# American Institute of Mathematical Sciences

## A multi-stage method for joint pricing and inventory model with promotion constrains

 1 Business School, Central South University, Changsha 410083, China 2 Department of Mathematics and Statistics, Curtin University, Perth, 6102, Australia 3 School of Economics and Management, Hunan University of Science and Engineering, Yongzhou 425199, China 4 School of Accountancy, Hunan University of Finance and Economics, Changsha 410205, China 5 Department of Mathematics and Statistics, Curtin University, Perth, 6102, Australia 6 Coordinated Innovation Center for Computable Modeling in Management Science, University of Finance and Economics, Tianjin 300222, China

* Corresponding author: Haiying Liu

Received  February 2018 Revised  August 2018 Published  September 2019

In this paper, we consider a joint pricing and inventory problem with promotion constrains over a finite planning horizon for a single fast-moving consumer good under monopolistic environment. The decision on the inventory is realized through the decision on inventory replenishment, i.e., decision on the quantity to be ordered. The demand function takes into account all reference price mechanisms. The main difficulty in solving this problem is how to deal with the binary logical decision variables. It is shown that the problem is equivalent to a quadratic programming problem involving binary decision variables. This quadratic programming problem with binary decision variables can be expressed as a series of conventional quadratic programming problems, each of which can be easily solved. The global optimal solution can then be obtained readily from the global solutions of the conventional quadratic programming problems. This method works well when the planning horizon is short. For longer planning horizon, we propose a multi-stage method for finding a near-optimal solution. In numerical simulation, the accuracy and efficiency of this multi-stage method is compared with a genetic algorithm. The results obtained validate the applicability of the constructed model and the effectiveness of the approach proposed. They also provide several interesting and useful managerial insights.

Citation: Li Deng, Wenjie Bi, Haiying Liu, Kok Lay Teo. A multi-stage method for joint pricing and inventory model with promotion constrains. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020097
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##### References:
The paths of optimal prices
The paths of reference prices
The optimal pricing paths of models without promotion constraints
Comparison of the optimal pricing paths ($1\leq M\leq8$)
Comparison of the optimal pricing paths ($9\leq M\leq19$)
The paths of optimal replenishment quantities
Change of the inventory level
Change of the backorder level
Comparison of the total profit obtained by different methods
Comparison of the computational time used by different methods
Relations between L/S and the total profit
Relations between L/S and the total profit with L/S cost being taken into consideration
Relations between T and the total profit
Relations between h/b and the total profit
Notations
