The Complex Dynamics of Bertrand-Stackelberg Pricing Models in a Risk-Averse Supply Chain

<table class="table-group" id="tab2"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td align="left">  Different periods </td><td align="center"> Stable period </td><td align="center"> Two-period-doubling bifurcation </td><td align="center"> Chaotic period </td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td align="left"><span style="width: 24.125px;"><svg height="12.125" id="M159" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 24.125 12.125" width="24.125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,15.775,11.962)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
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</svg></span></td><td align="center"> 8182.74 </td><td align="center"> 8190.65 </td><td align="center"> 8203.86 </td></tr><tr><td align="left"><span style="width: 27.65px;"><svg height="14.875" id="M160" style="vertical-align:-5.32956pt" version="1.1" viewbox="0 0 27.65 14.875" width="27.65" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.008,-0,0,-.008,22.263,14.813)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center"> 15539.95 </td><td align="center"> 14331.78 </td><td align="center"> 12443.96 </td></tr><tr><td align="left"><span style="width: 27.65px;"><svg height="14.875" id="M161" style="vertical-align:-5.32956pt" version="1.1" viewbox="0 0 27.65 14.875" width="27.65" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,10.113,11.963)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
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<g transform="matrix(.008,-0,0,-.008,22.263,14.813)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg></span></td><td align="center"> 8144.84 </td><td align="center"> 8136.82 </td><td align="center"> 8125.35 </td></tr><tr><td align="center" colspan="4"><hr/></td></tr><tr><td align="left"><span style="width: 25.75px;"><svg height="16.5625" id="M162" style="vertical-align:-3.2729pt" version="1.1" viewbox="0 0 25.75 16.5625" width="25.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.625)"><path d="M697 -46l-42 -182h-602v25l325 457l-308 460v25h585l12 -154l-28 -5q-17 63 -39 88q-12 15 -36.5 20.5t-80.5 5.5h-277l268 -396l-311 -440h382q51 0 75.5 19.5t47.5 83.5z" id="x2211"></path></g><g transform="matrix(.017,-0,0,-.017,15.633,12.625)"><path d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43
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</svg></span></td><td align="center"> 31867.63 </td><td align="center"> 30659.25 </td><td align="center"> 28773.17 </td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

The profit of the two manufacturers and retailer in different periods when <svg height="18.112499" id="M158" style="vertical-align:-5.32956pt" version="1.1" viewbox="0 0 72.449997 18.112499" width="72.449997" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,11.013,15.187)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
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</svg>.

Discrete Dynamics in Nature and Society

tab2

Table 2

Table 2: The Complex Dynamics of Bertrand-Stackelberg Pricing Models in a Risk-Averse Supply Chain 