]*L[2])<

=3.09*(((0.09*L[1]2)+(0.12*L[2]2)).5)

+340*L[1]+338*L[2];

  con cons18:

  19*(T[1]*L[1])+16*(T[2]*L[2])<

=3.09*(((0.1*L[1]2)+(0.10*L[2]2)).5)

+336*L[1]+335*L[2];

  con cons19:

  24*T[1]+36*T[2]+175<=1693.799;

  con cons20:

  8*T[1]+15*T[2]+222<=742.370;

  con cons21:

  60*T[1]+36*T[2]+238<=2084.284;

  con cons22:

  50.4*T[1]+25.5*T[2]<=1528.646;

  con cons23:

  486*T[1]+435*T[2]<=19074.383;

  /* starting point */

  L[1]=.09;

  L[2]=.09;

  T[1]=.09;

  T[2]=.09;

 solve with SQP / printfreq=4;

   print L[1] L[2] T[1] T[2];

  quit;

Additional information

Notes on contributors

Abolfazl Gharaei

Abolfazl Gharaei has a Ph.D. degree in Industrial Engineering at Kharazmi University, Iran. In addition, he was a Ph.D. visiting scholar at University of Toronto. Moreover, his Postdoctoral study was finished at University of Regina (Saskatchewan, Canada) on 2019, Feb. His research interests concentrate on inventory management of growing items, EGQ inventory model, Sustainability, RCPSP models, MRCPSP models, Job shop scheduling, Supply Chain (SC) modelling, Closed-loop SCs, Green SCs, and Decision-making methods. Optimization as another aspect of his fields of interests represents broad spectrum of Exact, Heuristic, and Meta-heuristic algorithms for solving the MINLP, NLP, MILP, and MIP models of SCs, inventory systems, RCPSP models, and Job shop scheduling. Furthermore, he has published more than 15 high-cited papers in his main interest fields.

Alireza Amjadian

Alireza Amjadian holds his M.Sc. in Industrial Engineering from Kharazmi University, Tehran, Iran. His research interests are inventory and supply chain modeling and optimization, which run the whole of Exact and Meta-heuristic algorithms. In addition, determining optimum Lot-sizing and Replenishment, in the integrated inventory systems such as EPQ, EOQ, and EGQ models in the form of MINLP, NLP, and MILP models make up an important part of his research interests.

Ali Shavandi

Ali Shavandi held his M.Sc. in Industrial Engineering from Sharif University of Technology, Tehran, Iran. His main fields of interest are Machine learning, Data mining, Data driven optimization, Heuristic & Meta-heuristic solution methods, and simulation. He is also interested in modelling and optimizing the supply chain and inventory models.