 Variable Description $T\; (T \in Z^+)$ length of the planning horizon $t\; (t \in \{1,\ldots,T\})$ time period $L\; (L \in \{1,\ldots,T\})$ the maximum total number of promotion times in the planning horizon $S\; (S \in \{0,\ldots,T-1\})$ the minimum separating period (a separating period is a period which spaces out the two successive promotions) $M\; (M \in \{0,\ldots,T-1\})$ length of consumers' memory window $\underline{u_1}\; (\underline{u_1}\in (0,\infty))$ permitted minimum promotion price $p_0\; (p_0\in (\underline{u_1},\infty))$ full/normal price without promotion $\overline{u_1}\; (\overline{u_1}\in (\underline{u_1},p_0))$ permitted maximum promotion price $u_1(t)\; (u_1(t) \in [\underline{u_1},\overline{u_1}]\; or\; u_1(t)=p_0)$ price in period $t$ $\underline{u_2}\; (\underline{u_2}\in (0,\infty))$ permitted minimum ordering quantity $\overline{u_2}\; (\overline{u_2} \in (\underline{u_2},\infty))$ permitted maximum ordering quantity $u_2(t)\; (u_2(t) \in [\underline{u_2},\overline{u_2}])$ ordering quantity in period $t$ $u_3(t)\; (u_3(t)\in \{0,1\})$ marking decision variable of promotion in period t, set $u_3(t)=1$ when there exists a price discount in period t and $u_3(t)=0$ when the item's price in period t equals to $p_0$ $y_{max}\; (y_{max}\in (0,\infty))$ permitted maximum inventory level in each period $y_t\; (y_t\in (-\infty,y_{max}])$ initial inventory level in period t, where $y_t <0$ means the back-order demand is $|y_t|$ at period $t$ $y_1\; (y_1\in [0,\infty))$ initial inventory level at the beginning of the planning horizon $c\; (c\in (0,\infty))$ per unit ordering cost $h\; (h\in (0,\infty))$ per unit inventory cost $b\; (b\in (0,\infty))$ per unit back-order cost $d_t\; (d_t\in [0,\infty))$ demand at period t $\gamma\; (\gamma\in (0,1))$ discount factor
 Variable Description $T\; (T \in Z^+)$ length of the planning horizon $t\; (t \in \{1,\ldots,T\})$ time period $L\; (L \in \{1,\ldots,T\})$ the maximum total number of promotion times in the planning horizon $S\; (S \in \{0,\ldots,T-1\})$ the minimum separating period (a separating period is a period which spaces out the two successive promotions) $M\; (M \in \{0,\ldots,T-1\})$ length of consumers' memory window $\underline{u_1}\; (\underline{u_1}\in (0,\infty))$ permitted minimum promotion price $p_0\; (p_0\in (\underline{u_1},\infty))$ full/normal price without promotion $\overline{u_1}\; (\overline{u_1}\in (\underline{u_1},p_0))$ permitted maximum promotion price $u_1(t)\; (u_1(t) \in [\underline{u_1},\overline{u_1}]\; or\; u_1(t)=p_0)$ price in period $t$ $\underline{u_2}\; (\underline{u_2}\in (0,\infty))$ permitted minimum ordering quantity $\overline{u_2}\; (\overline{u_2} \in (\underline{u_2},\infty))$ permitted maximum ordering quantity $u_2(t)\; (u_2(t) \in [\underline{u_2},\overline{u_2}])$ ordering quantity in period $t$ $u_3(t)\; (u_3(t)\in \{0,1\})$ marking decision variable of promotion in period t, set $u_3(t)=1$ when there exists a price discount in period t and $u_3(t)=0$ when the item's price in period t equals to $p_0$ $y_{max}\; (y_{max}\in (0,\infty))$ permitted maximum inventory level in each period $y_t\; (y_t\in (-\infty,y_{max}])$ initial inventory level in period t, where $y_t <0$ means the back-order demand is $|y_t|$ at period $t$ $y_1\; (y_1\in [0,\infty))$ initial inventory level at the beginning of the planning horizon $c\; (c\in (0,\infty))$ per unit ordering cost $h\; (h\in (0,\infty))$ per unit inventory cost $b\; (b\in (0,\infty))$ per unit back-order cost $d_t\; (d_t\in [0,\infty))$ demand at period t $\gamma\; (\gamma\in (0,1))$ discount factor
Comparison of the total profit obtained by different methods ($T\in \{x\mid x = 20+5y,\; y = 1,2,\ldots,7\}$)
 Conditions (T, L, S) Enumeration Method (＄) 2-stage Method (＄) 3-stage Stage (＄) 4-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄) T=20;S=4; 320.75 318.93 320.75 320.75 319.86 314.74 320.75 T=20;S=3; 325.35 324.92 320.75 325.16 325.16 324.34 315.19 T=20;S=2; 334.74 334.74 329.34 329.03 324.10 323.36 334.74 T=25;S=4; 354.58 354.05 353.16 354.42 348.62 348.11 352.96 T=25;S=3; 361.58 357.05 356.39 357.59 352.84 357.74 348.27 T=25;S=2; 364.49 363.36 363.31 362.25 364.33 361.73 352.20 T=30;S=4; 380.76 378.88 377.86 379.27 370.11 380.08 375.14 T=30;S=3; 385.20 384.27 383.96 380.66 373.42 384.22 382.70 T=30;S=2; 388.12 387.42 385.57 384.21 379.88 385.67 385.22 T=35;S=4; - 400.41 400.61 397.40 396.31 398.86 396.73 T=35;S=3; - 403.35 401.16 400.62 391.81 398.54 391.37 T=35;S=2; - 406.24 401.68 400.97 402.85 402.59 397.94 T=40;S=4; - 413.71 413.32 411.90 411.97 410.17 412.60 T=40;S=3; - 416.81 414.87 412.92 413.76 409.39 414.07 T=40;S=2; - 418.16 417.45 418.16 410.04 410.81 412.92 T=45;S=4; - 425.79 423.87 423.34 420.72 413.68 421.14 T=45;S=3; - 428.44 425.32 423.90 418.40 422.65 423.58 T=45;S=2; - 429.99 428.48 429.41 416.05 417.76 419.69 T=50;S=4; - 433.74 432.99 431.70 425.82 426.32 426.94 T=50;S=3; - 436.57 434.09 433.75 429.96 421.54 422.15 T=50;S=2; - 439.00 439.75 437.98 436.14 427.98 423.56 T=55;S=4; - 441.03 439.20 438.03 438.62 429.32 430.26 T=55;S=3; - 443.12 442.14 443.22 442.49 436.26 433.66 T=55;S=2; - 446.04 446.39 446.14 439.20 443.98 440.99
 Conditions (T, L, S) Enumeration Method (＄) 2-stage Method (＄) 3-stage Stage (＄) 4-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄) T=20;S=4; 320.75 318.93 320.75 320.75 319.86 314.74 320.75 T=20;S=3; 325.35 324.92 320.75 325.16 325.16 324.34 315.19 T=20;S=2; 334.74 334.74 329.34 329.03 324.10 323.36 334.74 T=25;S=4; 354.58 354.05 353.16 354.42 348.62 348.11 352.96 T=25;S=3; 361.58 357.05 356.39 357.59 352.84 357.74 348.27 T=25;S=2; 364.49 363.36 363.31 362.25 364.33 361.73 352.20 T=30;S=4; 380.76 378.88 377.86 379.27 370.11 380.08 375.14 T=30;S=3; 385.20 384.27 383.96 380.66 373.42 384.22 382.70 T=30;S=2; 388.12 387.42 385.57 384.21 379.88 385.67 385.22 T=35;S=4; - 400.41 400.61 397.40 396.31 398.86 396.73 T=35;S=3; - 403.35 401.16 400.62 391.81 398.54 391.37 T=35;S=2; - 406.24 401.68 400.97 402.85 402.59 397.94 T=40;S=4; - 413.71 413.32 411.90 411.97 410.17 412.60 T=40;S=3; - 416.81 414.87 412.92 413.76 409.39 414.07 T=40;S=2; - 418.16 417.45 418.16 410.04 410.81 412.92 T=45;S=4; - 425.79 423.87 423.34 420.72 413.68 421.14 T=45;S=3; - 428.44 425.32 423.90 418.40 422.65 423.58 T=45;S=2; - 429.99 428.48 429.41 416.05 417.76 419.69 T=50;S=4; - 433.74 432.99 431.70 425.82 426.32 426.94 T=50;S=3; - 436.57 434.09 433.75 429.96 421.54 422.15 T=50;S=2; - 439.00 439.75 437.98 436.14 427.98 423.56 T=55;S=4; - 441.03 439.20 438.03 438.62 429.32 430.26 T=55;S=3; - 443.12 442.14 443.22 442.49 436.26 433.66 T=55;S=2; - 446.04 446.39 446.14 439.20 443.98 440.99
Comparison of the computational time for using different methods ($T\in \{x\mid x = 20+5y,\; y = 1,2,\ldots,7\}$)
 Conditions (T, L, S) Enumeration Method (s) 2-stage Method (s) 3-stage Stage (s) 4-stage Method (s) GA-Roulette Wheel (s) GA-Tournament (s) GA-Random (s) T=20;S=4; 102.52 1.26 1.02 1.23 44.19 40.76 61.43 T=20;S=3; 117.19 3.94 1.15 1.32 80.02 84.25 54.55 T=20;S=2; 186.40 8.66 1.81 1.68 78.61 102.34 100.20 T=25;S=4; 3119.32 4.59 1.93 0.98 82.83 78.76 99.75 T=25;S=3; 3302.93 7.21 1.75 4.26 98.68 109.26 102.26 T=25;S=2; 4183.17 16.69 4.18 2.54 121.41 108.78 118.15 T=30;S=4; 98508.37 13.99 3.09 2.75 107.32 128.91 154.09 T=30;S=3; 100972.64 26.12 5.46 3.03 111.41 148.17 134.50 T=30;S=2; 104669.85 31.56 3.42 3.42 111.41 137.66 138.36 T=35;S=4; - 40.80 6.76 4.67 164.83 151.35 126.96 T=35;S=3; - 64.80 11.00 4.24 148.59 177.05 166.54 T=35;S=2; - 63.95 8.86 6.16 138.41 151.68 122.52 T=40;S=4; - 177.39 12.91 5.52 166.76 197.82 166.82 T=40;S=3; - 208.64 12.11 5.30 172.36 136.09 165.40 T=40;S=2; - 284.35 17.29 4.02 136.64 145.17 136.92 T=45;S=4; - 529.40 19.53 8.33 217.47 179.08 161.77 T=45;S=3; - 611.07 24.03 10.58 194.80 177.25 211.04 T=45;S=2; - 946.19 31.35 10.06 180.60 182.13 184.05 T=50;S=4; - 3810.19 23.29 9.68 160.34 142.00 130.60 T=50;S=3; - 4411.66 39.18 14.79 149.26 128.57 147.92 T=50;S=2; - 6273.47 71.00 19.41 146.46 124.79 147.17 T=55;S=4; - 14704.68 64.87 14.13 172.13 162.80 168.13 T=55;S=3; - 17253.20 84.41 30.32 163.70 164.13 163.31 T=55;S=2; - 23839.38 239.74 49.88 158.11 178.72 172.18
 Conditions (T, L, S) Enumeration Method (s) 2-stage Method (s) 3-stage Stage (s) 4-stage Method (s) GA-Roulette Wheel (s) GA-Tournament (s) GA-Random (s) T=20;S=4; 102.52 1.26 1.02 1.23 44.19 40.76 61.43 T=20;S=3; 117.19 3.94 1.15 1.32 80.02 84.25 54.55 T=20;S=2; 186.40 8.66 1.81 1.68 78.61 102.34 100.20 T=25;S=4; 3119.32 4.59 1.93 0.98 82.83 78.76 99.75 T=25;S=3; 3302.93 7.21 1.75 4.26 98.68 109.26 102.26 T=25;S=2; 4183.17 16.69 4.18 2.54 121.41 108.78 118.15 T=30;S=4; 98508.37 13.99 3.09 2.75 107.32 128.91 154.09 T=30;S=3; 100972.64 26.12 5.46 3.03 111.41 148.17 134.50 T=30;S=2; 104669.85 31.56 3.42 3.42 111.41 137.66 138.36 T=35;S=4; - 40.80 6.76 4.67 164.83 151.35 126.96 T=35;S=3; - 64.80 11.00 4.24 148.59 177.05 166.54 T=35;S=2; - 63.95 8.86 6.16 138.41 151.68 122.52 T=40;S=4; - 177.39 12.91 5.52 166.76 197.82 166.82 T=40;S=3; - 208.64 12.11 5.30 172.36 136.09 165.40 T=40;S=2; - 284.35 17.29 4.02 136.64 145.17 136.92 T=45;S=4; - 529.40 19.53 8.33 217.47 179.08 161.77 T=45;S=3; - 611.07 24.03 10.58 194.80 177.25 211.04 T=45;S=2; - 946.19 31.35 10.06 180.60 182.13 184.05 T=50;S=4; - 3810.19 23.29 9.68 160.34 142.00 130.60 T=50;S=3; - 4411.66 39.18 14.79 149.26 128.57 147.92 T=50;S=2; - 6273.47 71.00 19.41 146.46 124.79 147.17 T=55;S=4; - 14704.68 64.87 14.13 172.13 162.80 168.13 T=55;S=3; - 17253.20 84.41 30.32 163.70 164.13 163.31 T=55;S=2; - 23839.38 239.74 49.88 158.11 178.72 172.18
Comparison of the total profit obtained by different methods ($T\in \{x\mid x = 50+5y,\; y = 1,2,\ldots,7\}$)
 Conditions (T, L, S) 6-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄) T=50;S=4; 430.93 425.82 426.32 426.94 T=50;S=3; 432.92 429.96 421.54 422.15 T=50;S=2; 437.60 436.14 427.98 423.56 T=55;S=4; 436.84 438.62 429.32 430.26 T=55;S=3; 442.84 442.49 436.26 433.66 T=55;S=2; 445.07 439.20 443.98 440.99 T=60;S=4; 442.36 438.03 441.41 444.65 T=60;S=3; 447.21 440.00 436.47 444.08 T=60;S=2; 452.45 449.31 449.03 448.02 T=65;S=4; 446.70 446.36 446.56 440.53 T=65;S=3; 451.55 449.16 451.34 450.10 T=65;S=2; 456.79 450.53 449.03 447.76 T=70;S=4; 453.23 447.34 443.15 451.60 T=70;S=3; 453.79 448.39 450.38 453.16 T=70;S=2; 459.30 448.72 456.29 450.92 T=75;S=4; 454.83 450.42 450.39 455.13 T=75;S=3; 456.88 447.96 451.87 443.83 T=75;S=2; 461.11 461.54 453.54 450.14 T=80;S=4; 455.93 446.26 443.80 453.12 T=80;S=3; 461.12 456.58 447.33 448.42 T=80;S=2; 464.04 453.66 460.24 449.64 T=85;S=4; 456.66 449.92 448.51 455.99 T=85;S=3; 462.00 459.30 459.91 451.51 T=85;S=2; 465.04 460.63 452.55 454.77
 Conditions (T, L, S) 6-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄) T=50;S=4; 430.93 425.82 426.32 426.94 T=50;S=3; 432.92 429.96 421.54 422.15 T=50;S=2; 437.60 436.14 427.98 423.56 T=55;S=4; 436.84 438.62 429.32 430.26 T=55;S=3; 442.84 442.49 436.26 433.66 T=55;S=2; 445.07 439.20 443.98 440.99 T=60;S=4; 442.36 438.03 441.41 444.65 T=60;S=3; 447.21 440.00 436.47 444.08 T=60;S=2; 452.45 449.31 449.03 448.02 T=65;S=4; 446.70 446.36 446.56 440.53 T=65;S=3; 451.55 449.16 451.34 450.10 T=65;S=2; 456.79 450.53 449.03 447.76 T=70;S=4; 453.23 447.34 443.15 451.60 T=70;S=3; 453.79 448.39 450.38 453.16 T=70;S=2; 459.30 448.72 456.29 450.92 T=75;S=4; 454.83 450.42 450.39 455.13 T=75;S=3; 456.88 447.96 451.87 443.83 T=75;S=2; 461.11 461.54 453.54 450.14 T=80;S=4; 455.93 446.26 443.80 453.12 T=80;S=3; 461.12 456.58 447.33 448.42 T=80;S=2; 464.04 453.66 460.24 449.64 T=85;S=4; 456.66 449.92 448.51 455.99 T=85;S=3; 462.00 459.30 459.91 451.51 T=85;S=2; 465.04 460.63 452.55 454.77
Comparison of the computational time for using different methods ($T\in \{x\mid x = 50+5y,\; y = 1,2,\ldots,7\}$)
 Conditions (T, L, S) 6-stage Method (s) GA-Roulette Wheel (s) GA-Tournament (s) GA-Random (s) T=50;S=4; 5.66 160.34 142.00 130.60 T=50;S=3; 3.49 149.26 128.57 147.92 T=50;S=2; 4.43 146.46 124.79 147.17 T=55;S=4; 9.60 172.13 162.80 168.13 T=55;S=3; 5.13 163.70 164.13 163.31 T=55;S=2; 6.49 158.11 178.72 172.18 T=60;S=4; 5.51 184.39 219.30 212.04 T=60;S=3; 5.97 200.68 190.23 203.26 T=60;S=2; 4.93 192.55 196.28 199.34 T=65;S=4; 13.83 224.00 253.94 246.07 T=65;S=3; 14.60 217.75 217.02 225.30 T=65;S=2; 14.16 211.35 214.32 211.38 T=70;S=4; 16.32 283.38 274.18 273.71 T=70;S=3; 18.70 246.01 258.15 257.03 T=70;S=2; 18.78 238.08 237.44 276.35 T=75;S=4; 18.17 273.58 280.91 304.42 T=75;S=3; 21.14 284.10 289.44 301.87 T=75;S=2; 24.02 306.91 293.79 281.39 T=80;S=4; 22.20 349.83 333.17 338.17 T=80;S=3; 24.50 338.76 331.27 336.95 T=80;S=2; 34.04 355.38 344.48 342.70 T=85;S=4; 32.86 382.19 363.28 438.18 T=85;S=3; 31.13 413.49 386.06 378.90 T=85;S=2; 38.55 410.78 386.99 407.43
 Conditions (T, L, S) 6-stage Method (s) GA-Roulette Wheel (s) GA-Tournament (s) GA-Random (s) T=50;S=4; 5.66 160.34 142.00 130.60 T=50;S=3; 3.49 149.26 128.57 147.92 T=50;S=2; 4.43 146.46 124.79 147.17 T=55;S=4; 9.60 172.13 162.80 168.13 T=55;S=3; 5.13 163.70 164.13 163.31 T=55;S=2; 6.49 158.11 178.72 172.18 T=60;S=4; 5.51 184.39 219.30 212.04 T=60;S=3; 5.97 200.68 190.23 203.26 T=60;S=2; 4.93 192.55 196.28 199.34 T=65;S=4; 13.83 224.00 253.94 246.07 T=65;S=3; 14.60 217.75 217.02 225.30 T=65;S=2; 14.16 211.35 214.32 211.38 T=70;S=4; 16.32 283.38 274.18 273.71 T=70;S=3; 18.70 246.01 258.15 257.03 T=70;S=2; 18.78 238.08 237.44 276.35 T=75;S=4; 18.17 273.58 280.91 304.42 T=75;S=3; 21.14 284.10 289.44 301.87 T=75;S=2; 24.02 306.91 293.79 281.39 T=80;S=4; 22.20 349.83 333.17 338.17 T=80;S=3; 24.50 338.76 331.27 336.95 T=80;S=2; 34.04 355.38 344.48 342.70 T=85;S=4; 32.86 382.19 363.28 438.18 T=85;S=3; 31.13 413.49 386.06 378.90 T=85;S=2; 38.55 410.78 386.99 407.43
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